A counterexample concerning the relation between decoupling constants and UMD-constants
Authors: Geiss Stefan 
Published: July 2002

For Banach spaces

X
X

and

Y
Y

and a bounded linear operator

T
:
X
→
Y

T:X \rightarrow Y

we let

ρ
(
T
)
:=
inf
c

\rho (T):=\inf c

such that
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper A upper V Subscript theta Sub Subscript l Subscript equals plus-or-minus 1 Baseline double-vertical-bar sigma-summation Underscript l equals 1 Overscript normal infinity Endscripts theta Subscript l Baseline left-parenthesis sigma-summation Underscript k equals tau Subscript l minus 1 Baseline plus 1 Overscript tau Subscript l Baseline Endscripts h Subscript k Baseline upper T x Subscript k Baseline right-parenthesis double-vertical-bar Subscript upper L 2 Sub Superscript upper Y Subscript Superscript 2 Baseline right-parenthesis Superscript one half Baseline less-than-or-equal-to c double-vertical-bar sigma-summation Underscript k equals 1 Overscript normal infinity Endscripts h Subscript k Baseline x Subscript k Baseline double-vertical-bar Subscript upper L 2 Sub Superscript upper X">

(
A

V

θ
l

=
±
1

‖

∑

l
=
1

∞

θ
l

(

∑

k
=

τ

l
−
1

+
1

τ
l

h
k

T

x
k

)

‖

L
2
Y

2

)

1
2

≤
c

‖

∑

k
=
1

∞

h
k

x
k

‖

L
2
X

\left ( AV_{\theta _l = \pm 1} \left \|\sum \limits _{l=1}^\infty \theta _l \left ( \sum \limits _{k=\tau _{l-1}+1}^{\tau _l} h_k T x_k \right )\right \|_{L_2^Y}^2 \right )^{\frac {1}{2}} \le c \left \| \sum \limits _{k=1}^\infty h_k x_k \right \| _{L_2^X}

\]
for all finitely supported

(

x
k

)

k
=
1

∞

⊂
X

(x_k)_{k=1}^\infty \subset X

and all

0
=

τ
0

>

τ
1

>
⋯

0 = \tau _0 > \tau _1 > \cdots

, where

(

h
k

)

k
=
1

∞

⊂

L
1

[
0
,
1
)

(h_k)_{k=1}^\infty \subset L_1[0,1)

is the sequence of Haar functions. We construct an operator

T
:
X
→
X

T:X \rightarrow X

, where

X
X

is superreflexive and of type 2, with

ρ
(
T
)
>
∞

\rho (T)>\infty

such that there is no constant

c
>
0

c>0

with
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sup Underscript theta Subscript k Baseline equals plus-or-minus 1 Endscripts double-vertical-bar sigma-summation Underscript k equals 1 Overscript normal infinity Endscripts theta Subscript k Baseline h Subscript k Baseline upper T x Subscript k Baseline double-vertical-bar Subscript upper L 2 Sub Superscript upper X Subscript Baseline less-than-or-equal-to c double-vertical-bar sigma-summation Underscript k equals 1 Overscript normal infinity Endscripts h Subscript k Baseline x Subscript k Baseline double-vertical-bar Subscript upper L 2 Sub Superscript upper X Subscript Baseline period">

sup

θ
k

=
±
1

‖

∑

k
=
1

∞

θ
k

h
k

T

x
k

‖

L
2
X

≤
c

‖

∑

k
=
1

∞

h
k

x
k

‖

L
2
X

.

\sup _{\theta _k = \pm 1} \left \| \sum \limits _{k=1}^\infty \theta _k h_k T x_k \right \| _{L_2^X} \le c \left \| \sum \limits _{k=1}^\infty h_k x_k \right \| _{L_2^X}.

\]

In particular it turns out that the decoupling constants

ρ
(

I
X

)

\rho (I_X)

, where

I
X

I_X

is the identity of a Banach space

X
X

, fail to be equivalent up to absolute multiplicative constants to the usual

UMD
\operatorname {UMD}

–constants. As a by-product we extend the characterization of the non–superreflexive Banach spaces by the finite tree property using lower 2–estimates of sums of martingale differences.