bluuebunny/crossref_metadata_2025 huggingface dataset:
Group 𝐶*-algebras of real rank zero or one
Authors: Kaniuth Eberhard
Published: June 2010

Let

G
G

be a locally compact group and

C

∗

(
G
)

{C^{\ast }}(G)

its group

C

∗

{C^{\ast }}

-algebra, and denote by

RR
⁡
(

C

∗

(
G
)
)

\operatorname {RR} ({C^{\ast }}(G))

the real rank of

C

∗

(
G
)

{C^{\ast }}(G)

. This note is a first step towards relating

RR
⁡
(

C

∗

(
G
)
)

\operatorname {RR} ({C^{\ast }}(G))

to the structure of

G
G

. We identify the connected groups

G
G

with

RR
⁡
(

C

∗

(
G
)
)
=
0

\operatorname {RR} ({C^{\ast }}(G)) = 0

as precisely the compact connected ones and characterize the nilpotent groups whose

C

∗

{C^{\ast }}

-algebras have real rank zero or one.
Some characterizations of semi-Bloch functions
Authors: Aulaskari Rauno , Lappan Peter
Published: June 2010

A function

f
f

analytic in the unit disk is called a semi-Bloch function if, for each complex number

λ
\lambda

, the function

g
λ

(
z
)
=
exp
⁡
(
λ
f
(
z
)
)

{g_\lambda }(z) = \exp (\lambda f(z))

is a normal function. We give both an analytic and a geometric characterization of semi-Bloch functions, together with some examples to show that semi-Bloch functions are not closed under either addition or multiplication.
Involutions fixing products of circles
Authors: Stong R. E.
Published: June 2010

This paper determines the possible equivariant bordism classes of involutions having fixed set a union of products of circles.
Chain transitivity and rotation shadowing for annulus homeomorphisms
Authors: Botelho Fernanda , Chen Liang
Published: June 2010

We present a relation between the rotation of chain transitive sets and the rotation shadowing for annulus homeomorphisms isotopic to identity.
For right-angled Coxeter groups 𝑧^{|𝑔|} is a coefficient of a uniformly bounded representation
Authors: Januszkiewicz Tadeusz
Published: June 2010

A Coxeter group

Γ
\Gamma

is right angled if any exponent in the Coxeter diagram is either

2
2

or

∞
\infty

. Using the action of

Γ
\Gamma

on its Davis complex, we construct a family of cocycles that we use to perturb the left regular representation of

Γ
\Gamma

. In this way, we obtain a family

(

π
z

)

|

z

|

>
1

{({\pi _z})_{|z| > 1}}

of uniformly bounded representations of

Γ
\Gamma

, of which the function

g
→

|

g

|

g \to |g|

is a coefficient (where

|

g

|

|g|

denotes the word length of

g
∈
Γ

g \in \Gamma

).
Weak amenability of right-angled Coxeter groups
Authors: Valette Alain
Published: June 2010

We show that the representation-theoretic result of Januszkiewicz has an impact on harmonic analysis and operator algebras; more precisely, right-angled Coxeter groups are weakly amenable with Cowling-Haagerup constant11; as a consequence, von Neumann algebras with Cowling-Haagerup constant>1> 1are not embeddable into the von Neumann algebra of a right-angled Coxeter group.
Discreteness of some continuous spectrum eigenfunction expansions
Authors: Hinton Don B. , Kauffman Robert M.
Published: June 2010

We discuss replacing the integrals in continuous spectrum expansions by finite sums, in the special case of the Dirichlet problem for second-order ordinary differential operators on a half-line. The error is controlled in the operator norm ofB(Y,Z)B(Y,Z), whereYYandZZare natural Hilbert spaces for the problem.
A dichotomy theorem for the adjoint of a semigroup of operators
Authors: van Neerven J. M. A. M.
Published: June 2010

Let

T
(
t
)

T(t)

be a

C
0

{C_0}

-semigroup of linear operators on a Banach space

X
X

, and let

X
⊗

{X^ \otimes }

, resp.

X
⊙

{X^ \odot }

, denote the closed subspaces of

X

∗

{X^{\ast }}

consisting of all functionals

x

∗

{x^{\ast }}

such that the map

t
↦

T

∗

(
t
)

x

∗

t \mapsto {T^{\ast }}(t){x^{\ast }}

is strongly continuous for

t
>
0

t > 0

, resp.

t
⩾
0

t \geqslant 0

. Theorem. Every nonzero orbit of the quotient semigroup on

X

∗

/

X
⊗

{X^{\ast }}/{X^ \otimes }

is nonseparably valued. In particular, orbits in

X

∗

/

X
⊙

{X^{\ast }}/{X^ \odot }

are either zero for

t
>
0

t > 0

or nonseparable. It also follows that the quotient space

X

∗

/

X
⊗

{X^{\ast }}/{X^ \otimes }

is either zero or nonseparable. If

T
(
t
)

T(t)

extends to a

C
0

{C_0}

-group, then

X

∗

/

X
⊙

{X^{\ast }}/{X^ \odot }

is either zero or nonseparable. For the proofs we make a detailed study of the second adjoint of a

C
0

{C_0}

-semigroup.
A characterization of the second dual of 𝐶₀(𝑆,𝐴)
Authors: Choy Stephen T. L. , Wong James C. S.
Published: June 2010

Let

S
S

be a locally compact Hausdorff space, and let

A
A

be a Banach space. The space of the continuous functions from

S
S

to

A
A

vanishing at infinity is denoted by

C
0

(
S
,
A
)

{C_0}(S,A)

. Let

M
W
(
S
,

A

∗

)

MW(S,{A^{\ast }})

be the space of the representing measures of all the bounded linear functionals on

C
0

(
S
,
A
)

{C_0}(S,A)

. For

μ
∈
M
W
(
S
,

A

∗

)

\mu \in MW(S,{A^{\ast }})

let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript normal infinity Baseline left-parenthesis StartAbsoluteValue mu EndAbsoluteValue comma upper A Superscript asterisk asterisk Baseline comma upper A Superscript asterisk Baseline right-parenthesis equals left-brace f colon upper S right-arrow upper A Superscript asterisk asterisk Baseline vertical-bar f left-parenthesis dot right-parenthesis x Superscript asterisk Baseline element-of upper L Subscript normal infinity Baseline left-parenthesis StartAbsoluteValue mu EndAbsoluteValue right-parenthesis for-all x Superscript asterisk Baseline element-of upper A Superscript asterisk Baseline right-brace period">

L
∞

(

|

μ

|

,

A

∗

∗

,

A

∗

)
=
{
f
:
S
→

A

∗

∗

|

f
(
⋅
)

x

∗

∈

L
∞

(

|

μ

|

)
∀

x

∗

∈

A

∗

}
.

{L_\infty }(|\mu |,{A^{{\ast }{\ast }}},{A^{\ast }}) = \{ f:S \to {A^{{\ast }{\ast }}}|f( \cdot ){x^{\ast }} \in {L_\infty }(|\mu |)\forall {x^{\ast }} \in {A^{\ast }}\}.

\]
The second dual of

C
0

(
S
,
A
)

{C_0}(S,A)

is characterized in the general case by means of certain elements in the product linear space
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product StartSet upper L Subscript normal infinity Baseline left-parenthesis StartAbsoluteValue mu EndAbsoluteValue comma upper A Superscript asterisk asterisk Baseline comma upper A Superscript asterisk Baseline right-parenthesis colon mu element-of upper M upper W left-parenthesis upper S comma upper A Superscript asterisk Baseline right-parenthesis EndSet">

∏

{

L
∞

(

|

μ

|

,

A

∗

∗

,

A

∗

)
:
μ
∈
M
W
(
S
,

A

∗

)
}

\prod {\{ {L_\infty }(|\mu |,{A^{{\ast }{\ast }}},{A^{\ast }}):\mu \in MW(S,{A^{\ast }})\} }

.
A decomposition theorem for 𝑅ⁿ
Authors: Komjáth Péter
Published: June 2010

R

n

{{\mathbf {R}}^n}

is the union of countably many sets, none containing two points a rational distance apart.
Weak (1,1) estimate for oscillatory singular integrals with real-analytic phases
Authors: Pan Yibiao
Published: June 2010

In this paper, we prove the uniform weak

(
1
,

1
)

(1,\;1)

estimate for oscillatory singular integral operators with real-analytic phase functions in the one-dimensional case. Some partial results for the higher-dimensional case is also included.
Discrepancy of behavior of perturbed sequences in 𝐿^{𝑝} spaces
Authors: Reinhold-Larsson Karin
Published: June 2010

Given

p
∈
[
1
,
∞
)

p \in [1,\infty )

, examples of sequences

{

n
k

}

k
⊂

N

{\{ {n_k}\} _{k \subset \mathbb {N}}}

such that for any ergodic dynamical system

(
X
,
β
,
m
,
T
)

(X,\beta ,m,T)

the averages
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript upper N Baseline f left-parenthesis x right-parenthesis equals StartFraction 1 Over upper N EndFraction sigma-summation Underscript k equals 1 Overscript upper N Endscripts f left-parenthesis upper T Superscript n Super Subscript k Superscript Baseline x right-parenthesis">

A
N

f
(
x
)
=

1
N

∑

k
=
1

N

f
(

T

n
k

x
)

{A_N}f(x) = \frac {1} {N}\sum \limits _{k = 1}^N {f({T^{{n_k}}}x)}

\]
converge almost everywhere in all

L
q

(
X
)
,

q
>
p

{L^q}(X),\,q > p

, but fail to have a finite limit for some function in

L
p

(
X
)

{L^p}(X)

are shown. Also, sequences such that for all ergodic dynamical systems the averages

A
N

f
(
x
)

{A_N}f(x)

do not converge for some function

f
∈

L
p

(
X
)

f \in {L^p}(X)

for all

1
⩽
p
>
∞

1 \leqslant p > \infty

but do converge for all functions in

L
∞

(
X
)

{L^\infty }(X)

are shown.
On a counting formula of Djoković for elements of finite order in compact Lie groups
Authors: Destrempes F. , Pianzola A.
Published: June 2010

Given a compact connected simple Lie group

G

\mathfrak {G}

and a positive integer N relatively prime to the order of the Weyl group we give a counting formula for the number of conjugacy classes of elements x of order N in

G

\mathfrak {G}

with the property that the N-cyclotonic field when viewed as a Galois extension of the field of characters of x has Galois group containing a fixed chosen cyclic group

G

\mathcal {G}

. The case

G

=
{
1
}

\mathcal {G} = \{ 1\}

recovers a formula, due to Djoković, which counts the number of conjugacy classes of elements of order dividing N in

G

\mathfrak {G}

.
Character values at involutions
Authors: Gallagher P. X.
Published: June 2010

If

χ
1

,

χ
2

,

χ
3

{\chi _1},{\chi _2},{\chi _3}

are irreducible characters of a finite group

G
G

satisfying

∫
G

χ
1

χ
2

χ
3

≠
0

\int _G {{\chi _1}{\chi _2}{\chi _3} \ne 0}

and

σ
\sigma

is an involution in

G
G

, then the proportions of

−
1

- 1

’s among the eigenvalues of the corresponding representations at

σ
\sigma

are the sides of a triangle on a sphere of circumference

2
2

.
Involutions on algebras arising from locally compact groups
Authors: Patterson Paul L.
Published: June 2010

Two Banach algebras are naturally associated with a locally compact group G: the group algebra,

L
1

(
G
)

{L^1}(G)

, and the measure algebra,

M
(
G
)

M(G)

. For these two Banach algebras we determine all isometric involutions. Each of these Banach algebras has a natural involution. We will show that an isometric involution,

(
#

)

{(^\# })

, is the natural involution on

L
1

(
G
)

{L^1}(G)

if and only if the closure in the strict topology of the convex hull of the norm one unitaries in

M
(
G
)

M(G)

is equal to the unit ball of

M
(
G
)

M(G)

. There is a well-known relationship between the involutive representation theory of

L
1

(
G
)

{L^1}(G)

, with the natural involution, and the representation theory of G. We develop a similar theory for the other isometric involutions on

L
1

(
G
)

{L^1}(G)

. The main result is: if

(
#

)

{(^\# })

is an isometric involution on

L
1

(
G
)

{L^1}(G)

and T is an involutive representation of

(

L
1

(
G
)

,
#

)

({L^1}(G){,^\# })

, then T is also an involutive representation of

L
1

(
G
)

{L^1}(G)

with the natural involution.
A simple analytic proof of an inequality by P. Buser
Authors: Ledoux M.
Published: June 2010

We present a simple analytic proof of the inequality of P. Buser showing the equivalence of the first eigenvalue of a compact Riemannian manifold without boundary and Cheeger’s isoperimetric constant under a lower bound on the Ricci curvature. Our tools are the Li-Yau inequality and ideas of Varopoulos in his functional approach to isoperimetric inequalities and heat kernel estimates on groups and manifolds. The method is easily modified to yield a logarithmic isoperimetric inequality involving the hypercontractivity constant of the manifold.
Self-duality and differentiable structures on the connected sum of complex projective planes
Authors: Pedersen Henrik , Poon Yat Sun
Published: June 2010

It is proved that if the twistor space of a self-dual four-manifold of positive scalar curvature contains a real effective divisor of degree two, then the four-manifold is diffeomorphic to the connected sum

n

C

P
2

n\mathbb {C}{P^2}

of n complex projective planes for some n. It follows that if the four-manifold is known to be homeomorphic to

4

C

P
2

4\mathbb {C}{P^2}

, then it is also diffeomorphic to

4

C

P
2

4\mathbb {C}{P^2}

.
Positive definite operator sequences
Authors: Bisgaard T.
Published: June 2010

An example is given of a linear mapping from

C

[
x
]

{\mathbf {C}}[x]

to

M

2

(

C

)

{{\mathbf {M}}_2}({\mathbf {C}})

which is positive but not completely positive. It is shown that a positive linear mapping from

C

[
x
]

{\mathbf {C}}[x]

to

B

(

H

)

{\mathbf {B}}(\mathcal {H})

is completely positive if certain scalar moment sequences associated with it are determinate.
Characterizations and generalizations of continuity
Authors: Ash J. M. , Cohen J. , Freiling C. , Gluck L. , Rieders E. , Wang G.
Published: June 2010

The condition

f
(
x
+
2
h
)
−
2
f
(
x
+
h
)
+
f
(
x
)
=
o
(
1
)

f(x + 2h) - 2f(x + h) + f(x) = o(1)

(as

h
→
0

h \to 0

) at each x is equivalent to continuity for measurable functions. But there is a discontinuous function satisfying

2
f
(
x
+
2
h
)
−
f
(
x
+
h
)
−
f
(
x
)
=
o
(
1
)

2f(x + 2h) - f(x + h) - f(x) = o(1)

at each x. The question of which generalized Riemann derivatives of order 0 characterize continuity is studied. In particular, a measurable function satisfying

∑

i
=
1

n

α
i

f
(
x
+

β
i

h
)
≡
0

\sum \nolimits _{i = 1}^n {{\alpha _i}f(x + {\beta _i}h) \equiv 0}

must be a polynomial. On the other hand, for any Riemann derivative of order 0 and any

p
∈
[
1
,
∞
]

p \in [1,\infty ]

, generalized

L
p

{L^p}

continuity is equivalent to

L
p

{L^p}

continuity almost everywhere.
On the Sobolev class of a composite function
Authors: Cox Dennis D. , O’Sullivan Finbarr
Published: June 2010

It is shown that the Sobolev class of a function of the form

H
(
x
,
θ
(
x
)
)

H(x,\theta (x))

is the same as the Sobolev class of

θ
\theta

, for sufficiently smooth H. This result has applications in a perturbation analysis of a nonlinear system of differential equations considered elsewhere.
Noncommutative decomposition theorems in Riesz spaces
Authors: De Lucia Paolo , Morales Pedro
Published: June 2010

We show that an additive function defined on an orthomodular poset and taking its values in the positive cone of a normed Riesz space admits a Lebesgue Decomposition and a Yosida-Hewitt Decomposition.
Constants are definable in rings of analytic functions
Authors: Huuskonen Taneli
Published: June 2010

The analytic functions defined in a fixed domain form a ring with pointwise addition and multiplication. We describe a way to define constants in the ring using a first-order formula which is independent of the domain.
Postnikov sections of formal and hyperformal spaces
Authors: Smith Samuel B.
Published: June 2010

Let X be a simply connected CW complex and

X
n

{X_n}

its nth Postnikov section. We prove that X is formal provided

H
q

(

X
n

,

Q

)

{H^q}({X_n},\mathbb {Q})

is additively generated by decomposables for all q and n with

q
>
n

q > n

. Recall from [4] that a space X is said to be hyperformal if its rational cohomology algebra is the quotient of a free graded algebra by an ideal generated by a regular sequence. Using the main result of Felix and Halperin’s paper (Trans. Amer. Math. Soc. 270 (1982), 575-588) we show our sufficient condition for formality is actually equivalent to hyperformality.
A collaring theorem for codimension one manifolds
Authors: Daverman Robert J. , Tinsley Fred C.
Published: June 2010

The chief result implies that an

n
n

-manifold

S
S

embedded in the interior of an

(
n
+
1
)

(n + 1)

-manifold

M
M

as a closed, separating subset is locally flatly embedded if the embedding is well behaved in a locally peripheral sense and if

S
S

has arbitrarily close neighborhoods

Q
Q

such that the fundamental groups of appropriate components of

Q
∖
S

Q\backslash S

admit a uniform finite upper bound on the number of generators.
A note on the differentiability of convex functions
Authors: Wu Cong Xin , Cheng Li Xin
Published: June 2010

Every real-valued convex and locally Lipschitzian function f defined on a nonempty closed convex set D of a Banach space E is the local restriction of a convex Lipschitzian function defined on E. Moreover, if E is separable and

int
⁡
D
≠
∅

\operatorname {int} D \ne \emptyset

, then, for each Gateaux differentiability point x

(
∈
int
⁡
D
)

( \in \operatorname {int} D)

of f, there is a closed convex set

C
⊂
int
⁡
D

C \subset \operatorname {int} D

with the nonsupport points set

N
(
C
)
≠
∅

N(C) \ne \emptyset

and with

x
∈
N
(
C
)

x \in N(C)

such that

f
C

{f_C}

(the restriction of f on C) is Fréchet differentiable at x.
Groups with only resolvable group topologies
Authors: Comfort W. W. , van Mill Jan
Published: June 2010

Adapting terminology suggested by work of E. Hewitt [Duke Math. J. 10 (1943), 309-333], we say that a group

G
G

is strongly resolvable if for every nondiscrete Hausdorff group topology

I

\mathcal {I}

on

G
G

there is

D
⊆
G

D \subseteq G

such that both

D
D

and

G
∖
D

G\backslash D

are

I

\mathcal {I}

-dense in

G
G

. Theorem. Let

G
G

be an Abelian group. (a) If

G
G

contains no subgroup isomorphic to the group

⨁
ω

{
0.1
}

{ \bigoplus _\omega }\{ 0.1\}

, then

G
G

is strongly resolvable. (b) Assume MA. If

G
G

contains a copy of

⨁
ω

{
0
,
1
}

{ \bigoplus _\omega }\{ 0,1\}

, then

G
G

is not strongly resolvable. Our proof of (b) depends heavily on work of Malykhin.
On the relation between positive definite functions and generalized Toeplitz kernels
Authors: Friedrich J.
Published: June 2010

We show that extension problems for generalized Toeplitz kernels may be completely reduced to extension problems for positive definite functions, where the solution is well known. These considerations in particular imply that generalized Toeplitz kernels may be represented as Fourier transforms of positive operator-valued measures.
Approximation of normal elements in the multiplier algebra of an AF 𝐶*-algebra
Authors: Loring Terry A. , Spielberg Jack
Published: June 2010

It is shown that there is a simple separable AF algebra A such that

M
(

K

⊗
A
)

M(\mathcal {K} \otimes A)

does not have weak (FN) and such that the generalized Berg-Weylvon Neumann Theorem does not hold for

K

⊗
A

\mathcal {K} \otimes A

.
Local rings of relatively small type are Cohen-Macaulay
Authors: Kawasaki Takesi
Published: June 2010

Let A be a local ring of type n. It is known that if

n
=
1

n = 1

, then A is Cohen-Macaulay and that if

n
=
2

n = 2

and A is unmixed, then A is Cohen-Macaulay. Then let

n
≥
3

n \geq 3

. What makes A Cohen-Macaulay? We show that if A contains a field and

A
^

\hat A

satisfies

(

S

n
−
1

)

({S_{n - 1}})

, then A is Cohen-Macaulay.
On stratifiable and elastic spaces
Authors: Harris M. Jeanne
Published: June 2010

We show that every stratifiable space has a

σ
\sigma

-cushioned pair-base which is a function, and more generally, every linearly stratifiable space has a linearly cushioned pair-base which is a function. These results provide a correct proof for a theorem of Tamano and Vaughan.
On a class of Lipschitz continuous functions of several variables
Authors: Van Tran Duc , Son Nguyen Duy Thai
Published: June 2010

We establish an estimate via initial values for functions in a class of Lipschitz continuous functions of several variables. This estimate can be used to investigate the uniqueness of quasi-classical solutions of Cauchy problems for first-order nonlinear partial differential equations (PDEs). Particularly, we give an answer to an open problem posed by S. N. Kružkov.
𝐿-series and modular forms of half-integral weight
Authors: Hatcher Rhonda L.
Published: June 2010

Let f be a normalized Hecke eigenform of weight 2k, with k odd. The main result of this paper is an equation representing the value of

L
(
f
,
s
)
L
(
f
⊗
ε
,
s
)

L(f,s)L(f \otimes \varepsilon ,s)

at

s
=
k

s = k

in terms of the Fourier coefficients of a modular form of half-integral weight.
A remark on coincidence theorems
Authors: Tarafdar E. , Yuan Xian-Zhi
Published: June 2010

Using a different approach, we prove a general coincidence theorem of multivalued mappings which have contractible values in non-convex setting.
Base-invariance implies Benford’s law
Authors: Hill Theodore P.
Published: June 2010

A derivation of Benford’s Law or the First-Digit Phenomenon is given assuming only base-invariance of the underlying law. The only base-invariant distributions are shown to be convex combinations of two extremal probabilities, one corresponding to point mass and the other a log-Lebesgue measure. The main tools in the proof are identification of an appropriate mantissa

σ
\sigma

-algebra on the positive reals, and results for invariant measures on the circle.
𝜀-isometric embeddings
Authors: Qian Songwei
Published: June 2010

In this paper we study into

ε
\varepsilon

-isometries. We prove that if

φ
\varphi

is an

ε
\varepsilon

-isometry from

L
p

(

Ω
1

,

Σ
1

,

μ
1

)

{L^p}({\Omega _1},{\Sigma _1},{\mu _1})

into

L
p

(

Ω
2

,

Σ
2

,

μ
2

)

{L^p}({\Omega _2},{\Sigma _2},{\mu _2})

(for some

p
,
1
>
p
>
∞

p, 1 > p > \infty

), then there is a linear operator
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T colon upper L Superscript p Baseline left-parenthesis normal upper Omega 2 comma normal upper Sigma 2 comma mu 2 right-parenthesis right-arrow from bar upper L Superscript p Baseline left-parenthesis normal upper Omega 1 comma sigma 1 comma mu 1 right-parenthesis">

T
:

L
p

(

Ω
2

,

Σ
2

,

μ
2

)
↦

L
p

(

Ω
1

,

σ
1

,

μ
1

)

T:{L^p}({\Omega _2},{\Sigma _2},{\mu _2}) \mapsto {L^p}({\Omega _1},{\sigma _1},{\mu _1})

with

‖
T
‖

=
1

\left \| T \right \| = 1

such that

‖

T
∘
φ
(
f
)
−
f

‖

≤
6
ε

\left \| {T \circ \varphi (f) - f} \right \| \leq 6\varepsilon

for each

f
∈

L
p

(

Ω
1

,

Σ
1

,

μ
1

)

f \in {L^p}({\Omega _1},{\Sigma _1},{\mu _1})

. This forms a link between an into isometry result of Figiel and a surjective

ε
\varepsilon

-isometry result of Gevirtz in the case of

L
p

{L^p}

spaces.
On an elliptic equation with concave and convex nonlinearities
Authors: Bartsch Thomas , Willem Michel
Published: June 2010

We study the semilinear elliptic equation

−
Δ
u
=
λ

|

u

|

q
−
2

u
+
μ

|

u

|

p
−
2

u

- \Delta u = \lambda |u{|^{q - 2}}u + \mu |u{|^{p - 2}}u

in an open bounded domain

Ω
⊂

R

N

\Omega \subset {\mathbb {R}^N}

with Dirichlet boundary conditions; here

1
>
q
>
2
>
p
>

2
∗

1 > q > 2 > p > {2^ \ast }

. Using variational methods we show that for

λ
>
0

\lambda > 0

and

μ
∈

R

\mu \in \mathbb {R}

arbitrary there exists a sequence

(

v
k

)

({v_k})

of solutions with negative energy converging to 0 as

k
→
∞

k \to \infty

. Moreover, for

μ
>
0

\mu > 0

and

λ
\lambda

arbitrary there exists a sequence of solutions with unbounded energy. This answers a question of Ambrosetti, Brézis and Cerami. The main ingredient is a new critical point theorem, which guarantees the existence of infinitely many critical values of an even functional in a bounded range. We can also treat strongly indefinite functionals and obtain similar results for first-order Hamiltonian systems.
Some contributions to the theory of denumerable Markov chains
Authors: Derman Cyrus
Published: June 2010

§1 deals with the statistical regularity properties of a denumerable number of particles, all moving about the states of a Markov chain according to the same transition probabilities. §2 deals with the problem of obtaining a sharper version of a strong limit theorem proved independently by Harris and Lévy.
Some limit theorems for nonhomogeneous Markoff processes
Authors: Fuchs A.
Published: June 2010

We intend to study some problems related to the asymptotic behaviour of a physical system the evolution of which is markovian. The typical example of such an evolution is furnished by an homogeneous discrete chain with a finite number of possible states considered first by A. A. Markoff. In §1 we recall briefly the main results of this theory and in §2 we treat its obvious generalization to the continuous parameter case. In §3 we pass to the proper object of this paper and we establish a limit theorem for time-homogeneous Markoff processes. This limit theorem is then extended to the nonhomogeneous case under some supplementary conditions (§4). Finally we give an application of this theory to random functions connected with a Markoff process (§5).
Finite nilpotent characteristic nonverbal groups
Authors: Chein Orin
Published: June 2010

In this paper, we study nilpotent groups which are quotient groups of finitely generated free groups with respect to characteristic but nonverbal subgroups. We show that there are no abelian groups of the type in question. We also show that all such groups of nilpotence class 2 or 3 are finite and have minimal sets of two generators. In fact, formal presentations for all such groups are given. The direct product of two finite CNV groups (as the groups in question will be called) which have minimal sets of generators of the same size is shown to again be a CNV group, provided that the orders of the original two groups are relatively prime. Conversely, if a finite CNV group is a direct product of groups of relatively prime orders, then at least one of these direct factors is a CNV group. Several other related results are also obtained.
State spaces for Markov chains
Authors: Doob J. L.
Published: June 2010

If

p
(
t
,
i
,
j
)

p(t,i,j)

is the transition probability (from i to j in time t) of a continuous parameter Markov chain, with

p
(
0
+
,
i
,
i
)
=
1

p(0 + ,i,i) = 1

, entrance and exit spaces for p are defined. If

L
[

L
∗

]

L[{L^ \ast }]

is an entrance [exit] space, the function

p
(
⋅
,
⋅
,
j
)
[
p
(
⋅
,
i
,
⋅
)

/

h
(
⋅
)
]

p( \cdot , \cdot ,j)[p( \cdot ,i, \cdot )/h( \cdot )]

has a continuous extension to

(
0
,
∞
)
×
L
[
(
0
,
∞
)
×

L
∗

(0,\infty ) \times L[(0,\infty ) \times {L^ \ast }

, for a certain norming function h on

L
∗

{L^ \ast }

]. It is shown that there is always a space which is both an entrance and exit space. On this space one can define right continuous strong Markov processes, for the parameter interval [0, b], with the given transition function as conditioned by specification of the sample function limits at 0 and b.
Noncompact simplices
Authors: Simons S.
Published: June 2010

A bounded, but not necessarily closed, (Choquet) simplex in

R
n

{R^n}

with nonempty interior is the intersection of

n
+
1

n + 1

half-spaces. There is no bounded simplex with nonempty interior in an infinite dimensional Hausdorff real linear topological space.
A representation of the solutions of the Darboux equation in odd-dimensional spaces
Authors: Rhee H.
Published: June 2010

It is shown that determining a function from its averages over all spheres passing through the origin leads to an explicit representation of the even solutions of the Darboux equation in the exterior of the characteristic cones in terms of the hyperboloidal means of the boundary data on the cones.
Representations of certain compact semigroups by 𝐻𝐿-semigroups
Authors: Carruth J. H. , Clark C. E.
Published: June 2010

An HL-semigroup is defined to be a topological semigroup with the property that the Schützenberger group of each

H

\mathcal {H}

-class is a Lie group. The following problem is considered: Does a compact semigroup admit enough homomorphisms into HL-semigroups to separate points of S; or equivalently, is S isomorphic to a strict projective limit of HL-semigroups? An affirmative answer is given in the case that S is an irreducible semigroup. If S is irreducible and separable, it is shown that S admits enough homomorphisms into finite dimensional HL-semigroups to separate points of S.
Spaces of countable and point-countable type
Authors: Vaughan J. E.
Published: June 2010

These spaces were introduced by M. Henriksen and J. R. Isbell, and A. V. Arhangel’skiĭ, who proved results about the placement of such spaces in their compactifications. In the present paper, these results are consolidated using new terminology. In addition, further results concerning the heredity of these spaces are obtained. Generalizations of these spaces are introduced, and an analogous treatment is given for them. Finally, some examples are given of which one gives a solution to a problem raised by Arhangel’skiĭ by showing that the perfect image of a first countable space need not be of point-countable type.
Existence and convergence of probability measures in Banach spaces.
Authors: de Acosta Alejandro D.
Published: June 2010

Theorems of the Bochner-Sazonov type are proved for Banach spaces with a basis. These theorems give sufficient conditions of a topological nature under which a positive definite function is the characteristic functional of a probability measure. The conditions are, in a certain natural sense, best possible. Central limit theorems of the Lindeberg type for triangular systems of random variables taking values in a Banach space with a basis are obtained. Applications to

l
p

{l_p}

and

C
[
0
,
1
]

C[0,1]

are given.
Some remarks on self-dual locally compact Abelian groups
Authors: Corwin Lawrence
Published: June 2010

The main results of this paper are the construction of some new self-dual locally compact Abelian groups and the proof of a structure theorem for a certain class of such groups. The construction is based on an investigation of when the extension of a compact Abelian group by its dual yields a self-dual group. It turns out that such extensions can be described algebraically ; the structure theorem follows from an analysis of the algebraic description.
Two point boundary problems for second order matrix differential systems
Authors: Etgen Garret J.
Published: June 2010

This paper is concerned with second order matrix differential systems involving a parameter together with boundary conditions specified at two points. The object of the paper is to establish sufficient conditions for the existence of eigenvalues for the system. Although such problems have been considered using the results of and techniques from the calculus of variations, the methods and results here are entirely in the context of ordinary differential equations. Use is made of the matrix generalization of the polar coordinate transformation introduced by J. H. Barrett and the unitary transformation suggested by F. V. Atkinson and V. A. Jakubovič. The sufficient conditions for the existence of eigenvalues obtained here represent certain extensions of W. M. Whyburn’s work concerning linear and nonlinear boundary problems for second order differential systems.
Jacobi’s bound for the order of systems of first order differential equations
Authors: Lando Barbara A.
Published: June 2010

Let

A
1

,
…
,

A
n

{A_1}, \ldots ,{A_n}

be a system of differential polynomials in the differential indeterminates

y

(
1
)

,
…
,

y

(
n
)

{y^{(1)}}, \ldots ,{y^{(n)}}

, and let

M

\mathcal {M}

be an irreducible component of the differential variety

M

(

A
1

,
…
,

A
n

)

\mathcal {M}({A_1}, \ldots ,{A_n})

. If

dim
⁡

M

=
0

\dim \mathcal {M} = 0

, there arises the question of securing an upper bound for the order of

M

\mathcal {M}

in terms of the orders

r

i
j

{r_{ij}}

of the polynomials

A
i

{A_i}

in

y

(
j
)

{y^{(j)}}

. It has been conjectured that the Jacobi number
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J equals upper J left-parenthesis r Subscript i j Baseline right-parenthesis equals max left-brace sigma-summation Underscript i equals 1 Overscript n Endscripts r Subscript i j Sub Subscript i Subscript Baseline colon j 1 comma ellipsis comma j Subscript n Baseline is a permutation of 1 comma ellipsis comma n right-brace">

J
=
J
(

r

i
j

)
=
max
{

∑

i
=
1

n

r

i

j
i

:

j
1

,
…
,

j
n

 is a permutation of 1,

…
,
n
}

J = J({r_{ij}}) = \max \{\sum \limits _{i = 1}^n {{r_{i{j_i}}}} :{j_1}, \ldots ,{j_n}{\text { is a permutation of 1,}} \ldots ,n \}

\]
provides such a bound. In this paper

J
J

is obtained as a bound for systems consisting of first order polynomials. Differential kernels are employed in securing the bound, with the theory of kernels obtained in a manner analogous to that of difference kernels as given by R. M. Cohn.
Mesures associées aux fonctionnelles additives de Markov. I
Authors: Revuz D.
Published: June 2010

With each additive functional of Markov processes we associate a measure and characterize, under duality hypotheses, those which correspond to

σ
\sigma

-finite measures. This enables us to weaken the hypotheses of Meyer’s theorem on representation of potentials of measures as potentials of additive functional. We characterize also the measures which are associated with continuous additive functionals. This leads us to show that for each finite continuous additive functional of the process there exists a finite continuous additive functional of the dual process such that the corresponding time-changed processes are in duality. Similar results are also stated for subprocesses which generalize results by Hunt and Blumenthal and Getoor.
The curvature of level curves
Authors: Shaffer Dorothy Browne
Published: June 2010

Sharp bounds are derived for the curvature of level curves of analytic functions in the complex plane whose logarithmic derivative has the representation

c

/

(
w
−
g
(
w
)
)

c/(w - g(w))

, where

g
(
w
)

g(w)

is analytic for

|

w

|

>
a

|w| > a

and

|

g
(
w
)

|

≦
a
,
c

|g(w)| \leqq a,c

real. These results are applied in particular to lemniscates and sharpened for the level curves of lacunary polynomials. Extensions to the level curves of Green’s function and rational functions are indicated.
Concerning a bound problem in knot theory
Authors: Treybig L. B.
Published: June 2010

In a recent paper Treybig shows that if two knot functions

f
,
g

f,g

determine equivalent knots, then

f
,
g

f,g

are the ends of a simple sequence

x
x

of knot functions. In an effort to bound the length of

x
x

in terms of

f
f

and

g
g

(1) a bound is found for the moves necessary in moving one polyhedral disk onto another in the interior of a tetrahedron and (2) it is shown that two polygonal knots

K
,
L

K,L

in regular position can “essentially” be embedded as part of the

1
1

-skeleton of a triangulation

T
T

of a tetrahedron, where (1) all 3 cells which are unions of elements of

T
T

can be shelled and (2) the number of elements in

T
T

is determined by

K
,
L

K,L

. A number of “counting” lemmas are proved.
Differentiable monotone maps on manifolds. II
Authors: Church P. T.
Published: June 2010

Let

M
n

{M^n}

and

N
n

{N^n}

be closed manifolds, and let

G
G

be any (nonzero) module. (1) If

f
:

M
3

→

N
3

f:{M^3} \to {N^3}

is

C
3

{C^3}

G
G

-acyclic, then there is a closed

C
3

{C^3}

3
3

-manifold

K
3

{K^3}

such that

N
3

#

K
3

{N^3}\# {K^3}

is diffeomorphic to

M
3

{M^3}

, and

f

−
1

(
y
)

{f^{ - 1}}(y)

is cellular for all but at most

r
r

points

y
∈

N
3

y \in {N^3}

, where

r
r

is the number of nontrivial

G
G

-cohomology

3
3

-spheres in the prime decomposition of

K
3

{K^3}

. (2) If

f
:

M
3

→

M
3

f:{M^3} \to {M^3}

or

f
:

S
3

→

M
3

f:{S^3} \to {M^3}

is

G
G

-acyclic, then

f
f

is cellular. In case

G
G

is

Z
Z

or

Z
p

{Z_p}

(

p
p

prime), results analogous to (1) and (2) in the topological category have been proved by Alden Wright. (3) If

f
:

M
n

→

M
n

f:{M^n} \to {M^n}

or

f
:

S
n

→

M
n

f:{S^n} \to {M^n}

is real analytic monotone onto, then

f
f

is a homeomorphism.
An approach to the polygonal knot problem using projections and isotopies
Authors: Treybig L. B.
Published: June 2010

The author extends earlier work of Tait, Gauss, Nagy, and Penney in defining and developing properties of (1) the boundary collection of a knot function, and (2) simple sequences of knot functions or boundary collections. The main results are (1) if two knot functions have isomorphic boundary collections then the knots they determine are equivalent, and (2) if two knot functions determine equivalent knots, then the given functions (their boundary collections) are the ends of a simple sequence of knot functions (boundary collections). Matrices are also defined for knot functions.
Cyclic vectors and irreducibility for principal series representations.
Authors: Wallach Nolan R.
Published: June 2010

Canonical sets of cyclic vectors for principal series representations of semisimple Lie groups having faithful representations are found. These cyclic vectors are used to obtain estimates for the number of irreducible subrepresentations of a principal series representations. The results are used to prove irreducibility for the full principal series of complex semisimple Lie groups and for

S
L
(
2
n
+
1
,
R
)
,
n
≧
1

SL(2n + 1,R),n \geqq 1

.
Quasiconformal mappings and Royden algebras in space
Authors: Lewis Lawrence G.
Published: June 2010

On every open connected set

G
G

in Euclidean

n
n

-space

R
n

{R^n}

and for every index

p
>
1

p > 1

, we define the Royden

p
p

-algebra

M
p

(
G
)

{M_p}(G)

. We use results by F. W. Gehring and W. P. Ziemer to prove that two such sets

G
G

and

G
′

G’

are quasiconformally equivalent if and only if their Royden

n
n

-algebras are isomorphic as Banach algebras. Moreover, every such algebra isomorphism is given by composition with a quasiconformal homeomorphism between

G
G

and

G
′

G’

. This generalizes a theorem by M. Nakai concerning Riemann surfaces. In case

p
≠
n

p \ne n

, the only homeomorphisms which induce an isomorphism of the

p
p

-algebras are the locally bi-Lipschitz mappings, and for

1
>
p
>
n

1 > p > n

, every such isomorphism arises this way. Under certain restrictions on the domains, these results extend to the Sobolev space

H
p
1

(
G
)

H_p^1(G)

and characterize those homeomorphisms which preserve the

H
p
1

H_p^1

classes.
Involutory automorphisms of operator algebras
Authors: Davies E. B.
Published: June 2010

We develop the mathematical machinery necessary in order to describe systematically the commutation and anticommutation relations of the field algebras of an algebraic quantum field theory of the fermion type. In this context it is possible to construct a skew tensor product of two von Neumann algebras and completely describe its type in terms of the types of the constituent algebras. Mathematically the paper is a study of involutory automorphisms of

W
∗

{W^\ast }

-algebras, of particular importance to quantum field theory being the outer involutory automorphisms of the type III factors. It is shown that each of the hyperfinite type III factors studied by Powers has at least two outer involutory automorphisms not conjugate under the group of all automorphisms of the factor.
Arcwise connectedness of semiaposyndetic plane continua
Authors: Hagopian Charles L.
Published: June 2010

In a recent paper, the author proved that if a compact plane continuum

M
M

contains a finite point set

F
F

such that, for each point

x
x

in

M
−
F
,
M

M - F,M

is semi-locally-connected and not aposyndetic at

x
x

, then

M
M

is arcwise connected. The primary purpose of this paper is to generalize that theorem. Semiaposyndesis, a generalization of semi-local-connectedness, is defined and arcwise connectedness is established for certain semiaposyndetic plane continua.
Multipliers of 𝐿^{𝑝}_{𝐸}. I
Authors: Oberlin Daniel M.
Published: June 2010

Let X be an abelian group, the character group of a compact group G. For a subset E of X let

L
E
p

L_E^p

be the subspace of E-spectral functions in

L
p

(
G
)

{L^p}(G)

. We show that if X is infinite and

1
⩽
p
>
2

1 \leqslant p > 2

, then E can be chosen so that not every multiplier of

L
E
p

^

\widehat {L_E^p}

extends to a multiplier of

L
p

^

(
G
)

\widehat {{L^p}}(G)

.
Linking forms and maps of odd prime order
Authors: Alexander J. P. , Hamrick G. C. , Vick J. W.
Published: June 2010

A differentiable orientation preserving map of odd prime period on a closed oriented differentiable manifold gives rise to two invariants taking values in a Witt group of bilinear forms. One is globally defined in terms of the rational cohomology of the manifold and the other is locally defined in terms of the fixed point set and its normal bundle. We show that these two invariants are, in fact, equal and apply this result to relate the structure of the manifold to that of the fixed point set and the quotient space.
Presentations of 3-manifolds arising from vector fields
Authors: Percell Peter
Published: June 2010

A method is given for constructing a smooth, closed, orientable 3-manifold from the information contained in a combinatorial object called an abstract intersection sequence. An abstract intersection sequence of length n is just a cyclic ordering of the set

{
±
1
,
…
,
±
n
}

\{ \pm 1, \ldots , \pm n\}

plus a map

ν
:
{
1
,
…
,
n
}
→
{
±
1
}

\nu :\{ 1, \ldots ,n\} \to \{ \pm 1\}

. It is shown that up to diffeomorphism every closed, connected, orientable 3-manifold can be constructed by the method. This is proved by showing that compact, connected, orientable 3-manifolds with boundary the 2-sphere admit vector fields of a certain type. The intersection sequences arise as descriptions of the vector fields.
Some consequences of the algebraic nature of 𝑝(𝑒^{𝑖𝜃})
Authors: Quine J. R.
Published: June 2010

For polynomial p of degree n, the curve

p
(

e

i
θ

)

p({e^{i\theta }})

is a closed curve in the complex plane. We show that the image of this curve is a subset of an algebraic curve of degree 2n. Using Bézout’s theorem and taking into account imaginary intersections at infinity, we show that if p and q are polynomials of degree m and n respectively, then the curves

p
(

e

i
θ

)

p({e^{i\theta }})

and

q
(

e

i
θ

)

q({e^{i\theta }})

intersect at most 2mn times. Finally, let

U
k

{U_k}

be the set of points w, not on

p
(

e

i
θ

)

p({e^{i\theta }})

, such that

p
(
z
)
−
w

p(z) - w

has exactly k roots in

|

z

|

>
1

|z| > 1

. We prove that if L is a line then

L
∩

U
k

L \cap {U_k}

has at most

n
−
k
+
1

n - k + 1

components in L and in particular

U
n

{U_n}

is convex.
Nonimmersion of lens spaces with 2-torsion
Authors: Berrick A. J.
Published: June 2010

From a study of the equivariant unitary K-theory of the Stiefel manifold

V

k
+
1
,
2

(

C

)

{V_{k + 1,2}}({\mathbf {C}})

, it is shown that the lens space

L
k

(
n
)

{L^k}(n)

, with n a multiple of

2

2
k
−
1
−
α
(
k
−
1
)

{2^{2k - 1 - \alpha (k - 1)}}

, does not immerse in Euclidean space of dimension

4
k
−
2
α
(
k
)
−
2

4k - 2\alpha (k) - 2

.
Continuously perfectly normal spaces and some generalizations
Authors: Gruenhage Gary
Published: June 2010

In this work we continue the study of continuously perfectly normal, continuously normal, and continuously completely regular spaces which was begun by Phillip Zenor. Among other results, we prove that separable continuously completely regular spaces are metrizable, and provide an example of a nonmetrizable continuously perfectly normal space.
The regularity of the locally integrable and continuous solutions of nonlinear functional equations
Authors: Światak Halina
Published: June 2010

The purpose of this paper is to show a general method which allows one to find all the continuous (and sometimes also all the locally integrable) solutions of functional equations by considering solutions of class

C
m

{C^m}

. One can do it if one is assured that all the continuous (or all the locally integrable) solutions of a given equation are functions of class

C
m

{C^m}

or

C
∞

{C^\infty }

. Such a property is characteristic for the solutions

f
:

R
n

→
R

f:{R^n} \to R

of the equations
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row with Label left-parenthesis asterisk right-parenthesis EndLabel sigma-summation Underscript i equals 1 Overscript k Endscripts a Subscript i Baseline left-parenthesis x comma t right-parenthesis f left-parenthesis phi Subscript i Baseline left-parenthesis x comma t right-parenthesis right-parenthesis equals upper F left-parenthesis x comma f left-parenthesis lamda 1 left-parenthesis x right-parenthesis right-parenthesis comma ellipsis comma f left-parenthesis lamda Subscript s Baseline left-parenthesis x right-parenthesis right-parenthesis right-parenthesis plus b left-parenthesis x comma t right-parenthesis comma EndLayout">

(

∗

)

∑

i
=
1

k

a
i

(
x
,
t
)
f
(

ϕ
i

(
x
,
t
)
)
=
F
(
x
,
f
(

λ
1

(
x
)
)
,
…
,
f
(

λ
s

(
x
)
)
)
+
b
(
x
,
t
)
,

\begin{equation}\tag {$\ast $} \sum \limits _{i = 1}^k {{a_i}(x,t)f({\phi _i}(x,t)) = F(x,f({\lambda _1}(x)), \ldots ,f({\lambda _s}(x))) + b(x,t),} \end{equation}

where

x
∈

R
n

,
t
∈

R
r

,
n
⩾
1
,
r
⩾
1

x \in {R^n},t \in {R^r},n \geqslant 1,r \geqslant 1

and where the functions
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="phi Subscript i Baseline colon upper R Superscript n plus r Baseline right-arrow upper R Superscript n Baseline comma lamda Subscript j Baseline colon upper R Superscript n Baseline right-arrow upper R Superscript n Baseline comma a Subscript i Baseline colon upper R Superscript n plus r Baseline right-arrow upper R comma b colon upper R Superscript n plus r Baseline right-arrow upper R comma upper F colon upper R Superscript n plus s Baseline right-arrow upper R">

ϕ
i

:

R

n
+
r

→

R
n

,

λ
j

:

R
n

→

R
n

,

a
i

:

R

n
+
r

→
R
,
b
:

R

n
+
r

→
R
,
F
:

R

n
+
s

→
R

{\phi _i}:{R^{n + r}} \to {R^n},{\lambda _j}:{R^n} \to {R^n},{a_i}:{R^{n + r}} \to R,b:{R^{n + r}} \to R,F:{R^{n + s}} \to R

satisfy some regularity assumptions and the assumptions which guarantee that an equation obtained by differentiating

(
∗
)

(\ast )

and fixing t is of constant strength, hypoelliptic at a point

x
0

{x_0}

. A general theorem, concerning the regularity of the continuous and locally integrable solutions f of

(
∗
)

(\ast )

, is formulated and proved by the reduction to the corresponding problem for the distributional solutions of linear partial differential equations.
Duality theory for locally compact groups with precompact conjugacy classes. II. The dual space
Authors: Sund Terje
Published: June 2010

The present paper is concerned with the dual space Ĝ consisting of all unitary equivalence classes of continuous irreducible unitary representations of separable

[
F
C

]
−

{[FC]^ - }

groups (i.e. groups with precompact conjugacy classes). The main purpose of the paper is to extend certain results from the duality theory of abelian groups and [Z] groups to the larger class of

[
F
C

]
−

{[FC]^ - }

groups. In addition, we deal briefly with square-integrability for representations of

[
F
C

]
−

{[FC]^ - }

groups. Most of our results are proved for type I groups. Our key result is that Ĝ may be written as a disjoint union of abelian topological

T
4

{T_4}

groups, which are open in Ĝ.
Local Fatou theorem and area theorem for symmetric spaces of rank one
Authors: Korányi A. , Putz R. B.
Published: June 2010

The classical results for the unit disc mentioned in the title are extended to harmonic functions on symmetric spaces of rank one.
Modular forms for 𝐺₀(𝑁) and Dirichlet series
Authors: Razar Michael J.
Published: June 2010

A criterion is given for a function to be a modular form for

Γ
0

(
N
)

{\Gamma _0}(N)

. It is similar to the criterion given by Weil in his 1967 Math. Ann. paper Über die Bestimmung Dirichletscher Reihen durch Funktional-gleichungen in that it involves checking that certain twists of the associated Dirichlet series satisfy functional equations. It differs in the number and type of such equations which need to be satisfied.
Continuous dependence of solutions of operator equations. I
Authors: Artstein Zvi
Published: June 2010

Continuous dependence of the solutions of the operator equation

x
=
T
x
+
z

x = Tx + z

in a topological vector space is the main subject of the paper. We find sufficient and necessary conditions for the continuous dependence on the data (T, z) or on a parameter. We do it for the space of all closed operators. Equivalent conditions for particular subfamilies are discussed. Among other families we deal with compact operators, compact perturbations of the identity, condensing operators and demicompact operators.