Document:

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                                                                EXHIBIT 4.33

                   COMMON SECURITIES GUARANTEE AGREEMENT

          This COMMON SECURITIES GUARANTEE AGREEMENT (the "Common Securities
Guarantee"), dated as of May 19, 2000, is executed and delivered by GREATER BAY
BANCORP, a California corporation (the "Guarantor"), for the benefit of the
Holders (as defined herein) from time to time of the Common Securities (as
defined herein) of GBB CAPITAL IV, a Delaware business trust (the "Issuer").

          WHEREAS, pursuant to an Amended and Restated Declaration of Trust (the
"Declaration"), dated as of May 19, 2000, by and among the Trustees of the
Issuer named therein, the Guarantor, as sponsor, and the holders from time to
time of undivided beneficial interests in the assets of the Issuer, the Issuer
is issuing on the date hereof 1,268 common securities having an aggregate
liquidation amount of one million two hundred sixty-eight thousand dollars
($1,268,000), to meet the capital requirements of the Trust; such common
securities being designated the 10.75% Common Securities (the "Common
Securities").

          WHEREAS, as incentive for the Holders to purchase the Common
Securities, the Guarantor desires irrevocably and unconditionally to agree, to
the extent set forth in this Common Securities Guarantee, to pay the Guarantee
Payments (as defined herein) to the Holders of the Common Securities and to make
certain other payments on the terms and conditions set forth herein; and

          WHEREAS, the Guarantor is also executing and delivering the Series A
Capital Securities Guarantee Agreement, dated as of May 19, 2000 (the "Series A
Capital Securities Guarantee"), for the benefit of the holders of the Series A
Capital Securities (as defined in the Declaration) and upon consummation of the
Exchange Offer (as defined in the Declaration) will execute and deliver the
Series B Capital Securities Guarantee Agreement (the "Series B Capital
Securities Guarantee") for the benefit of the holders of the Series B Capital
Securities (as defined in the Declaration); it being understood that if an Event
of Default (as defined in the Declaration) has occurred and is continuing, the
rights of Holders of the Common Securities to receive Guarantee Payments under
this Common Securities Guarantee are subordinated, to the extent and in the
manner set forth herein, to the rights of holders of Capital Securities (as
defined in the Declaration) to receive Guarantee Payments under the Series A
Capital Securities Guarantee and the Series B Capital Securities Guarantee, as
the case may be.

          NOW, THEREFORE, in consideration of the purchase by each Holder of the
Common Securities, which purchase the Guarantor hereby acknowledges shall
benefit the Guarantor, the Guarantor executes and delivers this Common
Securities Guarantee for the benefit of the Holders.

                                   ARTICLE I
                        DEFINITIONS AND INTERPRETATION

SECTION 1.1 Definitions and Interpretation
            ------------------------------

            In this Common Securities Guarantee, unless the context otherwise
requires:

                                       1
<PAGE>

          (a)  capitalized terms used in this Common Securities Guarantee but
not defined in the preamble above have the respective meanings assigned to them
in this Section 1.1;

          (b)  terms defined in the Declaration as at the date of execution of
this Common Securities Guarantee have the same meaning when used in this Common
Securities Guarantee unless otherwise defined in this Common Securities
Guarantee;

          (c)  a term defined anywhere in this Common Securities Guarantee has
the same meaning throughout;

          (d)  all references to "the Common Securities Guarantee" or "this
Common Securities Guarantee" are references to this Common Securities Guarantee
as modified, supplemented or amended from time to time;

          (e)  all references in this Common Securities Guarantee to Articles
and Sections are references to Articles and Sections of this Common Securities
Guarantee unless otherwise specified;

          (f)  a term defined in the Trust Indenture Act has the same meaning as
in the Trust Indenture Act unless otherwise defined in this Common Securities
Guarantee; and

          (g)  a reference to the singular includes the plural and vice versa.

          "Guarantee Payments" means the following payments or distributions,
without duplication, with respect to the Common Securities, to the extent not
paid or made by or on behalf of the Issuer: (i) any accumulated and unpaid
Distributions (as defined in the Declaration) that are required to be paid on
such Common Securities, to the extent the Issuer has funds legally available
therefor at such time, (ii) the redemption price, including all accumulated and
unpaid Distributions to the date of redemption (the "Redemption Price"), to the
extent the Issuer has funds legally available therefor at such time, with
respect to any Common Securities called for redemption, and (iii) upon a
voluntary or involuntary dissolution, winding-up or liquidation of the Issuer
(other than in connection with the distribution of Debentures (as defined in the
Declaration) to the Holders in exchange for Common Securities or in connection
with the redemption of the Common Securities, in each case as provided in the
Declaration), the lesser of (a) the aggregate of the liquidation amount and all
accumulated and unpaid Distributions on the Common Securities to the date of
payment, to the extent the Issuer has funds legally available therefor at such
time, and (b) the amount of assets of the Issuer remaining available for
distribution to the Holders after satisfaction of liabilities to creditors of
the Issuer as required by applicable law (in either case, the "Liquidation
Distribution"). If an Event of Default under the Declaration, the Series A
Capital Securities Guarantee Agreement or the Series B Capital Securities
Guarantee Agreement has occurred and is continuing, no Guarantee Payments with
respect to the Common Securities shall be made until holders of Capital
Securities shall be paid in full the Guarantee Payments to which they are
entitled under the Series A Capital Securities Guarantee and the Series B
Capital Securities Guarantee, as the case may be.

          "Holder" means any holder, as registered on the books and records of
the Issuer, of any Common Securities.

                                       2
<PAGE>

          "Other Guarantees" means all guarantees, other than this Common
Securities Guarantee, to be issued by the Guarantor with respect to common
securities (if any) similar to the Common Securities issued by trusts other than
the Issuer to be established by the Guarantor (if any), in each case similar to
the Issuer.

                                  ARTICLE II
                                   GUARANTEE

SECTION 2.1    Guarantee
               ---------

               The Guarantor irrevocably and unconditionally agrees to pay in
full to the Holders the Guarantee Payments (without duplication of amounts
theretofore paid by the Issuer), as and when due, regardless of any defense,
right of set-off or counterclaim which the Issuer may have or assert. The
Guarantor fully, knowingly and unconditionally waives any right the Guarantor
may have to revoke this Guarantee under Section 2815 of the California Civil
Code or otherwise. The Guarantor's obligation to make a Guarantee Payment may be
satisfied by direct payment of the required amounts by the Guarantor to the
Holders or by causing the Issuer to pay such amounts to the Holders.

SECTION 2.2    Waiver of Notice and Demand
               ---------------------------

               The Guarantor hereby waives notice of acceptance of this Common
Securities Guarantee and of any liability to which it applies or may apply,
presentment, demand for payment, any right to require a proceeding first against
the Issuer or any other Person before proceeding against the Guarantor, protest,
notice of nonpayment, notice of dishonor, notice of redemption and all other
notices and demands.

SECTION 2.3    Obligations Not Affected
               ------------------------

               The obligations, covenants, agreements and duties of the
Guarantor under this Common Securities Guarantee shall in no way be affected or
impaired by reason of the happening from time to time of any of the following:

               (a)  the release or waiver, by operation of law or otherwise, of
the performance or observance by the Issuer of any express or implied agreement,
covenant, term or condition relating to the Common Securities to be performed or
observed by the Issuer;

               (b)  the extension of time for the payment by the Issuer of all
or any portion of the Distributions, Redemption Price, Liquidation Distribution
or any other sums payable under the terms of the Common Securities or the
extension of time for the performance of any other obligation under, arising out
of, or in connection with, the Common Securities;

               (c)  any failure, omission, delay or lack of diligence on the
part of the Holders to enforce, assert or exercise any right, privilege, power
or remedy conferred on the Holders pursuant to the terms of the Common
Securities, or any action on the part of the Issuer granting indulgence or
extension of any kind;

                                       3
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               (d)  the voluntary or involuntary liquidation, dissolution, sale
of any collateral, receivership, insolvency, bankruptcy, assignment for the
benefit of creditors, reorganization, arrangement, composition or readjustment
of debt of, or other similar proceedings affecting, the Issuer or any of the
assets of the Issuer;

               (e)  any invalidity of, or defect or deficiency in, the Common
Securities;

               (f)  the settlement or compromise of any obligation guaranteed
hereby or hereby incurred; or

               (g)  any other circumstance whatsoever that might otherwise
constitute a legal or equitable discharge or defense of a guarantor;

it being the intent of this Section 2.3 that the obligations of the Guarantor
with respect to the Guarantee Payments shall be absolute and unconditional under
any and all circumstances.

               There shall be no obligation of the Holders to give notice to, or
obtain consent of, the Guarantor with respect to the happening of any of the
foregoing.

SECTION 2.4    Rights of Holders
               -----------------

               The Guarantor expressly acknowledges that any Holder of the
Common Securities may institute a legal proceeding directly against the
Guarantor to enforce its rights under this Common Securities Guarantee, without
first instituting a legal proceeding against the Issuer or any other Person.

SECTION 2.5    Guarantee of Payment
               --------------------

               This Common Securities Guarantee creates a guarantee of payment
and not of collection.

SECTION 2.6    Subrogation
               -----------

               The Guarantor shall be subrogated to all (if any) rights of the
Holders of the Common Securities against the Issuer in respect of any amounts
paid to such Holders by the Guarantor under this Common Securities Guarantee;
provided, however, that the Guarantor shall not (except to the extent required
--------  -------
by mandatory provisions of law) be entitled to enforce or exercise any rights
which it may acquire by way of subrogation or any indemnity, reimbursement or
other agreement, in all cases as a result of payment under this Common
Securities Guarantee, if, at the time of any such payment, any amounts are due
and unpaid under this Common Securities Guarantee. If any amount shall be paid
to the Guarantor in violation of the preceding sentence, the Guarantor agrees to
hold such amount in trust for the Holders and to pay over such amount to the
Holders.

                                       4
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SECTION 2.7    Independent Obligations
               -----------------------

               The Guarantor acknowledges that its obligations hereunder are
independent of the obligations of the Issuer with respect to the Common
Securities and that the Guarantor shall be liable as principal and as debtor
hereunder to make Guarantee Payments pursuant to the terms of this Common
Securities Guarantee notwithstanding the occurrence of any event referred to in
subsections (a) through (g), inclusive, of Section 2.3 hereof.

                                  ARTICLE III
                   LIMITATION OF TRANSACTIONS; SUBORDINATION

SECTION 3.1    Limitation of Transactions
               --------------------------

               So long as any Common Securities remain outstanding, the
Guarantor will not (i) declare or pay any dividends or distribution on, or
redeem, purchase, acquire or make a liquidation payment with respect to any of
the Guarantor's capital stock, (ii) make any payment of principal of, or
interest or premium, if any, on or repay, repurchase or redeem any debt
securities of the Guarantor (including Other Debentures, as defined in the
Indenture) that rank pari passu with or junior in right of payment to the
Debentures or (iii) make any guarantee payments with respect to any guarantee by
the Guarantor of the debt securities of any subsidiary of the Guarantor
(including Other Guarantees, as defined in the Indenture) if such guarantee
ranks pari passu with or junior in right of payment to the Debentures (other
than (a) dividends or distributions in shares of, or options, warrants or rights
to subscribe for or purchase shares of, common stock of the Guarantor, (b) any
declaration of a dividend in connection with the implementation of a
shareholders' rights plan, or the issuance of stock under any such plan in the
future, or the redemption or repurchase of any such rights pursuant thereto, (c)
payments under the Series A Capital Securities Guarantee and the Series B
Capital Securities Guarantee, (d) as a result of a reclassification of the
Guarantor's capital stock or the exchange or the conversion of one class or
series of the Guarantor's capital stock for another class or series of the
Guarantor's capital stock, (e) the purchase of fractional interests in shares of
the Guarantor's capital stock pursuant to the conversion or exchange provisions
of such capital stock or the security being converted or exchanged, and (f)
purchases of common stock related to the issuance of common stock or rights
under any of the Guarantor's benefit or compensation plans for its directors,
officers or employees or any of the Guarantor's dividend reinvestment plans) if
at such time (1) there shall have occurred any event of which the Guarantor has
actual knowledge that (A) is, or with the giving of notice or the lapse of time,
or both, would be, an Event of Default and (B) in respect of which the Guarantor
shall not have taken reasonable steps to cure, (2) if such Debentures are held
by the Property Trustee, the Guarantor shall be in default with respect to its
payment of any obligations under the Series A Capital Securities Guarantee or
the Series B Capital Securities Guarantee or (3) the Guarantor shall have given
notice of its election of the exercise of its right to commence an Extended
Interest Payment Period as provided in the Indenture and shall not have
rescinded such notice, and such Extended Interest Payment Period, or an
extension thereof, shall have commenced and be continuing.

                                       5
<PAGE>

SECTION 3.2    Ranking
               -------

               This Common Securities Guarantee will constitute an unsecured
obligation of the Guarantor and will rank (i) subordinate and junior in right of
payment to the Senior Indebtedness (as defined in the Indenture), to the same
extent and in the same manner that the Debentures are subordinated to the Senior
Indebtedness pursuant to the Indenture, it being understood that the terms of
Article XV of the Indenture shall apply to the obligations of the Guarantor
under this Common Securities Guarantee as if such Article XV were set forth
herein in full, (ii) pari passu with the most senior preferred or preference
stock now or hereafter issued by the Guarantor and with any Other Guarantee and,
except as set forth herein or therein, the Series A Capital Securities
Guarantee, the Series B Capital Securities Guarantee and any guarantee now or
hereafter entered into by the Guarantor in respect of any preferred or
preference stock of any Affiliate of the Guarantor and (iii) senior to the
Guarantor's common stock.

                                  ARTICLE IV
                                  TERMINATION

SECTION 4.1    Termination
               -----------

               This Common Securities Guarantee shall terminate and be of no
further force or effect upon (i) full payment of the Redemption Price of all
Common Securities, (ii) the dissolution, winding-up or liquidation of the
Issuer, immediately following the full payment of the amounts payable in
accordance with the Declaration, or (iii) the distribution of all of the
Debentures to the Holders of the Trust Securities (as defined in the
Declaration). Notwithstanding the foregoing, this Common Securities Guarantee
will continue to be effective or will be reinstated, as the case may be, if at
any time any Holder of the Common Securities must restore payment of any sums
paid under the Common Securities or under this Common Securities Guarantee.

                                   ARTICLE V
                                 MISCELLANEOUS

SECTION 5.1    Successors and Assigns
               ----------------------

               All guarantees and agreements contained in this Common Securities
Guarantee shall bind the successors, assigns, receivers, trustees and
representatives of the Guarantor and shall inure to the benefit of the Holders
of the Common Securities then outstanding.

SECTION 5.2    Amendments
               ----------

               Except with respect to any changes which do not adversely affect
in any material respect the rights of the Holders (in which case no consent of
the Holders will be required), this Common Securities Guarantee may only be
amended with the prior approval of the Holders of a majority in liquidation
amount of all the outstanding Common Securities. The provisions of Section 12.2
of the Declaration with respect to meetings of Holders of the Trust Securities
apply to the giving of such approval.

SECTION 5.3    Notices
               -------

                                       6
<PAGE>

          All notices provided for in this Common Securities Guarantee shall be
in writing, duly signed by the party giving such notice, and shall be delivered,
telecopied or mailed by registered or certified mail, as follows:

          (a)  if given to the Issuer, in care of the Administrative Trustee at
the Issuer's mailing address set forth below (or such other address as the
Issuer may give notice of to the Holders of the Common Securities):

               GBB CAPITAL III
               c/o Greater Bay
               Bancorp
               2860 West Bayshore Road
               Palo Alto, California 94303
               Attention: Steven C. Smith
               Telephone:   (650) 813-8200
               Telecopier:  (650) 494-9193

          (b)  if given to the Guarantor, at the Guarantor's mailing address set
forth below (or such other address as the Guarantor may give notice of to the
Holders of the Common Securities):

               GREATER BAY BANCORP
               2860 West Bayshore Road
               Palo Alto, California 94303
               Attention: Steven C. Smith
               Telephone:   (650) 813-8200
               Telecopier:  (650) 494-9193

          (c)  if given to any Holder of the Common Securities, at the address
set forth on the books and records of the Issuer.

          All such notices shall be deemed to have been given when received in
person, telecopied with receipt confirmed, or mailed by first class mail,
postage prepaid except that if a notice or other document is refused delivery or
cannot be delivered because of a changed address of which no notice was given,
such notice or other document shall be deemed to have been delivered on the date
of such refusal or inability to deliver.

SECTION 5.4    Benefit
               -------

          This Common Securities Guarantee is solely for the benefit of the
Holders of the Common Securities and is not separately transferrable from the
Common Securities.

                                       7
<PAGE>

SECTION 5.5    Governing Law
               -------------

               THIS COMMON SECURITIES GUARANTEE SHALL BE GOVERNED BY, AND
CONSTRUED AND INTERPRETED IN ACCORDANCE WITH, THE LAWS OF THE STATE OF NEW YORK
WITHOUT REGARD TO CONFLICT OF LAW PRINCIPLES THEREOF.

               This Common Securities Guarantee is executed as of the day and
year first above written.

                              GREATER BAY BANCORP

                              By: /s/ Shawn E. Saunders
                                  ---------------------
                                  Name:  Shawn E. Saunders
                                  Title: Senior Vice President, Finance
                                         and Accounting

                                       8<PAGE>

                                                                    Exhibit 4.34

              ===================================================

                SERIES A CAPITAL SECURITIES GUARANTEE AGREEMENT

                              GREATER BAY BANCORP

                           Dated as of May 19, 2000

              ===================================================
<PAGE>

                               TABLE OF CONTENTS
                               -----------------

<TABLE>
<CAPTION>
                                                                                                                        Page
                                                                                                                        ----
<S>                                                                                                                     <C>
ARTICLE I        DEFINITIONS AND INTERPRETATION......................................................................      2
     SECTION 1.1        Definitions and Interpretation...............................................................      2
ARTICLE II       TRUST INDENTURE ACT.................................................................................      5
     SECTION 2.1        Trust Indenture Act; Application.............................................................      5
     SECTION 2.2        Lists of Holders of Securities...............................................................      5
     SECTION 2.3        Reports by the Capital Securities Guarantee Trustee..........................................      6
     SECTION 2.4        Periodic Reports.............................................................................      6
     SECTION 2.5        Evidence of Compliance with Conditions Precedent.............................................      6
     SECTION 2.6        Waiver of Events of Default..................................................................      6
     SECTION 2.7        Notice of Events of Default..................................................................      7
     SECTION 2.8        Conflicting Interests........................................................................      7
ARTICLE III      POWERS, DUTIES AND RIGHTS OF CAPITAL SECURITIES GUARANTEE TRUSTEE...................................      7
     SECTION 3.1        Powers and Duties of the Capital Securities Guarantee Trustee................................      7
     SECTION 3.2        Certain Rights of Capital Securities Guarantee Trustee.......................................      9
     SECTION 3.3        Not Responsible for Recitals or Issuance of Series A Capital Securities Guarantee............     11
ARTICLE IV       CAPITAL SECURITIES GUARANTEE TRUSTEE................................................................     11
     SECTION 4.1        Capital Securities Guarantee Trustee; Eligibility............................................     11
     SECTION 4.2        Appointment, Removal and Resignation of Capital Securities Guarantee Trustee.................     12
ARTICLE V        GUARANTEE...........................................................................................     13
     SECTION 5.1        Guarantee....................................................................................     13
     SECTION 5.2        Waiver of Notice and Demand..................................................................     13
     SECTION 5.3        Obligations Not Affected.....................................................................     13
     SECTION 5.4        Rights of Holders............................................................................     14
     SECTION 5.5        Guarantee of Payment.........................................................................     14
     SECTION 5.6        Subrogation..................................................................................     14
     SECTION 5.7        Independent Obligations......................................................................     15
ARTICLE VI       LIMITATION OF TRANSACTIONS; SUBORDINATION...........................................................     15
</TABLE>

                                       i
<PAGE>

                         TABLE OF CONTENTS(continued)
                         -----------------

<TABLE>
<CAPTION>

                                                                                                                        Page
                                                                                                                        ----
<S>                                                                                                                     <C>
     SECTION 6.1        Limitation of Transactions...................................................................     15
     SECTION 6.2        Ranking......................................................................................     16
ARTICLE VII      TERMINATION.........................................................................................     16
     SECTION 7.1        Termination..................................................................................     16
ARTICLE VIII     INDEMNIFICATION.....................................................................................     16
     SECTION 8.1        Exculpation..................................................................................     16
     SECTION 8.2        Compensation and Indemnification.............................................................     17
ARTICLE IX       MISCELLANEOUS.......................................................................................     17
     SECTION 9.1        Successors and Assigns.......................................................................     17
     SECTION 9.2        Amendments...................................................................................     17
     SECTION 9.3        Notices......................................................................................     18
     SECTION 9.4        Exchange Offer...............................................................................     19
     SECTION 9.5        Benefit......................................................................................     19
     SECTION 9.6        Governing Law................................................................................     20
</TABLE>

                                      ii
<PAGE>

                             CROSS REFERENCE TABLE

<TABLE>
<CAPTION>
Section of Trust
Indenture Act of                                                                  Section of Guarantee
1939, as amended                                                                        Agreement
----------------                                                                        ---------
<S>                                                                               <C>
       310(a)     ..............................................................          4.1(a)
       310(b)     ..............................................................       2.8, 4.1(c)
       310(c)     ..............................................................           N/A
       311(a)     ..............................................................          2.2(b)
       311(b)     ..............................................................          2.2(b)
       311(c)     ..............................................................           N/A
       312(a)     ..............................................................          2.2(a)
       312(b)     ..............................................................          2.2(b)
       312(c)     ..............................................................           N/A
       313        ..............................................................           2.3
       314(a)     ..............................................................           2.4
       314(b)     ..............................................................           N/A
       314(c)     ..............................................................           2.5
       314(d)     ..............................................................           N/A
       314(e)     ..............................................................      1.1, 2.5, 3.2
       314(f)     ..............................................................         2.1, 3.2
       315(a)     ..............................................................      3.1(d), 3.2(a)
       315(b)     ..............................................................           2.7
       315(c)     ..............................................................          3.1(c)
       315(d)     ..............................................................   3.1(d), 3.2(a), 8.1
       315(e)     ..............................................................           N/A
       316(a)     ..............................................................      1.1, 2.6, 5.4
       316(b)     ..............................................................         5.3, 5.4
       316(c)     ..............................................................           9.2
       317(a)     ..............................................................           N/A
       317(b)     ..............................................................           N/A
       318(a)     ..............................................................          2.1(a)
       318(b)     ..............................................................          2.1(b)
       318(c)     ..............................................................          2.1(b)
</TABLE>

-------------------------
*      This Cross-Reference Table does not constitute part of this Guarantee
       Agreement and shall not affect the interpretation of any of its terms
       or provisions.

                                      iii
<PAGE>

                SERIES A CAPITAL SECURITIES GUARANTEE AGREEMENT

     This SERIES A CAPITAL SECURITIES GUARANTEE AGREEMENT (the "Series A Capital
Securities Guarantee"), dated as of May 19, 2000 is executed and delivered by
GREATER BAY BANCORP, a California corporation (the "Guarantor"), and WILMINGTON
TRUST COMPANY, a Delaware banking corporation, as trustee (the "Capital
Securities Guarantee Trustee" or "Trustee"), for the benefit of the Holders (as
defined herein) from time to time of the Series A Capital Securities (as defined
herein) of GBB CAPITAL IV, a Delaware statutory business trust (the "Issuer").

     WHEREAS, pursuant to an Amended and Restated Declaration of Trust (the
"Declaration"), dated as of May 19, 2000, by and among the trustees of the
Issuer named therein, the Guarantor, as sponsor, and the Holders from time to
time of undivided beneficial interests in the assets of the Issuer, the Issuer
(i) is issuing on the date hereof 41,000 capital securities, having an aggregate
liquidation amount of $41,000,000, such capital securities being designated the
10.75% Capital Securities, Series A (collectively the "Series A Capital
Securities") and (ii) in connection with an Exchange Offer (as defined in the
Declaration), will execute and deliver the Series B Capital Securities Guarantee
(as defined in the Declaration) for the benefit of Holders of the Series B
Capital Securities (as defined in the Declaration).

     WHEREAS, as incentive for the Holders to purchase the Series A Capital
Securities, the Guarantor desires irrevocably and unconditionally to agree, to
the extent set forth in this Series A Capital Securities Guarantee, to pay the
Guarantee Payments (as defined herein) to the Holders of the Series A Capital
Securities, and the Guarantor agrees to make certain other payments on the terms
and conditions set forth herein.

     WHEREAS, the Guarantor is also executing and delivering the Common
Securities Guarantee Agreement, dated as of May 19, 2000 (the "Common Securities
Guarantee"), for the benefit of the holders of the Common Securities (as defined
herein), the terms of which provide that if an Event of Default (as defined in
the Declaration) has occurred and is continuing, the rights of holders of the
Common Securities to receive Guarantee Payments under the Common Securities
Guarantee are subordinated, to the extent and in the manner set forth in the
Common Securities Guarantee, to the rights of Holders of Series A Capital
Securities and the Series B Capital Securities to receive Guarantee Payments
under this Series A Capital Securities Guarantee and the Series B Capital
Securities Guarantee, as the case may be.

     NOW, THEREFORE, in consideration of the purchase by each Holder of the
Series A Capital Securities, which purchase the Guarantor hereby acknowledges
shall benefit the Guarantor, the Guarantor executes and delivers this Series A
Capital Securities Guarantee for the benefit of such Holders.

                                       1
<PAGE>

                                   ARTICLE I

                        DEFINITIONS AND INTERPRETATION

     SECTION 1.1    Definitions and Interpretation
                    ------------------------------

     In this Series A Capital Securities Guarantee, unless the context otherwise
requires:

     (a) capitalized terms used in this Series A Capital Securities Guarantee
but not defined in the preamble above have the respective meanings assigned to
them in this Section 1.1;

     (b) terms defined in the Declaration as in effect at the date of execution
of this Series A Capital Securities Guarantee have the same meaning when used in
this Series A Capital Securities Guarantee unless otherwise defined in this
Series A Capital Securities Guarantee;

     (c) a term defined anywhere in this Series A Capital Securities Guarantee
has the same meaning throughout;

     (d) all references to "the Series A Capital Securities Guarantee" or "this
Series A Capital Securities Guarantee" are references to this Series A Capital
Securities Guarantee as modified, supplemented or amended from time to time;

     (e) all references in this Series A Capital Securities Guarantee to
Articles and Sections are references to Articles and Sections of this Series A
Capital Securities Guarantee, unless otherwise specified;

     (f) a term defined in the Trust Indenture Act has the same meaning when
used in this Series A Capital Securities Guarantee, unless otherwise defined in
this Series A Capital Securities Guarantee or unless the context otherwise
requires; and

     (g) a reference to the singular includes the plural and vice versa.

     "Affiliate" has the same meaning as given to that term in Rule 405 under
      ---------
the Securities Act of 1933, as amended, or any successor rule thereunder.

     "Business Day" shall mean any day other than a Saturday, a Sunday, or a day
      ------------
on which banking institutions in Wilmington, Delaware, San Francisco, California
or New York, New York  are authorized or required by law or executive order to
remain closed.

     "Capital Securities Guarantee Trustee" shall mean Wilmington Trust Company,
      ------------------------------------
as Trustee under the Series A Capital Securities Guarantee, until a Successor
Capital Securities Guarantee Trustee has been appointed and has accepted such
appointment pursuant to the terms of this Series A Capital Securities Guarantee
and thereafter means each such Successor Capital Securities Guarantee Trustee.

     "Common Securities" shall mean the securities representing common undivided
      -----------------
beneficial interests in the assets of the Issuer.

                                       2
<PAGE>

     "Corporate Trust Office" shall mean the office of the Capital Securities
      ----------------------
Guarantee Trustee at which the corporate trust business of the Capital
Securities Guarantee Trustee shall, at any particular time, be principally
administered, which office at the date of execution of this Agreement is located
at Rodney Square North, 1100 North Market Street, Wilmington, Delaware 19890-
0001, Attention: Corporate Trust Administration.

     "Covered Person" shall mean any Holder or beneficial owner of Series A
      --------------
Capital Securities.

     "Debentures" shall mean the series of subordinated debt securities of the
      ----------
Guarantor designated the 10.75% Junior Subordinated Deferrable Interest
Debentures due June 1, 2030, Series A, held by the Property Trustee (as defined
in the Declaration) of the Issuer.

     "Event of Default" shall mean a default by the Guarantor on any of its
      ----------------
payment or other obligations under this Series A Capital Securities Guarantee;
provided, however, that, except with respect to default in respect of any
--------  -------
Guarantee Payment, no default by the Guarantor hereunder shall constitute an
Event of Default unless the Guarantor shall have received written notice of the
default and shall not have cured such default within 60 days after receipt
thereof.

     "Guarantee Payments" shall mean the following payments or distributions,
      ------------------
without duplication, with respect to the Series A Capital Securities, to the
extent not paid or made by or on behalf of the Issuer: (i) any accumulated and
unpaid Distributions (as defined in the Declaration) that are required to be
paid on such Series A Capital Securities, to the extent the Issuer has funds
legally available therefor at such time, (ii) the redemption price, including
all accumulated and unpaid Distributions to the date of redemption (the
"Redemption Price"), to the extent the Issuer has funds legally available
therefor at such time, with respect to any Series A Capital Securities called
for redemption, and (iii) upon a voluntary or involuntary dissolution, winding
up or liquidation of the Issuer (other than in connection with the distribution
of Debentures to the Holders in exchange for Series A Capital Securities or in
connection with the redemption of the Series A Capital Securities, in each case
as provided in the Declaration), the lesser of (a) the aggregate of the
liquidation amount and all accumulated and unpaid Distributions on the Series A
Capital Securities to the date of payment, to the extent the Issuer has funds
legally available therefor at such time, and (b) the amount of assets of the
Issuer remaining available for distribution to Holders after satisfaction of
liabilities to creditors of the Issuer as required by applicable law (in either
case, the "Liquidation Distribution"). If an Event of Default has occurred and
is continuing, no Guarantee Payments under the Common Securities Guarantee with
respect to the Common Securities or any guarantee payment under the Common
Securities Guarantee or any Other Common Securities Guarantee shall be made
until the Holders of the Series A Capital Securities shall be paid in full the
Guarantee Payments to which they are entitled under this Series A Capital
Securities Guarantee.

     "Holder" shall mean any holder, as registered on the books and records of
      ------
the Issuer, of any Series A Capital Securities; provided, however, that, in
                                                --------  -------
determining whether the holders of the requisite percentage of Series A Capital
Securities have given any request, notice, consent or waiver hereunder, "Holder"
shall not include the Guarantor or any Person actually known to a Responsible
Officer of the Capital Securities Guarantee Trustee to be an Affiliate of the
Guarantor.

                                       3
<PAGE>

     "Indemnified Person" shall mean the Capital Securities Guarantee Trustee
      ------------------
(including in its individual capacity), any Affiliate of the Capital Securities
Guarantee Trustee, or any officers, directors, shareholders, members, partners,
employees, representatives, nominees, custodians or agents of the Capital
Securities Guarantee Trustee.

     "Indenture" shall mean the Indenture, dated as of May 19, 2000, between
      ---------
Greater Bay Bancorp, as issuer of Debentures (the "Debenture Issuer"), and
Wilmington Trust Company, as trustee, pursuant to which the Debentures are to be
issued to the Property Trustee of the Issuer.

     "Majority in Liquidation Amount of the Series A Capital Securities" shall
      -----------------------------------------------------------------
mean, except as provided by the Trust Indenture Act, a vote by Holder(s) of the
Series A Capital Securities, voting separately as a class, of more than 50% of
the aggregate liquidation amount (including the amount that would be paid on
redemption, liquidation or otherwise, plus accumulated and unpaid Distributions
to but excluding the date upon which the voting percentages are determined) of
all outstanding Series A Capital Securities, excluding Series A Capital
Securities held by the Guarantor, the Issuer or any Affiliate thereof.

     "Officers' Certificate" shall mean, with respect to any Person, a
      ---------------------
certificate signed by the Chairman, the Chief Executive Officer, the President,
an Executive or Senior Vice President, a Vice President, the Chief Financial
Officer and the Secretary or an Assistant Secretary. Any Officers' Certificate
delivered with respect to compliance with a condition or covenant provided for
in this Series A Capital Securities Guarantee shall include:

     (a)  a statement that each officer signing the Officers' Certificate has
read the covenants or conditions and the definitions relating thereto;

     (b)  a statement that each such officer has made such examination or
investigation as, in such officer's opinion, is necessary to enable such officer
to express an informed opinion as to whether or not such covenant or condition
has been complied with; and

     (c)  a statement as to whether or not, in the opinion of each such officer,
such condition or covenant has been complied with.

     "Other Common Securities Guarantees" shall have the same meaning as "Other
      ----------------------------------
Guarantees" in the Common Securities Guarantee.

     "Other Debentures" shall mean all junior subordinated debentures, other
      ----------------
than the Debentures and the Series B Debentures (as defined in the Indenture),
issued by the Guarantor, from time to time and sold to trusts other than the
Issuer to be established by the Guarantor (if any), in each case similar to the
Issuer.

     "Other Guarantees" shall mean all guarantees, other than this Series A
      ----------------
Capital Securities Guarantee and the Series B Capital Securities Guarantee, to
be issued by the Guarantor with respect to capital securities (if any) similar
to the Series A Capital Securities, issued by trusts other than the Issuer to be
established by the Guarantor (if any), in each case similar to the Issuer.

                                       4
<PAGE>

     "Person" shall mean a legal person, including any individual, corporation,
      ------
estate, partnership, joint venture, association, joint stock company, limited
liability company, trust, unincorporated association, or government or any
agency or political subdivision thereof, or any other entity of whatever nature.

     "Registration Rights Agreement" shall mean the Registration Rights
      -----------------------------
Agreement, dated as of May 16, 2000, by and among the Guarantor, the Issuer and
the Initial Purchasers named therein, as such agreement may be amended, modified
or supplemented from time to time.

     "Responsible Officer" shall mean, with respect to a Person, any officer
      -------------------
with direct responsibility for the administration of any matters relating to
this Series A Capital Securities Guarantee.

     "Successor Capital Securities Guarantee Trustee" shall mean a successor
      ----------------------------------------------
Capital Securities Guarantee Trustee possessing the qualifications to act as
Capital Securities Guarantee Trustee under Section 4.1.

     "Trust Indenture Act" shall mean the Trust Indenture Act of 1939, as
      -------------------
amended.

     "Trust Securities" shall mean the Common Securities and the Series A
      ----------------
Capital Securities and Series B Capital Securities, collectively.

                                  ARTICLE II

                              TRUST INDENTURE ACT

     SECTION 2.1    Trust Indenture Act; Application
                    --------------------------------

     (a)  This Series A Capital Securities Guarantee is subject to the
provisions of the Trust Indenture Act that are required to be part of this
Series A Capital Securities Guarantee and shall, to the extent applicable, be
governed by such provisions.

     (b)  If and to the extent that any provision of this Series A Capital
Securities Guarantee limits, qualifies or conflicts with the duties imposed by
Section 310 to 317, inclusive, of the Trust Indenture Act, such imposed duties
shall control. If any provision of this Series A Capital Securities Guarantee
modifies or excludes any provision of the Trust Indenture Act that may be so
modified or excluded, the modified or excluded provision of the Trust Indenture
Act shall be deemed to apply to this Series A Capital Securities Guarantee as so
modified or excluded, as the case may be.

     SECTION 2.2    Lists of Holders of Securities
                    ------------------------------

     (a)  The Guarantor shall provide the Capital Securities Guarantee Trustee
(unless the Capital Securities Guarantee Trustee is otherwise the registrar of
the Series A Capital Securities) with a list, in such form as the Capital
Securities Guarantee Trustee may reasonably require, of the names and addresses
of the Holders of the Series A Capital Securities ("List of Holders") as of such
date, (i) within fourteen (14) days after each record date for payment of
Distributions (as defined in the Declaration), and (ii) at any other time within
30 days of receipt by the Guarantor

                                       5
<PAGE>

of a written request for a List of Holders as of a date no more than 14 days
before such List of Holders is given to the Capital Securities Guarantee
Trustee; provided, however, that the Guarantor shall not be obligated to provide
         --------  -------
such List of Holders at any time the List of Holders does not differ from the
most recent List of Holders given to the Capital Securities Guarantee Trustee by
the Guarantor. The Capital Securities Guarantee Trustee may destroy any List of
Holders previously given to it upon receipt of a new List of Holders.

     (b)  The Capital Securities Guarantee Trustee shall comply with its
obligations under Sections 31l(a), 31l(b) and Section 312(b) of the Trust
Indenture Act.

     SECTION 2.3    Reports by the Capital Securities Guarantee Trustee
                    ---------------------------------------------------

     By July 31 of each year, commencing July 31, 2001, the Capital Securities
Guarantee Trustee shall provide to the Holders of the Series A Capital
Securities such reports as are required by Section 313 of the Trust Indenture
Act, if any, in the form and in the manner provided by Section 313 of the Trust
Indenture Act. The Capital Securities Guarantee Trustee shall also comply with
the requirements of Section 313(d) of the Trust Indenture Act.

     SECTION 2.4    Periodic Reports
                    ----------------

     The Guarantor shall provide to the Capital Securities Guarantee Trustee
such documents, reports and information as are required by Section 314 (if any)
and the compliance certificate required by Section 314 of the Trust Indenture
Act in the form, in the manner and at the times required by Section 314 of the
Trust Indenture Act. Delivery of such reports, information and documents to the
Capital Securities Guarantee Trustee is for informational purposes only and the
Capital Securities Guarantee Trustee's receipt of such shall not constitute
constructive notice of any information contained therein or determinable from
information contained therein, including the Guarantor's compliance with any of
its covenants hereunder (as to which the Capital Securities Guarantee Trustee is
entitled to rely exclusively on Officers' Certificates).

     SECTION 2.5    Evidence of Compliance with Conditions Precedent
                    ------------------------------------------------

     The Guarantor shall provide to the Capital Securities Guarantee Trustee
such evidence of compliance with the conditions precedent, if any, provided for
in this Series A Capital Securities Guarantee that relate to any of the matters
set forth in Section 314(c) of the Trust Indenture Act. Any certificate or
opinion required to be given by an officer pursuant to Section 314(c)(1) may be
given in the form of an Officers' Certificate.

     SECTION 2.6    Waiver of Events of Default
                    ---------------------------

     The Holders of a Majority in Liquidation Amount of the Series A Capital
Securities may, by vote, on behalf of the Holders of all of the Series A Capital
Securities, waive any past Event of Default and its consequences. Upon such
waiver, any such Event of Default shall cease to exist, and any Event of Default
arising therefrom shall be deemed to have been cured, for every purpose of this
Series A Capital Securities Guarantee, but no such waiver shall extend to any
subsequent or other default or Event of Default or impair any right consequent
thereon.

                                       6
<PAGE>

     SECTION 2.7    Notice of Events of Default
                    ---------------------------

     (a)  The Capital Securities Guarantee Trustee shall, within 10 Business
Days after the occurrence of an Event of Default with respect to this Series A
Capital Securities Guarantee actually known to a Responsible Officer of the
Capital Securities Guarantee Trustee, transmit by mail, first class postage
prepaid, to all Holders of the Series A Capital Securities, notices of all such
Events of Default, unless such Events of Default have been cured before the
giving of such notice; provided, however, that, except in the case of an Event
                       --------  -------
of Default arising from the non-payment of any Guarantee Payment, the Capital
Securities Guarantee Trustee shall be protected in withholding such notice if
and so long as a Responsible Officer of the Capital Securities Guarantee Trustee
in good faith determines that the withholding of such notice is in the interests
of the Holders of the Series A Capital Securities.

     (b)  The Capital Securities Guarantee Trustee shall not be deemed to have
knowledge of any Event of Default unless the Capital Securities Guarantee
Trustee shall have received written notice, or a Responsible Officer of the
Capital Securities Guarantee Trustee charged with the administration of the
Declaration shall have obtained actual knowledge, of such Event of Default.

     SECTION 2.8    Conflicting Interests
                    ---------------------

     The Declaration shall be deemed to be specifically described in this Series
A Capital Securities Guarantee for the purposes of clause (i) of the first
proviso contained in Section 310(b) of the Trust Indenture Act.

                                  ARTICLE III

                         POWERS, DUTIES AND RIGHTS OF
                     CAPITAL SECURITIES GUARANTEE TRUSTEE

     SECTION 3.1    Powers and Duties of the Capital Securities Guarantee
                    -----------------------------------------------------
Trustee
-------

     (a)  This Series A Capital Securities Guarantee shall be held by the
Capital Securities Guarantee Trustee for the benefit of the Holders of the
Series A Capital Securities, and the Capital Securities Guarantee Trustee shall
not transfer this Series A Capital Securities Guarantee to any Person except a
Holder of the Series A Capital Securities exercising his or her rights pursuant
to Section 5.4(b) or to a Successor Capital Securities Guarantee Trustee on
acceptance by such Successor Capital Securities Guarantee Trustee of its
appointment to act as Successor Capital Securities Guarantee Trustee. The right,
title and interest of the Capital Securities Guarantee Trustee shall
automatically vest in any Successor Capital Securities Guarantee Trustee, and
such vesting and succession of title shall be effective whether or not
conveyancing documents have been executed and delivered pursuant to the
appointment of such Successor Capital Securities Guarantee Trustee.

     (b)  If an Event of Default actually known to a Responsible Officer of the
Capital Securities Guarantee Trustee has occurred and is continuing, the Capital
Securities Guarantee Trustee shall enforce this Series A Capital Securities
Guarantee for the benefit of the Holders of the Series A Capital Securities.

                                       7
<PAGE>

     (c)  The Capital Securities Guarantee Trustee, before the occurrence of any
Event of Default (of which, other than a default in respect of any Guarantee
Payment, a Responsible Officer of the Property Trustee has actual knowledge) and
after the curing of all such Events of Default that may have occurred, shall
undertake to perform only such duties as are specifically set forth in this
Series A Capital Securities Guarantee, and no implied covenants or obligations
shall be read into this Series A Capital Securities Guarantee against the
Capital Securities Guarantee Trustee. In case an Event of Default has occurred
(that has not been cured or waived pursuant to Section 2.6) and is actually
known to a Responsible Officer of the Capital Securities Guarantee Trustee, the
Capital Securities Guarantee Trustee shall exercise such of the rights and
powers vested in it by this Series A Capital Securities Guarantee, and use the
same degree of care and skill in its exercise thereof, as a prudent person would
exercise or use under the circumstances in the conduct of his or her own
affairs.

     (d)  No provision of this Series A Capital Securities Guarantee shall be
construed to relieve the Capital Securities Guarantee Trustee from liability for
its own negligent action, its own negligent failure to act, or its own willful
misconduct, except that:

          (i)  prior to the occurrence of any Event of Default (of which, other
     than a default in respect of any Guarantee Payment, a Responsible Officer
     of the Property Trustee has actual knowledge) and after the curing or
     waiving of all such Events of Default that may have occurred:

          (A)  the duties and obligations of the Capital Securities Guarantee
     Trustee shall be determined solely by the express provisions of this Series
     A Capital Securities Guarantee, and the Capital Securities Guarantee
     Trustee shall not be liable except for the performance of such duties and
     obligations as are specifically set forth in this Series A Capital
     Securities Guarantee, and no implied covenants or obligations shall be read
     into this Series A Capital Securities Guarantee against the Capital
     Securities Guarantee Trustee; and

          (B)  in the absence of bad faith on the part of the Capital Securities
     Guarantee Trustee, the Capital Securities Guarantee Trustee may
     conclusively rely, as to the truth of the statements and the correctness of
     the opinions expressed therein, upon any certificates or opinions furnished
     to the Capital Securities Guarantee Trustee and conforming to the
     requirements of this Series A Capital Securities Guarantee; provided,
                                                                 --------
     however, that in the case of any such certificates or opinions that by any
     -------
     provision hereof are specifically required to be furnished to the Capital
     Securities Guarantee Trustee, the Capital Securities Guarantee Trustee
     shall be under a duty to examine the same to determine whether or not on
     their face they conform to the requirements of this Series A Capital
     Securities Guarantee;

          (ii) the Capital Securities Guarantee Trustee shall not be liable for
     any errors of judgment made in good faith by a Responsible Officer of the
     Capital Securities Guarantee Trustee, unless it shall be proved that the
     Capital Securities Guarantee Trustee or such Responsible Officer was
     negligent in ascertaining the pertinent facts upon which such judgment was
     made;

                                       8
<PAGE>

          (iii)  the Capital Securities Guarantee Trustee shall not be liable
     with respect to any actions taken or omitted to be taken by it in good
     faith in accordance with the direction of the Holders of a Majority in
     Liquidation Amount of the Series A Capital Securities relating to the time,
     method and place of conducting any proceeding for any remedy available to
     the Capital Securities Guarantee Trustee, or exercising any trust or power
     conferred upon the Capital Securities Guarantee Trustee under this Series A
     Capital Securities Guarantee; and

          (iv)   no provision of this Series A Capital Securities Guarantee
     shall require the Capital Securities Guarantee Trustee to expend or risk
     its own funds or otherwise incur personal financial liability in the
     performance of any of its duties or in the exercise of any of its rights or
     powers, if the Capital Securities Guarantee Trustee shall have reasonable
     grounds for believing that the repayment of such funds or liability is not
     reasonably assured to it under the terms of this Series A Capital
     Securities Guarantee or indemnity, reasonably satisfactory to the Capital
     Securities Guarantee Trustee, against such risk or liability is not
     reasonably assured to it.

     SECTION 3.2    Certain Rights of Capital Securities Guarantee Trustee
                    ------------------------------------------------------

     (a)  Subject to the provisions of Section 3.1:

          (i)   the Capital Securities Guarantee Trustee may conclusively rely,
     and shall be fully protected in acting or refraining from acting, upon any
     resolution, certificate, statement, instrument, opinion, report, notice,
     request, direction, consent, order, bond, debenture, note, other evidence
     of indebtedness or other paper or document believed by it to be genuine and
     to have been signed, sent or presented by the proper party or parties;

          (ii)  any direction or act of the Guarantor contemplated by this
     Series A Capital Securities Guarantee may be sufficiently evidenced by an
     Officers' Certificate;

          (iii) whenever, in the administration of this Series A Capital
     Securities Guarantee, the Capital Securities Guarantee Trustee shall deem
     it desirable that a matter be proved or established before taking,
     suffering or omitting any action hereunder, the Capital Securities
     Guarantee Trustee (unless other evidence is herein specifically prescribed)
     may, in the absence of bad faith on its part, request and conclusively rely
     upon an Officers' Certificate, which, upon receipt of such request, shall
     be promptly delivered by the Guarantor;

          (iv)  the Capital Securities Guarantee Trustee shall have no duty to
     see to any recording, filing or registration of any instrument or other
     document (or any rerecording, refiling or registration thereof);

          (v)   the Capital Securities Guarantee Trustee may consult with
     counsel of its selection, and the advice or opinion of such counsel with
     respect to legal matters shall be full and complete authorization and
     protection in respect of any action taken, suffered or omitted by it
     hereunder in good faith and in accordance with such advice or opinion; and
     such counsel may be counsel to the Guarantor or any of its Affiliates and
     may include any of its employees; and the Capital Securities Guarantee
     Trustee shall have the right at

                                       9
<PAGE>

     any time to seek instructions concerning the administration of this Series
     A Capital Securities Guarantee from any court of competent jurisdiction;

          (vi)   the Capital Securities Guarantee Trustee shall be under no
     obligation to exercise any of the rights or powers vested in it by this
     Series A Capital Securities Guarantee at the request or direction of any
     Holder, unless such Holder shall have provided to the Capital Securities
     Guarantee Trustee such security and indemnity, reasonably satisfactory to
     the Capital Securities Guarantee Trustee, against the costs, expenses
     (including attorneys' fees and expenses and the expenses of the Capital
     Securities Guarantee Trustee's agents, nominees or custodians) and
     liabilities that might be incurred by it in complying with such request or
     direction, including such reasonable advances as may be requested by the
     Capital Securities Guarantee Trustee, provided, however, that nothing
                                           --------  -------
     contained in this Section 3.2(a)(vi) shall be taken to relieve the Capital
     Securities Guarantee Trustee, upon the occurrence of an Event of Default,
     of its obligation to exercise the rights and powers vested in it by this
     Series A Capital Securities Guarantee;

          (vii)  the Capital Securities Guarantee Trustee shall have no
     obligation to make any investigation into the facts or matters stated in
     any resolution, certificate, statement, instrument, opinion, report,
     notice, request, direction, consent, order, bond, debenture, note, other
     evidence of indebtedness or other paper or document, but the Capital
     Securities Guarantee Trustee, in its discretion, may make such further
     inquiry or investigation into such facts or matters as it may see fit;

          (viii) the Capital Securities Guarantee Trustee may execute any of
     the trusts or powers hereunder or perform any duties hereunder either
     directly or by or through agents, nominees, custodians or attorneys, and
     the Capital Securities Guarantee Trustee shall not be responsible for any
     misconduct or negligence on the part of any such person appointed with due
     care by it hereunder;

          (ix)   any action taken by the Capital Securities Guarantee Trustee or
     its agents hereunder shall bind the Holders of the Series A Capital
     Securities, and the signature of the Capital Securities Guarantee Trustee
     or its agents alone shall be sufficient and effective to perform any such
     action; and no third party shall be required to inquire as to the authority
     of the Capital Securities Guarantee Trustee to so act or as to its
     compliance with any of the terms and provisions of this Series A Capital
     Securities Guarantee, both of which shall be conclusively evidenced by the
     Capital Securities Guarantee Trustee's or its agent's taking such action;

          (x)    whenever in the administration of this Series A Capital
     Securities Guarantee the Capital Securities Guarantee Trustee shall deem it
     desirable to receive instructions with respect to enforcing any remedy or
     right or taking any other action hereunder, the Capital Securities
     Guarantee Trustee (i) may request instructions from the Holders of a
     Majority in Liquidation Amount of the Series A Capital Securities, (ii) may
     refrain from enforcing such remedy or right or taking such other action
     until such instructions are received, and (iii) shall be protected in
     conclusively relying on or acting in accordance with such instructions; and

                                       10
<PAGE>

          (xi) the Capital Securities Guarantee Trustee shall not be liable for
     any action taken, suffered, or omitted to be taken by it in good faith,
     without negligence, and reasonably believed by it to be authorized or
     within the discretion or rights or powers conferred upon it by this Series
     A Capital Securities Guarantee.

     (b)  No provision of this Series A Capital Securities Guarantee shall be
deemed to impose any duty or obligation on the Capital Securities Guarantee
Trustee to perform any act or acts or exercise any right, power, duty or
obligation conferred or imposed on it in any jurisdiction in which it shall be
illegal, or in which the Capital Securities Guarantee Trustee shall be
unqualified or incompetent in accordance with applicable law, to perform any
such act or acts or to exercise any such right, power, duty or obligation. No
permissive power or authority available to the Capital Securities Guarantee
Trustee shall be construed to be a duty.

     SECTION 3.3  Not Responsible for Recitals or Issuance of Series A Capital
                  ------------------------------------------------------------
                  Securities Guarantee
                  --------------------

     The recitals contained in this Series A Capital Securities Guarantee shall
be taken as the statements of the Guarantor, and the Capital Securities
Guarantee Trustee does not assume any responsibility for their correctness. The
Capital Securities Guarantee Trustee makes no representation as to the validity
or sufficiency of this Series A Capital Securities Guarantee.

                                  ARTICLE IV

                     CAPITAL SECURITIES GUARANTEE TRUSTEE

     SECTION 4.1    Capital Securities Guarantee Trustee; Eligibility
                    -------------------------------------------------

     (a)  There shall at all times be a Capital Securities Guarantee Trustee
that shall

          (i)  not be an Affiliate of the Guarantor; and

          (ii) be a corporation or other Person organized and doing business
     under the laws of the United States of America or any state or territory
     thereof or of the District of Columbia, or a corporation or other Person
     permitted by the Securities and Exchange Commission to act as an indenture
     trustee under the Trust Indenture Act, authorized under such laws to
     exercise corporate trust powers, having a combined capital and surplus of
     at least fifty million U.S. dollars ($50,000,000), and subject to
     supervision or examination by federal, state, territorial or District of
     Columbia authority; it being understood that if such corporation or other
     Person publishes reports of condition at least annually, pursuant to law or
     to the requirements of the supervising or examining authority referred to
     above, then, for the purposes of this Section 4.1(a)(ii) and to the extent
     permitted by the Trust Indenture Act, the combined capital and surplus of
     such corporation shall be deemed to be its combined capital and surplus as
     set forth in its most recent report of condition so published.

     (b)  If at any time the Capital Securities Guarantee Trustee shall cease to
be eligible to so act under Section 4.1(a), the Capital Securities Guarantee
Trustee shall immediately resign in the manner and with the effect set out in
Section 4.2(c).

                                       11
<PAGE>

     (c)  If the Capital Securities Guarantee Trustee has or shall acquire any
"conflicting interest" within the meaning of Section 310(b) of the Trust
Indenture Act, the Capital Securities Guarantee Trustee and Guarantor shall in
all respects comply with the provisions of Section 310(b) of the Trust Indenture
Act.

     SECTION 4.2  Appointment, Removal and Resignation of Capital Securities
                  ----------------------------------------------------------
                  Guarantee Trustee
                  -----------------

     (a)  Subject to Section 4.2(b), the Capital Securities Guarantee Trustee
may be appointed or removed without cause at any time by the Guarantor except
during an Event of Default.

     (b)  The Capital Securities Guarantee Trustee shall not be removed in
accordance with Section 4.2(a) until a Successor Capital Securities Guarantee
Trustee has been appointed and has accepted such appointment by written
instrument executed by such Successor Capital Securities Guarantee Trustee and
delivered to the Guarantor.

     (c)  The Capital Securities Guarantee Trustee shall hold office until a
Successor Capital Securities Guarantee Trustee shall have been appointed,
subject to Section 4.1, or until its removal or resignation. The Capital
Securities Guarantee Trustee may resign from office (without need for prior or
subsequent accounting) by an instrument in writing executed by the Capital
Securities Guarantee Trustee and delivered to the Guarantor, which resignation
shall not take effect until a Successor Capital Securities Guarantee Trustee has
been appointed, subject to Section 4.1, and has accepted such appointment by
instrument in writing executed by such Successor Capital Securities Guarantee
Trustee and delivered to the Guarantor and the resigning Capital Securities
Guarantee Trustee.

     (d)  If no Successor Capital Securities Guarantee Trustee shall have been
appointed and accepted appointment as provided in this Section 4.2 within 60
days after delivery of an instrument of removal or resignation, the Capital
Securities Guarantee Trustee resigning or being removed may petition any court
of competent jurisdiction for appointment of a Successor Capital Securities
Guarantee Trustee. Such court may thereupon, after prescribing such notice, if
any, as it may deem proper, appoint a Successor Capital Securities Guarantee
Trustee.

     (e)  No Capital Securities Guarantee Trustee shall be liable for the acts
or omissions to act of any Successor Capital Securities Guarantee Trustee.

     (f)  Upon termination of this Series A Capital Securities Guarantee or
removal or resignation of the Capital Securities Guarantee Trustee pursuant to
this Section 4.2, the Guarantor shall pay to the Capital Securities Guarantee
Trustee all amounts due to the Capital Securities Guarantee Trustee accrued to
the date of such termination, removal or resignation.

                                       12
<PAGE>

                                   ARTICLE V

                                   GUARANTEE
                                   ---------

     SECTION 5.1  Guarantee
                  ---------

     The Guarantor irrevocably and unconditionally agrees to pay in full to the
Holders the Guarantee Payments (without duplication of amounts theretofore paid
by the Issuer), as and when due, regardless of any defense, right of set-off or
counterclaim that the Issuer may have or assert. The Guarantor fully, knowingly
and unconditionally waives any right the Guarantor may have to revoke this
Guarantee under Section 2815 of the California Civil Code or otherwise. The
Guarantor's obligation to make a Guarantee Payment may be satisfied by direct
payment of the required amounts by the Guarantor to the Holders or by causing
the Issuer to pay such amounts to the Holders.

     SECTION 5.2  Waiver of Notice and Demand
                  ---------------------------

     The Guarantor hereby waives notice of acceptance of this Series A Capital
Securities Guarantee and of any liability to which it applies or may apply,
presentment, demand for payment, any right to require a proceeding first against
the Issuer or any other Person before proceeding against the Guarantor, protest,
notice of nonpayment, notice of dishonor, notice of redemption and all other
notices and demands.

     SECTION 5.3  Obligations Not Affected
                  ------------------------

     The obligations, covenants, agreements and duties of the Guarantor under
this Series A Capital Securities Guarantee shall in no way be affected or
impaired by reason of the happening from time to time of any of the following:

     (a)  the release or waiver, by operation of law or otherwise, of the
performance or observance by the Issuer of any express or implied agreement,
covenant, term or condition relating to the Series A Capital Securities to be
performed or observed by the Issuer;

     (b)  the extension of time for the payment by the Issuer of all or any
portion of the Distributions, Redemption Price, Liquidation Distribution or any
other sums payable under the terms of the Series A Capital Securities or the
extension of time for the performance of any other obligation under, arising out
of, or in connection with, the Series A Capital Securities;

     (c)  any failure, omission, delay or lack of diligence on the part of the
Holders to enforce, assert or exercise any right, privilege, power or remedy
conferred on the Holders pursuant to the terms of the Series A Capital
Securities, or any action on the part of the Issuer granting indulgence or
extension of any kind;

     (d)  the voluntary or involuntary liquidation, dissolution, sale of any
collateral, receivership, insolvency, bankruptcy, assignment for the benefit of
creditors, reorganization, arrangement, composition or readjustment of debt of,
or other similar proceedings affecting, the Issuer or any of the assets of the
Issuer;

                                       13
<PAGE>

     (e)  any invalidity of, or defect or deficiency in, the Series A Capital
Securities;

     (f)  the settlement or compromise of any obligation guaranteed hereby or
hereby incurred;

     (g)  the consummation of the Exchange Offer (subject to Section 7.1
hereof); or

     (h)  any other circumstance whatsoever that might otherwise constitute a
legal or equitable discharge or defense of a guarantor;

it being the intent of this Section 5.3 that the obligations of the Guarantor
with respect to the Guarantee Payments shall be absolute and unconditional under
any and all circumstances.

     There shall be no obligation of the Holders to give notice to, or obtain
consent of, the Guarantor with respect to the happening of any of the foregoing.

     SECTION 5.4  Rights of Holders
                  -----------------

     (a)  The Holders of a Majority in Liquidation Amount of the Series A
Capital Securities have the right to direct the time, method and place of
conducting any proceeding for any remedy available to the Capital Securities
Guarantee Trustee in respect of this Series A Capital Securities Guarantee or
exercising any trust or power conferred upon the Capital Securities Guarantee
Trustee under this Series A Capital Securities Guarantee.

     (b)  If the Capital Securities Guarantee Trustee fails to enforce this
Series A Capital Securities Guarantee, any Holder of the Series A Capital
Securities may institute a legal proceeding directly against the Guarantor to
enforce the Capital Securities Guarantee Trustee's rights under this Series A
Capital Securities Guarantee, without first instituting a legal proceeding
against the Issuer, the Capital Securities Guarantee Trustee or any other person
or entity. The Guarantor waives any right or remedy to require that any action
be brought first against the Issuer or any other person or entity before
proceeding directly against the Guarantor.

     SECTION 5.5  Guarantee of Payment
                  --------------------

     This Series A Capital Securities Guarantee creates a guarantee of payment
and not of collection.

     SECTION 5.6  Subrogation
                  -----------

     The Guarantor shall be subrogated to all (if any) rights of the Holders of
Series A Capital Securities against the Issuer in respect of any amounts paid to
such Holders by the Guarantor under this Series A Capital Securities Guarantee;
provided, however, that the Guarantor shall not (except to the extent required
--------  -------
by mandatory provisions of law) be entitled to enforce or exercise any right
that it may acquire by way of subrogation or any indemnity, reimbursement or
other agreement, in all cases as a result of payment under this Series A Capital
Securities Guarantee, if, at the time of any such payment, any amounts are due
and unpaid under this Series A Capital Securities Guarantee. If any amount shall
be paid to the Guarantor in violation of the preceding

                                       14
<PAGE>

sentence, the Guarantor agrees to hold such amount in trust for the Holders and
to pay over such amount to the Holders.

     SECTION 5.7  Independent Obligations
                  -----------------------

     The Guarantor acknowledges that its obligations hereunder are independent
of the obligations of the Issuer with respect to the Series A Capital
Securities, and that the Guarantor shall be liable as principal and as debtor
hereunder to make Guarantee Payments pursuant to the terms of this Series A
Capital Securities Guarantee notwithstanding the occurrence of any event
referred to in subsections (a) through (h), inclusive, of Section 5.3 hereof.

                                  ARTICLE VI

                   LIMITATION OF TRANSACTIONS; SUBORDINATION

     SECTION 6.1  Limitation of Transactions
                  --------------------------

     So long as any Capital Securities remain outstanding, the Guarantor shall
not (i) declare or pay any dividends or distributions on, or redeem, purchase,
acquire, or make a liquidation payment with respect to, any of the Guarantor's
capital stock, (ii) make any payment of principal of, or interest, if any, on or
repay, repurchase or redeem any debt securities of the Guarantor (including
Other Debentures) that rank pari passu with or junior in right of payment to the
Debentures or (iii) make any guarantee payments with respect to any guarantee by
the Guarantor of the debt securities of any subsidiary of the Guarantor
(including Other Guarantees) if such guarantee ranks pari passu with or junior
in right of payment to the Debentures (other than (a) dividends or distributions
in shares of, or options, warrants, rights to subscribe for or purchase shares
of, common stock of the Guarantor, (b) any declaration of a dividend in
connection with the implementation of a shareholders' rights plan, or the
issuance of stock under any such plan in the future, or the redemption or
repurchase of any such rights pursuant thereto, (c) payments under this Series A
Capital Securities Guarantee and the Series B Capital Securities Guarantee, (d)
as a result of a reclassification of the Guarantor's capital stock or the
exchange or the conversion of one class or series of the Guarantor's capital
stock for another class or series of the Guarantor's capital stock, (e) the
purchase of fractional interests in shares of the Guarantor's capital stock
pursuant to the conversion or exchange provisions of such capital stock or the
security being converted or exchanged, and (f) purchases of common stock related
to the issuance of common stock or rights under any of the Guarantor's benefit
or compensation plans for its directors, officers or employees or any of the
Guarantor's dividend reinvestment plans) if at such time (l) there shall have
occurred any event of which the Guarantor has actual knowledge that (A) is a
Default or Event of Default (each as defined in the Indenture) and (B) in
respect of which the Guarantor shall not have taken reasonable steps to cure,
(2) if the Debentures are held by the Property Trustee, the Guarantor shall be
in default with respect to its payment of any obligations under this Series A
Capital Securities Guarantee or (3) the Guarantor shall have given notice of its
election of the exercise of its right to commence an Extended Interest Payment
Period as provided in the Indenture and shall not have rescinded such notice,
and such Extended Interest Payment Period, or an extension thereof, shall have
commenced and be continuing.

                                       15
<PAGE>

     SECTION 6.2  Ranking
                  -------

     This Series A Capital Securities Guarantee will constitute an unsecured
obligation of the Guarantor and will rank (i) subordinate and junior in right of
payment to all Allocable Amounts (as defined in the Indenture) in respect of
Senior Indebtedness (as defined in the Indenture), to the same extent and in the
same manner that the Debentures are subordinated to Senior Indebtedness pursuant
to the Indenture, it being understood that the terms of Article XV of the
Indenture shall apply to the obligations of the Guarantor under this Series A
Capital Securities Guarantee as if such Article XV were set forth herein in
full, (ii) pari passu with the most senior preferred or preference stock now or
hereafter issued by the Guarantor and with the Series B Capital Securities
Guarantee, any Other Guarantee and, except to the extent set forth therein, the
Common Securities Guarantee, any Other Common Securities Guarantee, and any
guarantee now or hereafter entered into by the Guarantor in respect of any
preferred or preference stock of any Affiliate of the Guarantor, and (iii)
senior to the Guarantor's common stock.

                                  ARTICLE VII

                                  TERMINATION

     SECTION 7.1  Termination
                  -----------

     This Series A Capital Securities Guarantee shall terminate and be of no
further force and effect upon (i) full payment of the Redemption Price of all
Series A Capital Securities, (ii) the exchange of all Series A Capital
Securities for Series B Capital Securities pursuant to the Exchange Offer, (iii)
dissolution, winding up or liquidation of the Issuer, immediately following the
full payment of the amounts payable in accordance with the Declaration, or (iv)
the distribution of all of the Debentures to the Holders of the Trust
Securities. Notwithstanding the foregoing, this Series A Capital Securities
Guarantee will continue to be effective or will be reinstated, as the case may
be, if at any time any Holder of the Series A Capital Securities must restore
payment of any sums paid under the Series A Capital Securities or under this
Series A Capital Securities Guarantee.

                                 ARTICLE VIII

                                INDEMNIFICATION

     SECTION 8.1  Exculpation
                  -----------

     (a)  No Indemnified Person shall be liable, responsible or accountable in
damages or otherwise to the Guarantor or any Covered Person for any loss, damage
or claim incurred by reason of any act or omission performed or omitted by such
Indemnified Person in good faith in accordance with this Series A Capital
Securities Guarantee and in a manner that such Indemnified Person reasonably
believed to be within the scope of the authority conferred on such Indemnified
Person by this Series A Capital Securities Guarantee or by law, except that an
Indemnified Person shall be liable for any such loss, damage or claim incurred
by reason of such Indemnified Person's negligence or willful misconduct with
respect to such acts or omissions.

                                       16
<PAGE>

     (b)  An Indemnified Person shall be fully protected in relying in good
faith upon the records of the Guarantor and upon such information, opinions,
reports or statements presented to the Guarantor by any Person as to matters the
Indemnified Person reasonably believes are within such other Person's
professional or expert competence and who has been selected with reasonable care
by or on behalf of the Guarantor, including information, opinions, reports or
statements as to the value and amount of the assets, liabilities, profits,
losses, or any other facts pertinent to the existence and amount of assets from
which Distributions to Holders of Series A Capital Securities might properly be
paid.

     SECTION 8.2  Compensation and Indemnification
                  --------------------------------

     The Guarantor agrees to pay to the Capital Securities Guarantee Trustee
such compensation for its services as shall be mutually agreed upon by the
Guarantor and the Capital Securities Guarantee Trustee. The Guarantor shall
reimburse the Capital Securities Guarantee Trustee upon request for all
reasonable out-of-pocket expenses incurred by it, including the reasonable
compensation and expenses of the Capital Securities Guarantee Trustee's agents
and counsel, except any expense as may be attributable to the negligence or bad
faith of the Capital Securities Guarantee Trustee.

     The Guarantor agrees to indemnify each Indemnified Person for, and to hold
each Indemnified Person harmless against, any and all loss, liability, damage,
action, suit, claim or expense incurred without negligence or bad faith on its
part, arising out of or in connection with the acceptance or administration of
the trust or trusts hereunder, including the costs and expenses (including
reasonable legal fees and expenses) of defending itself against, or
investigating, any claim or liability in connection with the exercise or
performance of any of its powers or duties hereunder. The provisions of this
Section 8.2 shall survive the termination of this Series A Capital Securities
Guarantee and shall survive the resignation or removal of the Capital Securities
Guarantee Trustee.

                                   ARTICLE IX

                                 MISCELLANEOUS

     SECTION 9.1  Successors and Assigns
                  ----------------------

     All guarantees and agreements contained in this Series A Capital Securities
Guarantee shall bind the successors, assigns, receivers, trustees and
representatives of the Guarantor and shall inure to the benefit of the Holders
of the Series A Capital Securities then outstanding.  Except in connection with
any merger or consolidation of the Guarantor with or into another entity
permitted by Section 10.01 of the Indenture or any sale, transfer, conveyance or
other disposition of the property of the Guarantor permitted by Section 10.01 of
the Indenture, the Guarantor may not assign its rights or delegate its
obligations under this Capital Securities Guarantee.

     SECTION 9.2    Amendments
                    ----------

     Except with respect to any changes that do not materially adversely affect
the rights of Holders of the Series A Capital Securities (in which case no
consent of such Holders will be

                                       17
<PAGE>

required), this Series A Capital Securities Guarantee may only be amended with
the prior approval of the Holders of a Majority in Liquidation Amount of the
Series A Capital Securities. The provisions of Section 12.2 of the Declaration
with respect to meetings of Holders of the Trust Securities apply to the giving
of such approval. This Series A Capital Securities Guarantee may not be amended,
and no amendment hereof that affects the Capital Securities Guarantee Trustee's
rights, duties or immunities hereunder or otherwise, shall be effective, unless
such amendment is executed by the Capital Securities Guarantee Trustee (which
shall have no obligation to execute any such amendment, but may do so in its
sole discretion).

     SECTION 9.3  Notices
                  -------

     All notices provided for in this Series A Capital Securities Guarantee
shall be in writing, duly signed by the party giving such notice, and shall be
delivered, telecopied or mailed by first class mail, as follows:

     (a)  If given to the Issuer, in care of the Administrative Trustee at the
Issuer's mailing address set forth below (or such other address as the Issuer
may give notice of to the Capital Securities Guarantee Trustee and the Holders
of the Series A Capital Securities):

               GBB CAPITAL IV
               2860 West Bayshore Road
               Palo Alto, California 94303
               Attention:  Steven C. Smith
               Telephone:  (650) 813-8200
               Telecopier: (650) 494-9193

     (b)  If given to the Capital Securities Guarantee Trustee, at the Capital
Securities Guarantee Trustee's mailing address set forth below (or such other
address as the Capital Securities Guarantee Trustee may give notice of to the
Holders of the Series A Capital Securities):

               WILMINGTON TRUST COMPANY
               Rodney Square North
               1100 North Market Street
               Wilmington, Delaware 19890-0001
               Attention: Corporate Trust Administration
               Telephone:  (302) 651-1000
               Telecopier: (302) 651-882

                                       18
<PAGE>

     (c)  If given to the Guarantor, at the Guarantor's mailing address set
forth below (or such other address as the Guarantor may give notice of to the
Capital Securities Guarantee Trustee and the Holders of the Series A Capital
Securities):

               GREATER BAY BANCORP
               2860 West Bayshore Road
               Palo Alto, California 94303
               Attention:  Steven C. Smith
               Telephone:  (650) 813-8200
               Telecopier: (650) 494-9193

     (d)  If given to any Holder of the Series A Capital Securities, at the
address set forth on the books and records of the Issuer.

     All such notices shall be deemed to have been given when received in
person, telecopied with receipt confirmed, or mailed by first class mail,
postage prepaid except that if a notice or other document is refused delivery or
cannot be delivered because of a changed address of which no notice was given,
such notice or other document shall be deemed to have been delivered on the date
of such refusal or inability to deliver.

     SECTION 9.4  Exchange Offer
                  --------------

     In the event an Exchange Offer Registration Statement (as defined in the
Registration Rights Agreement) becomes effective and the Issuer exchanges any
Series B Capital Securities for Series A Capital Securities in the Exchange
Offer, the Guarantor will enter into the Series B Capital Securities Guarantee,
which will be in substantially the same form as this Series A Capital Securities
Guarantee, with respect to the Series B Capital Securities.  Each Holder, by its
acceptance hereof, agrees to be bound by the Registration Rights Agreement.

     SECTION 9.5  Benefit
                  -------

     This Series A Capital Securities Guarantee is solely for the benefit of the
Holders of the Series A Capital Securities and, subject to Section 3.1(a), is
not separately transferable from the Series A Capital Securities.

                                       19
<PAGE>

     SECTION 9.6  Governing Law
                  -------------

     THIS SERIES A CAPITAL SECURITIES GUARANTEE SHALL BE GOVERNED BY, AND
CONSTRUED AND INTERPRETED IN ACCORDANCE WITH, THE LAWS OF THE STATE OF NEW YORK
WITHOUT REGARD TO CONFLICT OF LAW PRINCIPLES THEREOF.

     This Series A Capital Securities Guarantee is executed as of  the day and
year first above written.

                              GREATER BAY BANCORP
                              as Guarantor

                              By: /S/ Shawn E. Saunders
                                 ______________________________
                                 Name:   Shawn E. Saunders
                                 Title:  Senior Vice President,
                                         Finance and Accounting

                              WILMINGTON TRUST COMPANY,
                              as Capital Securities Guarantee Trustee

                              By: /S/ Anita E. Dallago
                                 ______________________________
                                 Name: Anita E. Dallago
                                 Title: Administrative Account Manager

                                       20

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