bluuebunny/crossref_metadata_2025_split huggingface dataset:
The 𝛼-boundification of 𝛼
Authors: García-Ferreira Salvador , Tamariz-Mascarúa Angel 
Published: June 2010

A space

X
X

is

>
α

> \alpha

-bounded if for all

A
⊆
X

A \subseteq X

with

|

A

|

>
α

|A| > \alpha

,

cl
X

A

{\operatorname {cl} _X}\;A

is compact. Let

B
(
α
)

B(\alpha )

be the smallest

>
α

> \alpha

-bounded subspace of

β
(
α
)

\beta (\alpha )

containing

α
\alpha

. It is shown that the following properties are equivalent: (a)

α
\alpha

is a singular cardinal; (b)

B
(
α
)

B(\alpha )

is not locally compact; (c)

B
(
α
)

B(\alpha )

is

α
\alpha

-pseudocompact; (d)

B
(
α
)

B(\alpha )

is initially

α
\alpha

-compact. Define

B
0

(
α
)
=
α

{B^0}(\alpha ) = \alpha

and

B
n

(
α
)
=
{

cl

β
(
α
)

A
:
A
⊆

B

n
−
1

(
α
)
,

|

A

|

>
α
}

{B^n}(\alpha ) = \{ {\operatorname {cl} _{\beta (\alpha )}}A:A \subseteq {B^{n - 1}}(\alpha ),|A| > \alpha \}

for

0
>
n
>
ω

0 > n > \omega

. We also prove that

B
2

(
α
)
≠

B
3

(
α
)

{B^2}(\alpha ) \ne {B^3}(\alpha )

when

ω
=
cf
⁡
(
α
)
>
α

\omega = \operatorname {cf} (\alpha ) > \alpha

. Finally, we calculate the cardinality of

B
(
α
)

B(\alpha )

and prove that, for every singular cardinal

α
,

|

B
(
α
)

|

=

|

B
(
α
)

|

α

=

|

N
(
α
)

|

cf
⁡
(
α
)

\alpha ,\;|B(\alpha )| = |B(\alpha ){|^\alpha } = |N(\alpha ){|^{\operatorname {cf} (\alpha )}}

where

N
(
α
)
=
{
p
∈
β
(
α
)
:

there is

A
∈
p

with

|

A

|

>
α
}

N(\alpha ) = \{ p \in \beta (\alpha ):\;{\text {there is}}\;A \in p\;{\text {with}}\;|A| > \alpha \}

.
On the dimensional properties of Stone-Čech remainder of 𝑃₀-spaces
Authors: Attia H. 
Published: June 2010

A space X is called a

P
0

{P_0}

-space if there exists a perfect mapping f from X onto a metric space Y such that

dim
⁡
f
=
sup
{

f

−
1

(
y
)
:
y
∈
Y
}
=
0

\dim f = \sup \{ {f^{ - 1}}(y):y \in Y\} = 0

. We prove that the

P
0

{P_0}

-space X is almost weakly infinite dimensional iff the remainder

β
X
∖
X

\beta X\backslash X

of the Stone-Čech compactification

β
X

\beta X

of X is A-weakly infinite dimensional. Furthermore we prove that

Δ
(
β
X
∖
X
=

ind

(
β
X
∖
X
)
=

Ind

(
β
X
∖
X
)
=
dim
⁡
(
β
X
∖
X
)

\Delta (\beta X\backslash X = {\text {ind}}(\beta X\backslash X) = {\text {Ind}}(\beta X\backslash X) = \dim (\beta X\backslash X)

for the

P
0

{P_0}

-space X.
On a functional equation connected with Rao’s quadratic entropy
Authors: Chung J. K. , Ebanks B. R. , Ng C. T. , Sahoo P. K. 
Published: June 2010

We determine the general solution of the functional equation fxy,
\[

f

(

x
+
y

2

)

+
f

(

x
−
y

2

)

=
2
f

(

x
2

)

+
2
f

(

y
2

)

+
λ
f
(
x
)
f
(
y
)
,

f\left ( {\frac {{x + y}} {2}} \right ) + f\left ( {\frac {{x - y}} {2}} \right ) = 2f\left ( {\frac {x} {2}} \right ) + 2f\left ( {\frac {y} {2}} \right ) + \lambda f(x)f(y),

\]
/: [-

f
:
[
−
1
,
1
]
→

R

f:[ - 1,1] \to {\mathbf {R}}

. This equation was used by Lau in order to characterize Rao’s quadratic entropies. The general solution is obtained here as a special case of a more general result for

f
f

mapping a neighborhood of

0
0

in linear topological space into a field.
Parabolicity of a class of higher order abstract differential equations
Authors: Xio Ti Jun , Liang Jin 
Published: June 2010

Let

E
E

be a complex Banach space,

c
i

∈

C

(
1
⩽
i
⩽
n
−
1
)

{c_i} \in \mathbb {C}\;(1 \leqslant i \leqslant n - 1)

, and

A
A

be a nonnegative operator in

E
E

. We discuss the parabolicity of the higher-order abstract differential equations

(

∗

)

u

(
n
)

(
t
)
+

∑

i
=
1

n
−
1

c
i

A

k
i

u

(
n
−
i
)

(
t
)
+
A
u
(
t
)
=
0

\begin{equation} \tag {$\ast $} {u^{(n)}}(t) + \sum \limits _{i = 1}^{n - 1} {{c_i}{A^{{k_i}}}{u^{(n - i)}}(t) + Au(t) = 0} \end{equation}

and some perturbation cases of (

∗
\ast

). A sufficient and necessary condition for (

∗
\ast

) to be parabolic is obtained, provided

k
1

>

k
2

−

k
1

>
⋯
>
1
−

k

n
−
1

>
0
,

c
i

≠
0

(
1
⩽
i
⩽
n
−
1
)

{k_1} > {k_2} - {k_1} > \cdots > 1 - {k_{n - 1}} > 0,\;{c_i} \ne 0\;(1 \leqslant i \leqslant n - 1)

. For

A
A

strictly nonnegative (Definition 1.3),

n
=
3
,

c
1

,

c
2

⩾
0

n = 3,{c_1},{c_2} \geqslant 0

, a sharp criterion is given.
Fixed point iteration processes for asymptotically nonexpansive mappings
Authors: Tan Kok-Keong , Xu Hong Kun 
Published: June 2010

Let X be a uniformly convex Banach space which satisfies Opial’s condition or has a Fréchet differentiable norm, C a bounded closed convex subset of X, and

T
:
C
→
C

T:C \to C

an asymptotically nonexpansive mapping. It is then shown that the modified Mann and Ishikawa iteration processes defined by

x

n
+
1

=

t
n

T
n

x
n

+
(
1
−

t
n

)

x
n

{x_{n + 1}} = {t_n}{T^n}{x_n} + (1 - {t_n}){x_n}

and

x

n
+
1

=

t
n

T
n

(

s
n

T
n

x
n

+
(
1
−

s
n

)

x
n

)
+
(
1
−

t
n

)

x
n

{x_{n + 1}} = {t_n}{T^n}({s_n}{T^n}{x_n} + (1 - {s_n}){x_n}) + (1 - {t_n}){x_n}

, respectively, converge weakly to a fixed point of T.
𝐺_{2𝑛} spaces
Authors: Koehler Donald O. 
Published: June 2010

A complex normed linear space

X
X

will be called a

G

2
n

{G_{2n}}

space if and only if there is a mapping

⟨

⋅
,
…
,
⋅

⟩

\left \langle { \cdot , \ldots , \cdot } \right \rangle

from

X

2
n

{X^{2n}}

into the complex numbers such that:

x
k

→

⟨

x
1

,
…
,

x

2
n

⟩

{x_k} \to \left \langle {{x_1}, \ldots ,{x_{2n}}} \right \rangle

is linear for

k
=
1
,
…
,
n
;

⟨

x
1

,
…
,

x

2
n

⟩

=

⟨

x

2
n

,
…
,

x
1

⟩

−

k = 1, \ldots ,n;\left \langle {{x_1}, \ldots ,{x_{2n}}} \right \rangle = {\left \langle {{x_{2n}}, \ldots ,{x_1}} \right \rangle ^ - }

; and

⟨

x
,
…
,
x

⟩

1

/

2
n

=

|

|

x

|

|

{\left \langle {x, \ldots ,x} \right \rangle ^{1/2n}} = ||x||

. The basic models are the

L

2
n

{L^{2n}}

spaces, but one also has that every inner product space is a

G

2
n

{G_{2n}}

space for every integer

n
n

. Hence

G

2
n

{G_{2n}}

spaces of a given cardinality need not be isometrically isomorphic. It is shown that a complex normed linear space is a

G

2
n

{G_{2n}}

space if and only if the norm satisfies a generalized parallelogram law. From the proof of this characterization it follows that a linear map

U
U

from

X
X

to

X
X

is an isometry if and only if

⟨

U
(

x
1

)
,
…
,
U
(

x

2
n

)

⟩

=

⟨

x
1

,
…
,

x

2
n

⟩

\left \langle {U({x_1}), \ldots ,U({x_{2n}})} \right \rangle = \left \langle {{x_1}, \ldots ,{x_{2n}}} \right \rangle

for all

x
1

,
…
,

x

2
n

{x_1}, \ldots ,{x_{2n}}

. This then provides a way to construct all of the isometries of a finite dimensional

G

2
n

{G_{2n}}

space. Of particular interest are the

CBS
⁡

G

2
n

\operatorname {CBS} {G_{2n}}

spaces in which

|

⟨

x
1

,
…
,

x

2
n

⟩

|

≦

|

|

x
1

|

|

⋯

|

|

x

2
n

|

|

|\left \langle {{x_1}, \ldots ,{x_{2n}}} \right \rangle | \leqq ||{x_1}|| \cdots ||{x_{2n}}||

. These spaces have many properties similar to inner product spaces. An operator

A
A

on a complete

CBS
⁡

G

2
n

\operatorname {CBS} {G_{2n}}

space is said to be symmetric if and only if

⟨

x
1

,
…
,
A
(

x
i

)
,
…
,

x

2
n

⟩

=

⟨

x
1

,
…
,
A
(

x
j

)
,
…
,

x

2
n

⟩

\left \langle {{x_1}, \ldots ,A({x_i}), \ldots ,{x_{2n}}} \right \rangle = \left \langle {{x_1}, \ldots ,A({x_j}), \ldots ,{x_{2n}}} \right \rangle

for all

i
i

and

j
j

. It is easy to show that these operators are scalar and that on

L

2
n

,
n
>
1

{L^{2n}},n > 1

, they characterize multiplication by a real

L
∞

{L^\infty }

function. The interest in nontrivial symmetric operators is that they exist if and only if the space can be decomposed into the direct sum of nontrivial

G

2
n

{G_{2n}}

spaces.
Absolute Tauberian constants for Cesàro means of a function
Authors: Sherif Soraya 
Published: June 2010

This paper is concerned with introducing two estimates of the forms

F
⩽
C

A
k

(
α
)
,
F
⩽
D

B
k

(
α
)
,
(
α
>
0
)

F \leqslant C{A_k}(\alpha ),F \leqslant D{B_k}(\alpha ),(\alpha > 0)

, where

F
=

∫
0
∞

|

d
{
f
(
α
x
)
−

σ
k

(
x
)
}

|

,

σ
k

(
x
)

F = \smallint _0^\infty {|d\{ f(\alpha x) - {\sigma _k}(x)\} |,{\sigma _k}(x)}

denote the Cesàro transform of order k of the function

f
(
x
)
=

∫
0
x

g
(
t
)

d
t
,
g
(
t
)

f(x) = \smallint _0^x {g(t)\;dt,g(t)}

is a function of bounded variation in every finite interval of

t
⩾
0
,

A
k

(
α
)
,

B
k

(
α
)

t \geqslant 0,{A_k}(\alpha ),{B_k}(\alpha )

are absolute Tauberian constants,

C
=

∫
0
∞

|

d
{
t
g
(
t
)
}

|

>
∞
,
D
=

∫
0
∞

|

d
{
ϕ
(
t
)
}

|

>
∞

C = \smallint _0^\infty {|d\{ tg(t)\} | > \infty ,D = \smallint _0^\infty {|d\{ \phi (t)\} | > \infty } }

and

ϕ
(
t
)
=

t

−
1

∫
0
t

u
g
(
u
)
d
u

\phi (t) = {t^{ - 1}}\smallint _0^t {ug(u)du}

. The constants

A
k

(
α
)
,

B
k

(
α
)

{A_k}(\alpha ),{B_k}(\alpha )

will be determined.
Conjugate points of vector-matrix differential equations
Authors: Lewis Roger T. 
Published: June 2010

The system of equations
\[

∑

k
=
0

n

(
−
1
)

n
−
k

(

P
k

(
x
)

y

(
n
−
k
)

(
x
)

)

(
n
−
k
)

=
0

(
0
⩽
x
>
∞
)

\sum \limits _{k = 0}^n {{{( - 1)}^{n - k}}{{\left ( {{P_k}(x){y^{(n - k)}}(x)} \right )}^{(n - k)}}} = 0\quad (0 \leqslant x > \infty )

\]
is considered where the coefficients are real, continuous, symmetric matrices, y is a vector, and

P
0

(
x
)

{P_0}(x)

is positive definite. It is shown that the well-known quadratic functional criterion for existence of conjugate points for this system can be further utilized to extend results of the associated scalar equation to the vector-matrix case, and in some cases the scalar results are also improved. The existence and nonexistence criteria for conjugate points of this system are stated in terms of integral conditions on the eigenvalues or norms of the coefficient matrices.
The asymptotic behavior of the first eigenvalue of differential operators degenerating on the boundary
Authors: Devinatz Allen , Friedman Avner 
Published: June 2010

When L is a second order ordinary or elliptic differential operator, the principal eigenvalue for the Dirichlet problem and the corresponding principal (positive) eigenfunction u are known to exist and u is unique up to normalization. If further L has the form

ε
Σ

a

i
j

∂
2

/

∂

x
i

∂

x
i

+
Σ

b
i

∂

/

∂

x
i

\varepsilon \Sigma {a_{ij}}{\partial ^2}/\partial {x_i}\partial {x_i} + \Sigma {b_i}\partial /\partial {x_i}

then results are known regarding the behavior of the principal eigenvalue

λ
=

λ
ε

\lambda = {\lambda _\varepsilon }

as

ε
↓
0

\varepsilon \downarrow 0

. These results are very sharp in case the vector

(

b
i

)

({b_i})

has a unique asymptotically stable point in the domain

ω
\omega

where the eigenvalue problem is considered. In this paper the case where L is an ordinary differential operator degenerating on the boundary of

ω
\omega

is considered. Existence and uniqueness of a principal eigenvalue and eigenfunction are proved and results on the behavior of

λ
ε

{\lambda _\varepsilon }

as

ε
↓
0

\varepsilon \downarrow 0

are established.
Convergence of random processes without discontinuities of the second kind and limit theorems for sums of independent random variables
Authors: Grinblat L. Š. 
Published: June 2010

Let

ξ
1

(
t
)
,
…
,

ξ
n

(
t
)
,
…

{\xi _1}(t), \ldots ,{\xi _n}(t), \ldots

and

ξ
(
t
)

\xi (t)

be random processes on the interval [0, 1], without discontinuities of the second kind. A. V. Skorohod has given necessary and sufficient conditions under which the distribution of

f
(

ξ
n

(
t
)
)

f({\xi _n}(t))

converges to the distribution of

f
(
ξ
(
t
)
)

f(\xi (t))

as

n
→
∞

n \to \infty

for any functional f continuous in the Skorohod metric. In the following we shall consider only stochastically right-continuous processes without discontinuities of the second kind, i.e., processes such that the space X of their sample functions is the space of all right-continuous functions

x
(
t
)
(
0
⩽
t
⩽
1
)

x(t)(0 \leqslant t \leqslant 1)

without discontinuities of the second kind. For a set

T
=
{

t
1

,
…

t
n

,
…
}
⊂
[
0
,
1
]

T = \{ {t_1}, \ldots {t_n}, \ldots \} \subset [0,1]

the metric

ρ
T

{\rho _T}

is defined on X as in 2.3. The metric

ρ
T

{\rho _T}

defines on the X the minimal topology in which all functional continuous in Skorohod’s metric and also the functional

x
(

t
1

−
0
)
,
x
(

t
1

)
,
…
,
x
(

t
n

−
0
)
,
x
(

t
n

)
,
…

x({t_1} - 0),x({t_1}), \ldots ,x({t_n} - 0),x({t_n}), \ldots

are continuous. We will give necessary and sufficient conditions under which the distribution of

f
(

ξ
n

(
t
)
)

f({\xi _n}(t))

converges to the distribution of

f
(
ξ
(
t
)
)

f(\xi (t))

as

n
→
∞

n \to \infty

for any completely continuous functional f, i.e. for any functional f which is continuous in any of the metrics

ρ
T

{\rho _T}

defined in 2.3.
The Bergman norm and the Szegő norm
Authors: Saitoh Saburou 
Published: June 2010

Let G denote an arbitrary bounded regular region in the plane and

H
2

(
G
)

{H_2}\left ( G \right )

the analytic Hardy class on G with index 2. We show that the generalized isoperimetric inequality

1
π

∬
G

|

φ

(
z
)

ψ

(
z
)

|

2

d
x

d
y

⩽

1

2
π

∫

∂
G

|
φ
(
z
)
|

2

|
d
z
|

1

2
π

∫

∂
G

|
ψ
(
z
)
|

2

|
d
z
|

(
z

=

x

+

i
y
)

\begin{multline} \frac {1}{\pi }\,\iint \limits _G {{{\left | {\varphi \left ( z \right )\psi \left ( z \right )} \right |}^{2\,}}dx\,dy\,\, \leqslant }\,\frac {1}{{2\pi }}\,\int _{\partial G}{{{\left | \varphi (z) \right |}^{2}}}\left | dz \right |\,\frac {1}{2\pi }\,\int _{\partial G}{{{\left | \psi (z) \right |}^{2}}\,\left | dz \right |}\,\,\,\,\,\,\,(z\,=\,x\,+\,iy) \end{multline}

holds for any

φ
\varphi

and

ψ

∈

H
2

(
G
)

\psi \, \in \,{H_2}\left ( G \right )

. We also determine necessary and sufficient conditions for equality.
Derivation alternator rings with idempotent
Authors: Hentzel Irvin R. , Smith Harry F. 
Published: June 2010

A nonassociative ring is called a derivation alternator ring if it satisfies the identities

(
y
z
,

x
,

x
)

=

y
(
z
,

x
,

x
)

+

(
y
,

x
,

x
)
z
,

(
x
,

x
,

y
z
)

=

y
(
x
,

x
,

z
)

+

(
x
,

x
,

y
)
z

(yz,\,x,\,x)\, = \,y(z,\,x,\,x)\, + \,(y,\,x,\,x)z,\,(x,\,x,\,yz)\, = \,y(x,\,x,\,z)\, + \,(x,\,x,\,y)z

and

(
x
,

x
,

x
)

=
0

(x,\,x,\,x)\, = 0

. Let R be a prime derivation alternator ring with idempotent

e
≠
1

e \ne 1

and characteristic

≠
2

\ne 2

. If R is without nonzero nil ideals of index 2, then R is alternative.
On the Wall finiteness obstruction for the total space of certain fibrations
Authors: Munkholm Hans J. , Pedersen Erik Kjaer 
Published: June 2010

The problem of computing the Wall finiteness obstruction for the total space of a fibration

p
:

E

→

B

p:\,E\, \to \,B

in terms of that for the base and homological data of the fiber has been considered by D. R. Anderson and by E. K. Pedersen and L. R. Taylor. We generalize their results and show how the problem is related to the algebraically defined transfer map

φ

∗

:

K
0

(

Z

π
1

(
B
)
)

→

K
0

(

Z

π
1

(
E
)
)

{\varphi ^{\ast }}:\,{K_0}({\textbf {Z}}{\pi _1}(B))\, \to \,{K_0}({\textbf {Z}}{\pi _1}(E))

,

φ

=

p

∗

:

π
1

(
E
)

→

π
1

(
B
)

\varphi \, = \,{p_{\ast }}:\,{\pi _1}(E)\, \to \,{\pi _1}(B)

, whenever the latter is defined.
Arithmetic of elliptic curves upon quadratic extension
Authors: Kramer Kenneth 
Published: June 2010

This paper is a study of variations in the rank of the Mordell-Weil group of an elliptic curve

E
E

defined over a number field

F
F

as one passes to quadratic extensions

K
K

of

F
F

. Let

S
(
K
)

S(K)

be the Selmer group for multiplication by

2
2

on

E
(
K
)

E(K)

. In analogy with genus theory, we describe

S
(
K
)

S(K)

in terms of various objects defined over

F
F

and the local norm indices

i
υ

=

dim

F

2

E
(

F
υ

)

/

Norm
{
E
(

K
w

)
}

{i_\upsilon } = {\dim _{{{\mathbf {F}}_2}}}E({F_\upsilon })/\text {Norm} \{ E({K_w})\}

for each completion

F
υ

{F_\upsilon }

of

F
F

. In particular we show that

dim
⁡
S
(
K
)
+
dim
⁡
E

(
K

)
2

\dim S(K) + \dim E{(K)_2}

has the same parity as

Σ

i
υ

\Sigma {i_\upsilon }

. We compute

i
υ

{i_\upsilon }

when

E
E

has good or multiplicative reduction modulo

υ
\upsilon

. Assuming that the

2
2

-primary component of the Tate-Shafarevitch group

Ш

(
K
)

\Sha (K)

is finite, as conjectured, we obtain the parity of rank

E
(
K
)

E(K)

. For semistable elliptic curves defined over

Q

{\mathbf {Q}}

and parametrized by modular functions our parity results agree with those predicted analytically by the conjectures of Birch and Swinnerton-Dyer.
A tangential convergence for bounded harmonic functions on a rank one symmetric space
Authors: Cygan Jacek 
Published: June 2010

Let

u
u

be a bounded harmonic function on a noncompact rank one symmetric space

M
=
G

/

K
≈

N
−

A
,

N
−

A
K

M = G/K \approx {N^ - }A,{N^ - }AK

being a fixed Iwasawa decomposition of

G
G

. We prove that if for an

a
0

∈
A

{a_0} \in A

there exists a limit

u
(
n

a
0

)
≡

c
0

u(n{a_0}) \equiv {c_0}

, as

n
∈

N
−

n \in {N^ - }

goes to infinity, then for any

a
∈
A

a \in A

,

u
(
n
a
)
=

c
0

u(na) = {c_0}

. For

M
=
S
U
(
n
,
1
)

/

S
(
U
(
n
)
×
U
(
1
)
)
=

B
n

M = SU(n,1)/S(U(n) \times U(1)) = {B^n}

, the unit ball in

C

n

{{\mathbf {C}}^n}

with the Bergman metric, this is a result of Hulanicki and Ricci, and in this case it reads (via the Cayley transformation) as a theorem on convergence of a bounded harmonic function to a boundary value at a fixed boundary point, along appropriate, tangent to

∂

B
n

\partial {B^n}

, surfaces.
Markov processes with Lipschitz semigroups
Authors: Bass Richard 
Published: June 2010

For

f
f

a function on a metric space, let
\[

Lip
⁡
f
=

sup

x
≠
y

|

f
(
x
)
−
f
(
y
)

|

/

d
(
x
,

y
)
,

\operatorname {Lip} f = \sup \limits _{x \ne y} |f(x) - f(y)|/d(x,\,y),

\]
and say that a semigroup

P
t

{P_t}

is Lipschitz if

Lip
⁡
(

P
t

f
)
⩽

e

K
t

Lip
⁡
f

\operatorname {Lip} ({P_t}f) \leqslant {e^{Kt}}\operatorname {Lip} f

for all

f
f

,

t
t

, where

K
K

is a constant. If one has two Lipschitz semigroups, then, with some additional assumptions, the sum of their infinitesimal generators will also generate a Lipschitz semigroup. Furthermore a sequence of uniformly Lipschitz semigroups has a subsequence which converges in the strong operator topology. Examples of Markov processes with Lipschitz semigroups include all diffusions on the real line which are on natural scale whose speed measures satisfy mild conditions, as well as some jump processes. One thus gets Markov processes whose generators are certain integro-differential operators. One can also interpret the results as giving some smoothness conditions for the solutions of certain parabolic partial differential equations.
A Poisson summation formula for integrals over quadratic surfaces
Authors: Strichartz Robert S. 
Published: June 2010

Let

S
(
t
)

S(t)

denote Lebesgue measure on the sphere of radius

t
>
0

t > 0

in

R

n

{{\mathbf {R}}^n}

, and
\[

S
k

(
t
)
=

(

∂

∂
t

1
t

)

k

S
(
t
)
.

{S_k}(t) = {\left ( {\frac {\partial } {{\partial t}}\quad \frac {1} {t}} \right )^k}S(t).

\]
Let

P

∑
k

=

S
k

(
0
)
+
2

∑

m
=
1

∞

S
k

(
m
)

P{\sum _k} = {S_k}(0) + 2\sum _{m = 1}^\infty {S_k}(m)

. Theorem. If

n
n

is odd and

j
j

and

k
k

are nonnegative integers with

j
+
k
=
(
n
−
1
)

/

2

j + k = (n - 1) / 2

, then the Fourier transform of

P

∑
j

P{\sum _j}

is

(
2
π

)

j
−
k

P

∑
k

{(2\pi )^{j - k}}P{\sum _k}

. There is an analogous, although slightly different, identity involving integrals over hyperboloids in odd dimensions. These results were inspired by recent work of M. Vergne.
Algebras generated by a subnormal operator
Authors: Olin Robert F. , Thomson James E. 
Published: June 2010

We use the notion of generalized Toeplitz operators to obtain some basic results concerning the

C

∗

{C^{\ast }}

-algebra generated by a subnormal operator. We apply these results to problems concerning the intersection of

C

∗

(
S
)

{C^{\ast }}(S)

with rationally closed algebras generated by

S
S

. In particular, we prove that

C

∗

(
S
)
∩

W

(
S
)
=
{
f
(
S
)
:
f
∈
R
(

σ

W

(
S
)

(
S
)
)
}

{C^{\ast }}(S) \cap \mathcal {W}(S) = \{ f(S):f \in R({\sigma _{\mathcal {W}(S)}}(S))\}

. The spectral inclusion property for generalized Toeplitz operators with symbols in

P
∞

(
μ
)
+
C
(
σ
(
N
)
)

{P^\infty }(\mu ) + C(\sigma (N))

is also considered.
On weighted norm inequalities for the Hilbert transform of functions with moments zero
Authors: Adams Ernst 
Published: June 2010

Let

f
~

\tilde f

denote the Hilbert transform of

f
f

, i.e.
\[

f
~

(
x
)
=

p

.
v

.

∫

f
(
t
)

x
−
t

d
t

\tilde f(x) = {\rm {p}}{\rm {.v}}{\rm {.}}\int {\frac {{f(t)}}{{x - t}}dt}

\]
and let

1
>
p
>
∞

1 > p > \infty

. A weight function

w
w

is shown to satisfy
\[

∫

|

f
~

(
x
)

|

p

w
(
x
)
d
x
≤
C

∫

|

f
(
x
)

|

p

w
(
x
)
d
x

\int {|\tilde f(x)} {|^p}w(x)dx \le C{\int {|f(x)|} ^p}w(x)dx

\]
for all

f
f

with the first

N
N

moments zero, if and only if it is of the form

w
(
x
)
=

|

q
(
x
)

|

p

U
(
x
)

w(x) = |q(x){|^p}U(x)

, where

q
q

is a polynomial of degree at most

N
N

and

U
∈

A
p

U \in {A_p}

.
Almost convergent and weakly almost periodic functions on a semigroup
Authors: Dzinotyiweyi Heneri A. M. 
Published: June 2010

Let

S
S

be a topological semigroup,

US

(
S
)

{\text {US}}(S)

the set of all bounded uniformly continuous functions on

S
,

WAP(

S
)

S,{\text {WAP(}}S)

the set of all (bounded) weakly almost periodic functions on

S
,

E
0

(
S
)
:=
{
f
∈

UC(

S
)
:
m
(

|

f

|

)
=
0

S,{E_0}(S): = \{ f \in {\text {UC(}}S):m(|f|) = 0

for each left and right invariant mean

m
m

on

UC(

S
)
}

{\text {UC(}}S)\}

and

W
0

(
S
)
:=
{
f
∈

WAP

(
S
)
:

m
(

|

f

|

)
=
0

{W_0}(S): = \{ f \in {\text {WAP}}(S):\:m(|f|) = 0

for each left and right invariant mean

m
m

on

WAP(

S
)
}

{\text {WAP(}}S)\}

. Among other results, for a large class of noncompact locally compact topological semigroups

S
S

, we show that the quotient space

E
0

(
S
)

/

W
0

(
S
)

{E_0}(S)/{W_0}(S)

contains a linear isometric copy of

l
∞

{l^\infty }

and so is nonseparable.
An extension of the Fuglede commutativity theorem modulo the Hilbert-Schmidt class to operators of the form ∑𝑀_{𝑛}𝑋𝑁_{𝑛}
Authors: Weiss Gary 
Published: June 2010

We study the operators

Δ
(
X
)
=

∑
1
n

M
n

X

N
n

\Delta (X) = \sum \nolimits _1^n {{M_n}X{N_n}}

and

Δ

∗

(
X
)
=

∑
1
n

M
n

∗

X

N
n

∗

{\Delta ^{\ast }}(X) = \sum \nolimits _1^n {M_n^{\ast }XN_n^{\ast }}

which map the algebra of all bounded linear operators on a separable Hubert space to itself, where

⟨

M
n

⟩
1
m

\langle {M_n}\rangle _1^m

and

⟨

N
n

⟩
1
m

\langle {N_n}\rangle _1^m

are separately commuting sequences of normal operators. We prove that (1) when

m
⩽
2

m \leqslant 2

, the Hilbert-Schmidt norms of

Δ
(
X
)

\Delta (X)

and

Δ

∗

(
X
)

{\Delta ^{\ast }}(X)

are equal (finite or infinite); (2) for

m
⩾
3

m \geqslant 3

, if

Δ
(
X
)

\Delta (X)

and

Δ

∗

(
X
)

{\Delta ^{\ast }}(X)

are Hilbert-Schmidt operators, then their Hilbert-Schmidt norms are equal; (3) if

Δ
,

Δ

∗

\Delta ,{\Delta ^{\ast }}

have the property that for each

X
,
Δ
(
X
)
=
0

X,\Delta (X) = 0

implies

Δ

∗

(
X
)
=
0

{\Delta ^{\ast }}(X) = 0

, then for each

X
X

, if

Δ
(
X
)

\Delta (X)

is a Hilbert-Schmidt operator then

Δ

∗

2

(
X
)

{\Delta ^{\ast }}^2(X)

is also and the latter has the same Hilbert-Schmidt norm as

Δ
2

(
X
)

{\Delta ^2}(X)

. Note that Fuglede’s Theorem is immediate from

(
1
)

(1)

in the case

m
=
2
,

M
1

=

N
2

m = 2,{M_1} = {N_2}

and

N
1

=
I
=
−

M
2

{N_1} = I = - {M_2}

. The proofs employ the duality between the trace class and the class of all bounded linear operators and, unlike the early proofs of Fuglede’s Theorem, they are free of complex function theory.