An algebra $A$ is said to be directly finite if each left invertible element
in the (conditional) unitization of $A$ is right invertible. It has long been
known that the complex group algebra of a discrete group is directly finite. We
extend this result, using some Hilbert algebra techniques, and show that the
reduced group $C^\ast$-algebra of a unimodular group is directly finite.

An isomorphic (\ell_1)-predual space (X) is constructed such that neither (X)
is isomorphic to a subspace of (c_0), nor (C(\omega^\omega)) is isomorphic to a
subspace of (X). It follows that (X) is not isomorphic to a complemented
subspace of a (C(K)) space.

Two-direction multiscaling functions $\boldphi$ and two-direction
multiwavelets $\boldpsi$ associated with $\boldphi$ are a more general and more
flexible setting than one-direction multiscaling functions and multiwavelets.
In this paper, we derive two methods for computing continuous moments of
orthogonal two-direction multiscaling functions $\boldphi$ and orthogonal
two-direction multiwavelets $\boldpsi$ associated with $\boldphi$. The first
method is by doubling and the second method is by separation. Two examples for
both methods are given.

Let $(K_{n})_{n=1}^{\infty}$ be the optimal constants satisfying the
multilinear (real or complex) Bohnenblust--Hille inequality. The exact values
of the constants $K_{n}$ are still waiting to be discovered since eighty years
ago; recently, it was proved that $(K_{n})_{n=1}^{\infty}$ has a subexponential
growth. In this note we go a step further and address the following question:
Is it true that \[ \lim_{n\rightarrow\infty}(K_{n}-K_{n-1}) =0?

Natural images are typically a composition of cartoon and texture structures.
A medical image might, for instance, show a mixture of gray matter and the
skull cap. One common task is to separate such an image into two single images,
one containing the cartoon part and the other containing the texture part.
Recently, a powerful class of algorithms using sparse approximation and
$\ell_1$ minimization has been introduced to resolve this problem, and numerous
inspiring empirical results have already been obtained.

Many emerging applications involve sparse signals, and their processing is a
subject of active research. We desire a large class of sensing matrices which
allow the user to discern important properties of the measured sparse signal.
Of particular interest are matrices with the restricted isometry property
(RIP). RIP matrices are known to enable efficient and stable reconstruction of
sufficiently sparse signals, but the deterministic construction of such
matrices has proven very difficult.

We establish a fixed point theorem for mappings of square matrices of all
sizes which respect the matrix size and direct sums of matrices. The
conclusions are stronger if such a mapping also respects matrix similarities,
i.e., is a noncommutative function. As a special case, we prove the
corresponding contractive mapping theorem which can be viewed as a
generalization of the classical Banach Fixed Point Theorem.

An absolute continuity approach to quasinormality which relates the operator
in question to the spectral measure of its modulus is developed. Algebraic
characterizations of some classes of operators that emerged in this context are
invented. Various examples and counterexamples illustrating the concepts of the
paper are constructed by means of weighted shifts on directed trees.
Generalizations of these results that cover the case of q-quasinormal operators
are established.

Differentiations of operator algebras over non-archimedean spherically
complete fields are investigated. Theorems about a differentiation being
internal are demonstrated.

This paper is concerned with an important matrix condition in compressed
sensing known as the restricted isometry property (RIP). We demonstrate that
testing whether a matrix satisfies RIP is hard for NP under randomized
polynomial-time reductions. Our reduction is from the NP-complete clique
decision problem, and it uses ideas from matroid theory. As a consequence of
our result, it is impossible to efficiently test for RIP provided NP
\not\subseteq BPP, an assumption which is slightly stronger than P \neq NP.

In this article we develop a theory for frames in tensor product of Hilbert
spaces. We show that like bases if Y_1, Y_2, \cdot \cdot \cdot, Y_n are frames
for H_1,H_2, \cdot \cdot \cdot, H_n, respectively, then
Y_1\otimesY_2\otimes...\otimesY_n is a frame for H_\otimes1H_2\otimes \cdot
\cdot \cdot \otimesH_n. Moreover we consider the canonical dual frame in tensor
product space. We further obtain a relation between the dual frames in Hilbert
spaces, and their tensor product.

We prove that $b$ is in $Lip_{\bz}(\bz)$ if and only if the commutator
$[b,L^{-\alpha/2}]$ of the multiplication operator by $b$ and the general
fractional integral operator $L^{-\alpha/2}$ is bounded from the weighed Morrey
space $L^{p,k}(\omega)$ to $L^{q,kq/p}(\omega^{1-(1-\alpha/n)q},\omega)$, where
$0<\beta<1$, $0<\alpha+\beta<n, 1<p<{n}/({\alpha+\beta})$,
${1}/{q}={1}/{p}-{(\alpha+\beta)}/{n},$ $0\leq k<{p}/{q},$ $\omega^{{q}/{p}}\in
A_1$ and $ r_\omega> \frac{1-k}{p/q-k},$ and here $r_\omega$ denotes the
critical index of $\omega$ for the reverse H\"{o}lder co

We consider the linear Dirac operator with a (-1)-homogeneous locally
periodic potential that varies with respect to a small parameter. Using the
notation of G-convergence for positive self-adjoint operators in Hilbert spaces
we prove G-compactness in the strong resolvent sense for families of
projections of Dirac operators. We also prove convergence of the corresponding
point spectrum in the spectral gap.

In this paper, we define inductive limit cone topologies by the equivalence
relations which leads to the notion with the linear mappings. The definition
involves the quotient cones, direct sums, as well as the locally convex
topological vector spaces. In particular, the duality properties and
barreldness of the inductive limit cones have been discussed.

We present a new, elementary proof of Boyd's interpolation theorem. Our
approach naturally yields a vector-valued as well as a noncommutative version
of this result and even allows for the interpolation of certain operators on
$l^1$-valued noncommutative symmetric spaces. By duality we may interpolate
several well-known noncommutative maximal inequalities. In particular we obtain
a version of Doob's maximal inequality and the dual Doob inequality for
noncommutative symmetric spaces.

We introduce the higher order spreading models associated to a Banach space
$X$. Their definition is based on $\ff$-sequences $(x_s)_{s\in\ff}$ with $\ff$
a regular thin family and the plegma families. We show that the higher order
spreading models of a Banach space $X$ form an increasing transfinite hierarchy
$(\mathcal{SM}_\xi(X))_{\xi<\omega_1}$. Each $\mathcal{SM}_\xi (X)$ contains
all spreading models generated by $\ff$-sequences $(x_s)_{s\in\ff}$ with order
of $\ff$ equal to $\xi$. We also provide a study of the fundamental properties
of the hierarchy.

Let $p:\R\to(1,\infty)$ be a globally log-H\"older continuous variable
exponent and $w:\R\to[0,\infty]$ be a weight. We prove that the Cauchy singular
integral operator $S$ is bounded on the weighted variable Lebesgue space
$L^{p(\cdot)}(\R,w)=\{f:fw\in L^{p(\cdot)}(\R)\}$ if and only if the weight $w$
satisfies \[ \sup_{-\infty<a<b<\infty}
\frac{1}{b-a}\|w\chi_{(a,b)}\|_{p(\cdot)}\|w^{-1}\chi_{(a,b)}\|_{p'(\cdot)}<\infty
\quad (1/p(x)+1/p'(x)=1). \]

We obtain a new proof of Bobkov's lower bound on the first positive
eigenvalue of the (negative) Neumann Laplacian (or equivalently, the Cheeger
constant) on a bounded convex domain $K$ in Euclidean space. Our proof avoids
employing the localization method or any of its geometric extensions. Instead,
we deduce the lower bound by invoking a spectral transference principle for
log-concave measures, comparing the uniform measure on $K$ with an
appropriately scaled Gaussian measure which is conditioned on $K$.

Reflexive cones in Banach spaces are cones with weakly compact intersection
with the unit ball. In this paper we study the structure of this class of
cones. We investigate the relations between the notion of reflexive cones and
the properties of their bases. This allows us to prove a characterization of
reflexive cones in term of the absence of a subcone isomorphic to the positive
cone of \ell_{1}. Moreover, the properties of some specific classes of
reflexive cones are investigated.

For $d = 2, 3, \ldots$ and $p \in [1, \infty),$ we define a class of
representations $\rho$ of the Leavitt algebra $L_d$ on spaces of the form $L^p
(X, \mu),$ which we call the spatial representations. We prove that for fixed
$d$ and $p,$ the Banach algebra ${{\mathcal{O}}_{d}^{p}}$ obtained as the
closure of the image of $L_d$ under the representation $\rho$ is the same for
all spatial representations $\rho.$ When $p = 2,$ we recover the usual Cuntz
algebra ${\mathcal{O}}_{d}.$ We give a number of equivalent conditions for a
representation to be spatial.

The Gauss-Jordan elimination algorithm is extended to reduce a row-finite
$\omega\times\omega$ matrix to lower row-reduced form, founded on a strategy of
rightmost pivot elements. Such reduced matrix form preserves row equivalence,
unlike the dominant (upper) row-reduced form. This algorithm provides a
constructive alternative to an earlier existence and uniqueness result for
Quasi-Hermite forms based on the axiom of countable choice.

This paper is a continuation of our previous investigation on the truncated
matrix trigonometric moment problem in Ukrainian Math. J., 2011, 63, no.6,
786-797. In the present paper we obtain a Nevanlinna-type formula for this
moment problem in a general case. We only assume that we have more than one
moment, the moment problem is solvable and the problem has more than one
solution. The coefficients of the corresponding matrix linear fractional
transformation are explicitly expressed by the prescribed moments. Easy
conditions for the determinacy of the moment problem are given.

We study properties of representing and absolutely representing systems of
subspaces in Banach spaces. We also present sufficient conditions for the
system of subspaces to be a representing system of subspaces.

We characterize the metric spaces whose free space has the bounded
approximation property through a Lipschitz analogue of the local reflexivity
principle. We show that there exist compact metric spaces whose free spaces
fail the approximation property.

In this paper we give a combinatorial characterization of tight fusion frame
(TFF) sequences using Littlewood-Richardson skew tableaux. The equal rank case
has been solved recently by Casazza, Fickus, Mixon, Wang, and Zhou. Our
characterization does not have this limitation. We also develop some methods
for generating TFF sequences. The basic technique is a majorization principle
for TFF sequences combined with spatial and Naimark dualities. We use these
methods and our characterization to give necessary and sufficient conditions
which are satisfied by the first three highest ranks.

We prove results on weak convergence for the alternating split Bregman
algorithm in infinite dimensional Hilbert spaces. We also show convergence of
an approximate split Bregman algorithm, where errors are allowed at each step
of the computation. To be able to treat the infinite dimensional case, our
proofs focus mostly on the dual problem. We rely on Svaiter's theorem on weak
convergence of the Douglas-Rachford splitting algorithm and on the relation
between the alternating split Bregman and Douglas-Rachford splitting algorithms
discovered by Setzer.

The aim of this paper is two folded. Firstly, we study the validity of the
Pohozaev-type identity for the Schr\"{o}dinger operator $$A_\la:=-\D
-\frac{\la}{|x|^2}, \q \la\in \rr,$$ in the situation where the origin is
located on the boundary of a smooth domain $\Omega\subset \rr^N$, $N\geq 1$.
The problem we address is very much related to optimal Hardy-Poincar\'{e}
inequality with boundary singularities which has been investigated in the
recent past in various papers. In view of that, the proper functional framework
is described and explained.

Each Bernoulli convolution measure (\mu) with scaling factor 1/(2n) has at
least one associated orthonormal basis of exponential functions (ONB) for
L^2(\mu). In the particular case where the scaling constant for the Bernoulli
convolution measure is 1/4 and two specific ONBs are selected for L^2(\mu),
there is a unitary operator U defined by mapping one ONB to the other. This
paper focuses on the case in which one ONB (\Gamma) is the original
Jorgensen-Pedersen ONB for the Bernoulli convolution measure (\mu) and the
other ONB is is 5\Gamma.

We show that a best rank one approximation to a real symmetric tensor, which
in principle can be nonsymmetric, can be chosen symmetric.

Furthermore, a symmetric best rank one approximation to a symmetric tensor is
unique if the tensor does not lie on a certain real algebraic variety.

We introduce a new wavelet transform suitable for analyzing functions on
point clouds and graphs. Our construction is based on a generalization of the
average interpolating refinement scheme of Donoho. The most important
ingredient of the original scheme that needs to be altered is the choice of the
interpolant. Here, we define the interpolant as the minimizer of a smoothness
functional, namely a generalization of the Laplacian energy, subject to the
averaging constraints.

Using methods from the theory of commutative graded Banach algebras, we
obtain a generalization of the two dimensional Borsuk-Ulam theorem as follows:
Let $\phi:S^{2} \rightarrow S^{2}$ be a homeomorphism of order n and
$\lambda\neq 1$ be an nth root of the unity, then for every complex valued
continuous function $f$ on $S^{2}$ the function $\sum_{i=0}^{n-1}
\lambda^{i}f(\phi^{i}(x))$ must be vanished at some point of $S^{2}$. We also
discuss about some noncommutative versions of the Borsuk- Ulam theorem

We study the question: When are Lipschitz mappings dense in the Sobolev space
$W^{1,p}(M,H^n)$? Here $M$ denotes a compact Riemannian manifold with or
without boundary, while $H^n$ denotes the $n$th Heisenberg group equipped with
a sub-Riemannian metric. We show that Lipschitz maps are dense in
$W^{1,p}(M,H^n)$ for all $1\le p<\infty$ if $\dim M \le n$, but that Lipschitz
maps are not dense in $W^{1,p}(M,H^n)$ if $\dim M \ge n+1$ and $n\le p<n+1$.
The proofs rely on the construction of smooth horizontal embeddings of the
sphere $S^n$ into $H^n$.

We investigate in the paper general (not necessarily definite) canonical
systems of differential equation in the framework of extension theory of
symmetric linear relations. For this aim we first introduce the new notion of a
boundary relation $\G:\gH^2\to\HH$ for $A^*$, where $\gH$ is a Hilbert space,
$A$ is a symmetric linear relation in $\gH, \cH_0$ is a boundary Hilbert space
and $\cH_1$ is a subspace in $\cH_0$.

In this paper we study two separate problems on interpolation. We first give
a new proof of Stout's Theorem on necessary and sufficient conditions for a
sequence of points to be an interpolating sequence for the multiplier algebra
and for an associated Hilbert space. We next turn our attention to the question
of interpolation for reproducing kernel Hilbert spaces on the polydisc and
provide a collection of equivalent statements about when it is possible to
interpolation in the Schur-Agler class of the associated reproducing kernel
Hilbert space.

In this paper we consider two problems of frame theory. On the one hand,
given a fixed frame ${\mathcal F}$ we describe explicitly the spectral and
geometric structure of optimal frames ${\mathcal W}$ that are in duality with
${\mathcal F}$ and such that the Frobenius norms of their analysis operators is
bounded from below by a fixed constant, where optimality is measured with
respect to submajorization.

Let $S$ be a semi direct product $S=N\rtimes A$ where $N$ is a connected and
simply connected, non-abelian, nilpotent meta-abelian Lie group and $A$ is
isomorphic with $\R^k,$ $k>1.$ We consider a class of second order
left-invariant differential operators on $S$ of the form $\mathcal
L_\alpha=L^a+\Delta_\alpha,$ where $\alpha\in\R^k,$ and for each $a\in\R^k,$
$L^a$ is left-invariant second order differential operator on $N$ and
$\Delta_\alpha=\Delta-<\alpha,\nabla>,$ where $\Delta$ is the usual Laplacian
on $\R^k.$ Using some probabilistic techniques (e.g., skew-product formulas f

The well know conjecture of {\it Coburn} [{\it L.A. Coburn, {On the
Berezin-Toeplitz calculus}, Proc. Amer. Math. Soc. 129 (2001) 3331-3338.}]
proved by {\it Lo} [{\it M-L. Lo, {The Bargmann Transform and Windowed Fourier
Transform}, Integr. equ. oper. theory, 27 (2007), 397-412.}] and {\it Englis}
[{\it M. Engli$\check{s}$, Toeplitz Operators and Localization Operators,
Trans. Am.

To every bounded linear operator $A$ between Hilbert spaces $\mathcal{H}$ and
$\mathcal{K}$ three cardinals $\iota_r(A)$, $\iota_i(A)$ and $\iota_f(A)$ and a
binary number $\iota_b(A)$ are assigned in terms of which the descriptions of
the norm closures of the orbits $\{G A L^{-1}:\ L \in \mathcal{G}_1,\ G \in
\mathcal{G}_2\}$ are given for $\mathcal{G}_1$ and $\mathcal{G}_2$ (chosen
independently) being the trivial group, the unitary group or the group of all
invertible operators on $\mathcal{H}$ and $\mathcal{K}$, respectively.

For a collection of reproducing kernels k which includes those for the Hardy
space of the polydisk and ball and for the Bergman space, k is a complete Pick
kernel if and only if the multiplier algebra of the Hilbert space H^2(k)
associated to k has the Douglas property. Consequences for solving the operator
equation AX=Y are examined.

The main observation of this note is that the Lebesgue measure $\mu$ in the
Tur\'an-Nazarov inequality for exponential polynomials can be replaced with a
certain geometric invariant $\omega \ge \mu$, which can be effectively
estimated in terms of the metric entropy of a set, and may be nonzero for
discrete and even finite sets. While the frequencies (the imaginary parts of
the exponents) do not enter the original Tur\'an-Nazarov inequality, they
necessarily enter the definition of $\omega$.

We improve upon on a limit theorem for numerical index for large classes of
Banach spaces including vector valued $\ell_p$-spaces and $\ell_p$-sums of
Banach spaces where $1\leq p \leq \infty$. We first prove $ n_1(X) =
\displaystyle \lim_m n_1(X_m)$ for a modified numerical index $n_1(\, .\,)$.
Later, we establish if a norm on $X$ satisfies the local characterization
condition, then $n(X) = \displaystyle\lim_m n(X_m).$ We also present an example
of a Banach space where the local characterization condition is satisfied.

Some unfortunate errors from our paper math/0505591 are corrected.

The purpose of this paper is to establish the weighted norm inequalities of
one-sided oscillatory integral operators by the aid of interpolation of
operators with change of measures.

In this short note we prove the result stated in the title; that is, for
every $p>0$ there exists an infinite dimensional closed linear subspace of
$L_{p}[0,1]$ every nonzero element of which does not belong to
$\bigcup\limits_{q>p} L_{q}[0,1]$. This answers in the positive a question
raised in 2010 by R. M. Aron on the spaceability of the above sets (for both,
the Banach and quasi-Banach cases). We also complete some recent results from
\cite{BDFP} for subsets of sequence spaces.

We give the symmetric version of five lemmas which are essential for the
theory of tensor products (and norms). These are: the approximation, extension,
embedding, density and local technique lemmas. Some application of these tools
to the metric theory of symmetric tensor products and to the theory of
polynomials ideals are given.

A notion of band limited functions is considered in the case of the
hyperbolic plane in its Poincare upper half-plane $\mathbb{H}$ realization. The
concept of band-limitedness is based on the existence of the Helgason-Fourier
transform on $\mathbb{H}$. An iterative algorithm is presented, which allows to
reconstruct band-limited functions from some countable sets of their values. It
is shown that for sufficiently dense metric lattices a geometric rate of
convergence can be guaranteed as long as the sampling density is high enough
compared to the band-width of the sampled function.

Let $\M$ be a smooth connected non-compact manifold endowed with a smooth
measure $\mu$ and a smooth locally subelliptic diffusion operator $L$
satisfying $L1=0$, and which is symmetric with respect to $\mu$. We show that
if $L$ satisfies, with a non negative curvature parameter $\rho_1$, the
generalized curvature inequality in \eqref{CD} below, then the Riesz transform
is bounded in $L^p (\bM)$ for every $p>1$, that is \[\|
\sqrt{\Gamma((-L)^{-1/2}f)}\|_p \le C_p \| f \|_p, \quad f \in C^\infty_0(\bM),
\] where $\Gamma$ is the \textit{carr\'e du champ} associated to $L$.

For 2-variable weighted shifts W_{(\alpha,\beta)}(T_1, T_2) we study the
invariance of (joint) k- hyponormality under the action (h,\ell) ->
W_{(\alpha,\beta)}^{(h,\ell)}(T_1, T_2):=(T_1^k,T_2^{\ell}) (h,\ell >=1). We
show that for every k >= 1 there exists W_{(\alpha,\beta)}(T_1, T_2) such that
W_{(\alpha,\beta)}^{(h,\ell)}(T_1, T_2) is k-hyponormal (all h>=2,\ell>=1) but
W_{(\alpha,\beta)}(T_1, T_2) is not k-hyponormal. On the positive side, for a
class of 2-variable weighted shifts with tensor core we find a computable
necessary condition for invariance.

In this paper we give a new construction of parametric families of complex
Hadamard matrices of square orders, and connect them to equiangular tight
frames. The results presented here generalize some of the recent ideas of
Bodmann et al. and extend the list of known equiangular tight frames. In
particular, a (36,21) frame coming from a nontrivial cube root signature matrix
is obtained for the first time.

We give a necessary and sufficient condition for non-local functionals on
vector-valued Lebesgue spaces to be weakly sequentially lower semi-continuous.
Here a non-local functional shall have the form of a double integral of a
density which depends on the function values at two different points.

The characterisation we get is essentially that the density has to be convex
in one variable if we integrate over the other one with an arbitrary test
function in it.

We show that if K is Rosenthal compact which can be represented by functions
with countably many discontinuities then every Radon measure on K is countably
determined. We also present an alternative proof of the result stating that
every Radon measure on an arbitrary Rosenthal compactum is of countable type.
Our approach is based on some caliber-type properties of measures,
parameterized by separable metrizable spaces.

We show that every nonempty compact and convex space M of probability Radon
measures either contains a measure which has `small' local character in M or
else M contains a measure of `large' Maharam type. Such a dichotomy is related
to several results on Radon measures on compact spaces and to some properties
of Banach spaces of continuous functions.

In this paper we present a method to obtain Banach spaces of universal and
almost-universal disposition with respect to a given class $\mathfrak M$ of
normed spaces. The method produces, among other, the Gurari\u{\i} space
$\mathcal G$ (the only separable Banach space of almost-universal disposition
with respect to the class $\mathfrak F$ of finite dimensional spaces), or the
Kubis space $\mathcal K$ (under {\sf CH}, the only Banach space with the
density character the continuum which is of universal disposition with respect
to the class $\mathfrak S$ of separable spaces).

We prove that the set of all complex symmetric operators on a separable,
infinite-dimensional Hilbert space is not norm closed.

We study divergence properties of Fourier series on Cantor-type fractal
measures, also called mock Fourier series. We show that in some cases the
$L^1$-norm of the corresponding Dirichlet kernel grows exponentially fast, and
therefore the Fourier series are not even pointwise convergent. We apply these
results to the Lebesgue measure to show that a certain rearrangement of the
exponential functions, which we call scrambled Fourier series, have a
corresponding Dirichlet kernel whose $L^1$-norm grows exponentially fast, which
is much worse than the known logarithmic bound.

Let $(\Omega,{\cal F},P)$ be a probability space and $L^{0}({\cal F},R)$ the
algebra of equivalence classes of real-valued random variables on
$(\Omega,{\cal F},P)$. When $L^{0}({\cal F},R)$ is endowed with the topology of
convergence in probability, we prove an intermediate value theorem for a
continuous local function from $L^{0}({\cal F},R)$ to $L^{0}({\cal F},R)$.

Let $B_Y$ denote the unit ball of a normed linear space $Y$. A symmetric,
bounded, closed, convex set $A$ in a finite dimensional normed linear space $X$
is called a {\it sufficient enlargement} for $X$ if, for an arbitrary isometric
embedding of $X$ into a Banach space $Y$, there exists a linear projection
$P:Y\to X$ such that $P(B_Y)\subset A$. Each finite dimensional normed space
has a minimal-volume sufficient enlargement which is a parallelepiped, some
spaces have "exotic" minimal-volume sufficient enlargements.

In this survey, at first we review to many examples which have been made on
cone metric spaces to verify some properties of cones on real Banach spaces and
cone metrics and second, in continue like as examples that sandwich theorem
doesn't hold and we shall present an other example that comparison test doesn't
hold with an example for normal cones.

We study Toeplitz-type operators with respect to specific wavelets whose
Fourier transforms are related to Laguerre polynomials. This choice of wavelets
underlines the fact that these operators acting on wavelet subspaces share many
properties with the classical Toeplitz operators acting on the Bergman and
Bergman-type spaces. Restricting to symbols depending only on vertical variable
in the upper half-plane of the complex plane these operators are unitarily
equivalent to a multiplication operator with a certain function.

We present an example in ZFC of a locally compact, scattered Hausdorff
non-Gruenhage space $D$ having a $\G_delta$-diagonal. This answers a question
posed by Orihuela, Troyanski and the author in a study of strictly convex norms
on Banach spaces. In addition, we show that the Banach space of continuous
functions $C_0(D)$ admits a $C^\infty$-smooth bump function.

Compactly supported shearlets have been studied in both theory and
applications. In this paper, we construct symmetric compactly supported
shearlet systems based on pseudo splines of type II. Specially, using
B-splines, we construct shearlet frame having explicit analytical forms which
is important for applications. The shearlet systems based on B-splines also
provide optimally sparse approximation within cartoon-liked image.

Ivan Dobrakov has initiated a theory of non-additive set functions defined on
a ring of sets intended to be a non-additive generalization of the theory of
finite non-negative countably additive measures. These set functions are now
known as the Dobrakov submeasures. In this paper we extend Dobrakov's
considerations to vector-valued submeasures defined on a ring of sets. The
extension of such submeasures in the sense of Drewnowski is also given.

The notion of a firmly nonexpansive mapping is central in fixed point theory
because of attractive convergence properties for iterates and the
correspondence with maximal monotone operators due to Minty. In this paper, we
systematically analyze the relationship between properties of firmly
nonexpansive mappings and associated maximal monotone operators. Dual and
self-dual properties are also identified. The results are illustrated through
several examples.

We develop an information-theoretic perspective on some questions in convex
geometry, providing for instance a new equipartition property for log-concave
probability measures, some Gaussian comparison results for log-concave
measures, an entropic formulation of the hyperplane conjecture, and a new
reverse entropy power inequality for log-concave measures analogous to V.
Milman's reverse Brunn-Minkowski inequality.

We introduce and study the notion of null-orbit reflexivity, which is a
slight perturbation of the notion of orbit-reflexivity. Positive results for
orbit reflexivity and the recent notion of $\mathbb{C}$-orbit reflexivity both
extend to null-orbit reflexivity. Of the two known examples of operators that
are not orbit-reflexive, one is null-orbit reflexive and the other is not. The
class of null-orbit reflexive operators includes the classes of hyponormal,
algebraic, compact, strictly block-upper (lower) triangular operators, and
operators whose spectral radius is not 1.

As established by R T. Rockafellar, real valued convex-concave functions are
generically differentiable. It this paper we shall show that for a
convex-concave function defined on an open convex set $C \times D,$ there exist
dense subsets ${\cal N}$ of $C$ and ${\cal M}$ of $D$ such that the partial
derivative with respect to the first variable (resp. second variable) exists on
${\cal N} \times D$ (resp. $C \times {\cal M}$) and therefore the function is
differentiable on ${\cal N} \times {\cal M}$.

In this paper, we introduce a generalized quadratic functional equation $f(rx
+ sy) = rf(x) + sf(y) - rsf(x - y)$ where $r, s$ are nonzero real numbers with
$r + s = 1.$ We show that this functional equation is quadratic if $r, s$ are
rational numbers. We also investigate its stability problem on restricted
domains. These results are applied to study of an asymptotic behavior of these
generalized quadratic mappings.

Given commuting families of Hermitian matrices {A1, ..., Ak} and {B1, ....,
Bk}, conditions for the existence of a completely positive map L, such that
L(Aj) = Bj for j = 1, ...,k, are studied. Additional properties such as unital
or / and trace preserving on the map ? are also considered. Connections of the
study to dilation theory, matrix inequalities, unitary orbits, and quantum
information science are mentioned.

I present an inverse function theorem for differentiable maps between Frechet
spaces which contains the classical theorem of Nash and Moser as a particular
case. In contrast to the latter, the proof does not rely on the Newton
iteration procedure, but on Lebesgue's dominated convergence theorem and
Ekeland's variational principle. As a consequence, the assumptions are
substantially weakened: the map F to be inverted is not required to be C^2, or
even C^1, or even Frechet-differentiable.

For Banach left and right module actions, we will establish the relationships
between topological centers of module actions with some result in the weak
amenability of Banach algebras.

In this note, we study the Arens regularity of projective tensor product
$A\hat{\otimes}B$ whenever $A$ and $B$ are Arens regular. We establish some new
conditions for showing that the Banach algebras $A$ and $B$ are Arens regular
if and only if $A\hat{\otimes}B$ is Arens regular. We also introduce some new
concepts as left-weak$^*$-weak convergence property [$Lw^*wc-$property] and
right-weak$^*$-weak convergence property [$Rw^*wc-$property] and for Banach
algebra $A$, suppose that $A^*$ and $A^{**}$, respectively, have
$Rw^*wc-$property and $Lw^*wc-$property.

Sampling and reconstruction of functions is a central tool in science. A key
result is given by the sampling theorem for bandlimited functions attributed to
Whittaker, Shannon, Nyquist, and Kotelnikov. We develop an analogous sampling
theory for operators which we call bandlimited if their Kohn-Nirenberg symbols
are bandlimited. We prove sampling theorems for such operators and show that
they are extensions of the classical sampling theorem.

In this paper we prove the weak and strong convergence of the implicit
iterative process with errors to a common fixed point of a finite family
$\{T_j\}_{i=1}^N$ of asymptotically quasi $I_j-$nonexpansive mappings as well
as a family of $\{I_j\}_{j=1}^N$ of asymptotically quasi nonexpansive mappings
in the framework of Banach spaces.

Let $l[y]$ be a formally selfadjoint differential expression of an even order
on the interval $[0,b> \;(b\leq \infty)$ and let $L_0$ be the corresponding
minimal operator. By using the concept of a decomposing boundary triplet we
consider the boundary problem formed by the equation $l[y]-\l y=f\;(f\in
L_2[0,b>)$ and the Nevanlinna $\l$-depending boundary conditions with constant
values at the regular endpoint 0.

Interpretations of the Beurling-Lax-Halmos Theorem on invariant subspaces of
the unilateral shift are explored using the language of Hilbert modules.
Extensions and consequences are considered in both the one and multivariate
cases with an emphasis on the classical Hardy, Bergman and Drury-Arveson
spaces.

Shearlet theory has become a central tool in analyzing and representing 2D
data with anisotropic features. Shearlet systems are systems of functions
generated by one single generator with parabolic scaling, shearing, and
translation operators applied to it, in much the same way wavelet systems are
dyadic scalings and translations of a single function, but including a precise
control of directionality. Of the many directional representation systems
proposed in the last decade, shearlets are among the most versatile and
successful systems.

For an operator $T \in B(X,Y)$, we denote by $a_m(T)$, $c_m(T)$, $d_m(T)$,
and $t_m(T)$ its approximation, Gelfand, Kolmogorov, and absolute numbers. We
show that, for any infinite dimensional Banach spaces $X$ and $Y$, and any
sequence $\alpha_m \searrow 0$, there exists $T \in B(X,Y)$ for which the
inequality $$ 3 \alpha_{\lceil m/6 \rceil} \geq a_m(T) \geq \max\{c_m(t),
d_m(T)\} \geq \min\{c_m(t), d_m(T)\} \geq t_m(T) \geq \alpha_m/9 $$ holds for
every $m \in \N$. Similar results are obtained for other $s$-scales.

We study a minimum problem and associated maximum problem for finite,
complex, self-adjoint Toeplitz matrices. If $A$ is such a matrix, of size
$(N+1)$-by-$(N+1)$, we identify $A$ with the operator it represents on $P_N$,
the space of complex polynomials of degrees at most $N$, with the usual Hilbert
space structure it inherits as a subspace of $L^2$ of the unit circle. The
operator $A$ is the compression to $P_N$ of the multiplication operator on
$L^2$ induced by any function in $L^{\infty}$ whose Fourier coefficients of
indices between $-N$ and $N$ match the matrix entries of $A$.

In this paper, we investigate the roles of compact sets in the space of
tempered distributions $\mathscr{S}^{\prime}$. The key notion is "k-spaces",
which constitute a fairly general class of topological spaces. In a k-space,
the system of compact sets controls continuous functions and Borel measures.

In this paper we consider spherically symmetric trees endowed with the usual
combinatorial metric (SSTs). Using a simple geometric argument we show how to
determine decent upper bounds on the generalized roundness of finite SSTs that
depend only on the downward degree sequence of the tree in question. By
considering limits it follows that if the downward degree sequence $(d_{0},
d_{1}, d_{2}, \ldots)$ of a SST $(T,\rho)$ satisfies $|\{ j \, | \, d_{j} > 1
\}| = \aleph_{0}$, then $(T,\rho)$ has generalized roundness one.

In this paper, we study unitary Gaussian processes with independent
increments with which the unitary equivalence to a Hudson-Parthasarathy
evolution systems is proved. This gives a generalization of results in [16] and
[17] in the absence of the stationarity condition.

Let $A$ be a self-adjoint operator acting over a space $X$ endowed with a
partition. We give lower bounds on the energy of a mixed state $\rho$ from its
distribution in the partition and the spectral density of $A$. These bounds
improve with the refinement of the partition, and generalize inequalities by
Li-Yau and Lieb--Thirring for the Laplacian in $\R^n$. They imply an
uncertainty principle, giving a lower bound on the sum of the spatial entropy
of $\rho$, as seen from $X$, and some spectral entropy, with respect to its
energy distribution.

The existence of a ground state of the Nelson Hamiltonian with a perturbation
is considered. The self-adjointness of the Hamiltonian and the existence of a
ground state are proven for arbitrary values of coupling constants.

For the periodic matrix-valued Jacobi operator $J$ we obtain the estimate of
Lebesgue measure of spectra: $|\s(J)|\le2\pi \min_n\|a_n\|\rank a_n$, where
$a_n$ are off-diagonal elements of $J$.

Dual pseudo splines constitute a new class of refinable functions with
B-splines as special examples, which was introduced in \cite{DHSS}. In this
paper, we shall construct Riesz wavelet associated with dual pseudo splines.
Furthermore, we use dual pseudo splines to construct tight frame systems with
desired approximation order by applying the unitary extension principle.

We introduce the notion of theta-antieigenvalue and theta-antieigenvector of
a bounded linear operator on complex Hilbert space and study the realtion
between theta-antieigenvalue and centre of mass of a bounded linear operator.
We show that the notion of real antieigenvalue, imaginary antieigenvalue and
symmetric antieigenvalue follows as a special case of theta-antieigenvalue. We
also show how the notion of total antieigenvalue is related to the
theta-antieigenvalue. In fact, we show that all the notions of antieigenvalues
studied so far follows from the concept of theta- antieigenvalue.

We introduce another new type of combinations of Bernstein operators in this
paper, which can be used to approximate the functions with inner singularities.
The direct and inverse results of the weighted approximation of this new type
combinations are obtained.

We study in this article the Improved Sobolev inequalities with Muckenhoupt
weights within the framework of stratified Lie groups. This family of
inequalities estimate the Lq norm of a function by the geometric mean of two
norms corresponding to Sobolev spaces W(s;p) and Besov spaces B(-b, infty,
infty). When the value p which characterizes Sobolev space is strictly larger
than 1, the required result is well known in R^n and is classically obtained by
a Littlewood-Paley dyadic blocks manipulation. For these inequalities we will
develop here another totally different technique.

We prove that certain non-local functionals defined on measurable sets
Gamma-converge to the perimeter in the sense of De Giorgi.

In this paper we introduce a complete naive theory of operators on complex
Hilbert spaces, which achieve the norm in the unit sphere. We prove some
important results concerning the characterization of the AN operators, see
Definition 1.2. The class of AN operators contains the algebra of the compact
ones. Furthermore, we also study the operators that attains their minimum in
the unit sphere. At the end of this analysis, we propose a Conjecture referring
to the structure of positive AN operators.

In this article, linear operators satisfying anti-commutation relations are
considered. It is proven that an anti-commutative type of the
Glimm-Jaffe-Nelson commutator theorem follows.

In this paper we obtain an algorithm towards solving the two-dimensional
moment problem. This algorithm gives the necessary and sufficient conditions
for the solvability of the moment problem. It is shown that all solutions of
the moment problem can be constructed using this algorithm.

We obtain weighted $L^p$ inequalities for pseudo-differential operators with
smooth symbols and their commutators by using a class of new weight functions
which include Muckenhoupt weight functions. Our results improve essentially
some well-known results.

A method of proving local continuity of concave functions on convex set
possessing the $\mu$-compactness property is presented. This method is based on
a special approximation of these functions.

The class of $\mu$-compact sets can be considered as a natural extension of
the class of compact metrizable subsets of locally convex spaces, to which
particular results well known for compact sets can be generalized.

Applications of the obtained continuity conditions to analysis of different
entropic characteristics of quantum systems and channels are considered.

A generalization of Young's inequality for convolution with sharp constant is
conjectured for scenarios where more than two functions are being convolved,
and it is proven for certain parameter ranges. The conjecture would provide a
unified proof of recent entropy power inequalities of Barron and Madiman, as
well as of a (conjectured) generalization of the Brunn-Minkowski inequality. It
is shown that the generalized Brunn-Minkowski conjecture is true for convex
sets; an application of this to the law of large numbers for random sets is
described.

Recently Flaminio and Montagna, \cite{FlMo}, extended the language of
MV-algebras by adding a unary operation, called a state-operator. This notion
is introduced here also for effect algebras. Having it, we generalize the
Loomis--Sikorski Theorem for monotone $\sigma$-complete effect algebras with
internal state. In addition, we show that the category of divisible
state-morphism effect algebras satisfying (RDP) and countable interpolation
with an order determining system of states is dual to the category of Bauer
simplices $\Omega$ such that $\partial_e \Omega$ is an F-space.

We study frequent hypercyclicity in the context of strongly continuous
semigroups of operators. More precisely, we give a criterion (sufficient
condition) for a semigroup to be frequently hypercyclic, whose formulation
depends on the Pettis integral. This criterion can be verified in certain cases
in terms of the infinitesimal generator of semigroup.

We introduce the notion of C-orbit reflexivity and study its properties. An
operator on a finite-dimensional space is C-orbit reflexive if and only if the
two largest blocks in its Jordan form corresponding to nonzero eigenvalues with
the largest modulus differ in size by at most one. Most of the proofs of our
results in infinite dimensions are obtained from purely algebraic results we
obtain from linear-algebraic analogs of C-orbit reflexivity.

The most important open problem in Monotone Operator Theory concerns the
maximal monotonicity of the sum of two maximal monotone operators provided that
Rockafellar's constraint qualification holds. In this paper, we provide a new
maximal monotonicity result for the sum of two maximal monotone operators $A$
and $B$ in this setting satisfying that $A+N_{\bar{\dom B}}$ is of type (FPV)
and $\dom A\cap\bar{\dom B}\subseteq\dom B$. The proof relies on some results
on the Fitzpatrick function.

We study the problem of whether $\mathcal{P}_w(^nE)$, the space of
$n$-homogeneous polynomials which are weakly continuous on bounded sets, is an
$M$-ideal in the space of continuous $n$-homogeneous polynomials
$\mathcal{P}(^nE)$. We obtain conditions that assure this fact and present some
examples. We prove that if $\mathcal{P}_w(^nE)$ is an $M$-ideal in
$\mathcal{P}(^nE)$, then $\mathcal{P}_w(^nE)$ coincides with
$\mathcal{P}_{w0}(^nE)$ ($n$-homogeneous polynomials that are weakly continuous
on bounded sets at 0).

We provide a simple proof of the radial symmetry of any nonnegative minimizer
for a general class of quasi-linear minimization problems

Let $T:X\to X$ be a linear power bounded operator on Banach space. Let $X_0$
is a subspace of vectors tending to zero under iterating of $T$. We prove that
if $X_0$ is not equal to $X$ then there exists $\lambda$ in Sp(T) such that,
for every $\epsilon>0$, there is $x$ such that $|Tx-\lambda x|<\epsilon $ but
$|T^nx|>1-\epsilon$ for all $n$. The technique we develop enables us to
establish that if $X$ is reflexive and there exists a compactum $K$ in $X$ such
that for every norm-one $x\in X$ $\rho\{T^nx, K\}<\alpha (T)<1$ for some
$n=n_1, n_2,...$ then $codim(X_0)<\infty$.

This note records some dilation theorems about contraction semigroups on a
Hilbert space - all of which fall into the categories "known" or "probably
known" - that I proved while working on my PhD in mathematics (under the
supervision of Baruch Solel). It is convenient to have them recorded for
reference.

The proximal average of two convex functions has proven to be a useful tool
in convex analysis. In this note, we express Goebel's self-dual smoothing
operator in terms of the proximal average, which allows us to give a simple
proof of self duality. We also provide a novel self-dual smoothing operator.
Both operators are illustrated by smoothing the norm.

For $\alpha$ an ordinal and $1<p<\infty$, we determine a necessary and
sufficient condition for an $\ell_p$-direct sum of operators to have Szlenk
index not exceeding $\omega^\alpha$. It follows from our results that the
Szlenk index of an $\ell_p$-direct sum of operators is determined in a natural
way by the behaviour of the $\epsilon$-Szlenk indices of its summands. Our
methods give similar results for $c_0$-direct sums.

For $\alpha$ an ordinal, we investigate the class $\mathscr{SZ}_\alpha$
consisting of all operators whose Szlenk index is an ordinal not exceeding
$\omega^\alpha$. Our main result is that $\mathscr{SZ}_\alpha$ is a closed,
injective, surjective operator ideal for each $\alpha$. We also study the
relationship between the classes $\mathscr{SZ}_\alpha$ and several well-known
closed operator ideals.

Two Bernstein-type inequalities in the standard Bergman space L_{a}^{2} of
the unit disc \mathbb{D}={z\in\mathbb{C}: |z|<1}, for rational functions in
\mathbb{D} having at most n poles all outside of \frac{1}{r}\mathbb{D}, 0

A short proof of the "Rigidity theorem" using the sheaf theoretic model for
Hilbert modules over polynomial rings is given. The joint kernel for a large
class of submodules is described.

We prove the Ehrenfest theorem of quantum mechanics under sharp assumptions
on the operators involved.

The first and second representation theorems for sign-indefinite, not
necessarily semi-bounded quadratic forms are revisited. New straightforward
proofs of these theorems are given. A number of necessary and sufficient
conditions ensuring the second representation theorem to hold is proved. A new
simple and explicit example of a self-adjoint operator for which the second
representation theorem does not hold is also provided.

We prove that if $X$ is a subspace of $L_p$ $(2<p<\infty)$, then either $X$
embeds isomorphically into $\ell_p \oplus \ell_2$ or $X$ contains a subspace
$Y,$ which is isomorphic to $\ell_p(\ell_2)$. We also give an intrinsic
characterization of when $X$ embeds into $\ell_p \oplus \ell_2$ in terms of
weakly null trees in $X$ or, equivalently, in terms of the "infinite asymptotic
game" played in $X$. This solves problems concerning small subspaces of $L_p$
originating in the 1970's.

On a smoothly bounded domain $\Omega\subset\R{2m}$ we consider a sequence of
positive solutions $u_k\stackrel{w}{\rightharpoondown} 0$ in $H^m(\Omega)$ to
the equation $(-\Delta)^m u_k=\lambda_k u_k e^{mu_k^2}$ subject to Dirichlet
boundary conditions, where $0<\lambda_k\to 0$. Assuming that
$$\Lambda:=\lim_{k\to\infty}\int_\Omega u_k(-\Delta)^m u_k dx<\infty,$$ we
prove that $\Lambda$ is an integer multiple of
$\Lambda_1:=(2m-1)!\vol(S^{2m})$, the total $Q$-curvature of the standard
$2m$-dimensional sphere.

Notes from a course taught by Palle Jorgensen in the fall semester of 2009.
The course covered central themes in functional analysis and operator theory,
with an emphasis on topics of special relevance to such applications as
representation theory, harmonic analysis, mathematical physics, and stochastic
integration.

We consider the Schr\"odinger operators on zigzag nanoribbons

It is shown that every separable reflexive Banach space is a quotient of a
reflexive Hereditarily Indecomposable space, which yields that every separable
reflexive Banach is isomorphic to a subspace of a reflexive Indecomposable
space. Furthermore, every separable reflexive Banach space is a quotient of a
reflexive complementably $\ell_p$ saturated space with $1<p<\infty$ and of a
$c_0$ saturated space.

We present some new examples of separable (\mathcal_\infty) spaces which are
(\ell_r) saturated for some (1 < r < \infty).

We show that two smooth elements in the kernel of certain underdetermined
linear elliptic operators P can be glued in a chosen region in order to obtain
a new smooth solution. This new solution is exactly equal to the starting
elements outside the gluing region. This result completely contrasts with the
usual unique continuation for determined or overdetermined elliptic operators.
As a corollary we obtain compactly supported solutions in the kernel of P and
also solutions vanishing in a chosen relatively compact open region.

The main purpose of this note is to show that the question posed in the paper
of Sinha D.P. and Karn A.K.("Compact operators which factor through subspaces
of $l_p$ Math. Nachr. 281, 2008, 412-423; see the very end of that paper) has a
negative answer, and that the answer was known, essentially, in 1985 after the
papers "Approximation properties of order p and the existence of non-p-nuclear
operators with p-nuclear second adjoints" (Math. Nachr.

We introduce a technique for handling Whitney decompositions in Gaussian
harmonic analysis and apply it to the study of Gaussian analogues of the
classical tent spaces $T^{1,q}$ of Coifman, Meyer and Stein.

In 1968, Gohberg and Krupnik found a Fredholm criterion for singular integral
operators of the form $aP+bQ$, where $a,b$ are piecewise continuous functions
and $P,Q$ are complementary projections associated to the Cauchy singular
integral operator, acting on Lebesgue spaces over Lyapunov curves. We extend
this result to the case of Nakano spaces (also known as variable Lebesgue
spaces) with certain weights having finite sets of discontinuities on arbitrary
Carleson curves.

We describe all solutions of the matrix Stieltjes moment problem in the
general case (no conditions besides solvability are assumed). We use Krein's
formula for the generalized $\Pi$-resolvents of positive Hermitian operators in
the form of V.A Derkach and M.M. Malamud.

Paper deals with the singular Sturm-Liouville expressions $$l(y) = -(py')' +
qy$$ on a finite interval with coefficients $$q = Q', \quad 1/p, Q/p, Q^2/p \in
L_1,$$ where derivative of the function $Q$ is understood in the sense of
distributions. Due to a new regularization corresponding operators are
correctly defined as quasi-differential. Their resolvent approximation is
investigated and all self-adjoint and maximal dissipative extensions and
generalized resolvents are described in terms of homogeneous boundary
conditions of the canonic form.

A basic problem of interest in connection with the study of Schauder frames
in Banach spaces is that of characterizing those Schauder frames which can
essentially be regarded as Schauder bases. In this paper, we give a solution to
this problem using the notion of the minimal-associated sequence spaces and the
minimal-associated reconstruction operators for Schauder frames. We prove that
a Schauder frame is a near-Schauder basis if and only if the kernel of the
minimal-associated reconstruction operator contains no copy of $c_0$.

In the last decade, methods based on various kinds of spherical wavelet bases
have found applications in virtually all areas where analysis of spherical data
is required, including cosmology, weather prediction, and geodesy. In
particular, the so-called needlets (=band-limited Parseval frames) have become
an important tool for the analysis of Cosmic Microwave Background (CMB)
temperature data.

In the spirit of the work of Grothendieck, we introduce and study natural
symmetric n-fold tensor norms. We prove that there are exactly six natural
symmetric tensor norms for $n\ge 3$, a noteworthy difference with the 2-fold
case in which there are four. Using a symmetric version of a result of Carne we
also describe which natural symmetric tensor norms preserve Banach algebras.

It is a translation of an old paper of mine. We describe the topology tau_p
in the space Pi_p(Y,X), for which the closures of convex sets in tau_p and in
*-weak topology of the space Pi_p(Y,X) are coincident. Thereafter, we
investigate some properties of the space Pi_p, related to this new topology.
2010-remark: Occasionally, the topology is coincides with the lambda_p-topology
from the paper "Compact operators which factor through subspaces of l_p", Math.
Nachr. 281(2008), 412-423 by Deba Prasad Sinha and Anil Kumar Karn.

A zone diagram is a relatively new concept which has emerged in computational
geometry and is related to Voronoi diagrams. Formally, it is a fixed point of a
certain mapping, and neither its uniqueness nor its existence are obvious in
advance. It has been studied by several authors, starting with T. Asano, J.
Matousek and T. Tokuyama, who considered the Euclidean plane with singleton
sites, and proved the existence and uniqueness of zone diagrams there.