We establish some identities or inequalities for the Hausdorff measure of noncompactness for operators $L\in\mathcal{B}(X,Y)$ when $X=\ell_{p}$ $(p \in [1, \infty))$ and $Y=c$; $X=\ell_{p}$ $(p \in (1, \infty))$ and $Y=\ell_{\infty}$; $X=bv_{0}$ and $Y=c$; $X=c_{0}(\Delta),c(\Delta),\ell_{\infty}(\Delta)$ and $Y=\ell_{\infty}$. These identities and estimates are used to establish necessary and sufficient conditions for the operators to be compact. Furthermore, we apply a result by Sargent to establish necessary and sufficient conditions for operators in $\mathcal{B}(bv_{0},\ell_{\infty})$ and $\mathcal{B}(\ell_{1},Y)$ to be compact, where $Y=w_{\infty},v_{\infty}, [c]_{\infty}$.

We propose necessary and sufficient conditions for a distribution (generalized function) $f$ of several variables to be positive definite. For this purpose, certain analytic extensions of $f$ to tubular domains in complex space $\mathbb{C}^n$ are studied. The main result is given in terms of completely monotonic functions on convex cones in $\mathbb{R}^n$.

Let $\alpha$ and $\beta$ be orientation-preserving diffeomorphisms (shifts) of $\mathbb{R}_+=(0,\infty)$ onto itself with the only fixed points $0$ and $\infty$, where the derivatives $\alpha'$ and $\beta'$ may have discontinuities of slowly oscillating type at $0$ and $\infty$. For $p\in(1,\infty)$, we consider the weighted shift operators $U_\alpha$ and $U_\beta$ given on the Lebesgue space $L^p(\mathbb{R}_+)$ by $U_\alpha f=(\alpha')^{1/p}(f\circ\alpha)$ and $U_\beta f= (\beta')^{1/p}(f\circ\beta)$. For $i,j\in\mathbb{Z}$ we study the simplest weighted singular integral operators with two shifts $A_{ij}=U_\alpha^i P_\gamma^++U_\beta^j P_\gamma^-$ on $L^p(\mathbb{R}_+)$, where $P_\gamma^\pm=(I\pm S_\gamma)/2$ are operators associated to the weighted Cauchy singular integral operator $$ (S_\gamma f)(t)=\frac{1}{\pi i}\int_{\mathbb{R}_+} \left(\frac{t}{\tau}\right)^\gamma\frac{f(\tau)}{\tau-t}d\tau $$ with $\gamma\in\mathbb{C}$ satisfying $01/p+\Re\gamma\in (0,1)$. We prove that the operator $A_{ij}$ is a Fredholm operator on $L^p(\mathbb{R}_+)$ and has zero index if \[ \frac{1}{p}+\Re\gamma+\frac{1}{2\pi}\inf_{t\in\mathbb{R}_+}(\omega_{ij}(t)\Im\gamma),\,\, \frac{1}{p}+\Re\gamma+\frac{1}{2\pi}\sup_{t\in\mathbb{R}_+}(\omega_{ij}(t)\Im\gamma)\in (0,1), \] where $\omega_{ij}(t)=\log[\alpha_i(\beta_{-j}(t))/t]$ and $\alpha_i$, $\beta_{-j}$ are iterations of $\alpha$, $\beta$. %%% This statement extends an earlier result obtained by the author, Yuri Karlovich, and Amarino Lebre for $\gamma=0$.

The purpose of this paper is to introduce the concept of $\mathrm{E}-$frames for a separable Hilbert space ${\mathcal H},$ where $\mathrm{E}$ is an invertible infinite matrix mapping on the Hilbert space $\mathop \oplus \limits_{n = 1}^\infty {{ \mathcal H}}$. We investigate and study some properties of $\mathrm{E}-$frames and characterize all $\mathrm{E}-$frames for ${\mathcal H}$. Further more, we characterize all dual $\mathrm{E}-$frames associated with a given $\mathrm{E}-$frame. A similar characterization is also established for $\mathrm{E}-$orthonormal bases, $\mathrm{E}-$Riesz bases and dual $\mathrm{E}-$Riesz bases. In continue we obtain a lower estimate for the lower bound of some matrix operators on the $p-$bounded variation sequence space $bv_p$ and Euler weighted sequence space $e_{w,p}^\theta$. Then we deal with several types of $\mathrm{E}-$frames such as $\mathbf{\Delta}-$frames and Euler frames for ${\mathcal H}$ which are related to the Hilbert spaces $bv_2$ and $e_{2}^\theta$, respectively.

Key-Words: E-frame; E-orthonormal basis; E-Bessel sequence; E-Riesz basis; Direct sum of Hilbert spaces; Euler frame; $\Delta-$frame.

Let $T: A \longrightarrow B$ be a surjective operator between two unital semisimple commutative Banach algebras $A$ and $B$ with $T1=1$. We show that if $T$ satisfies the peripheral multiplicativity condition $\sigma_\pi(Tf.Tg) = \sigma_\pi (f.g)$ for all $f$ and $g$ in $A$, where $\sigma_\pi(f)$ shows the peripheral spectrum of $f$, then $T$ is a composition operator in modulus on the $\check{S}$ilov boundary of $A$ in the sense that $|f(x)|=|Tf(\tau(x))|,$ for each $f\in A$ and $x\in \partial(A)$ where $\tau: \partial (A) \longrightarrow \partial (B)$ is a homeomorphism between $\check{S}$ilov boundaries of $A$ and $B$.

In this paper, inspired by some results concerning the double difference property, we show that the class $C^p(\mathbb{R},\mathbb{R})$ of $p$-times continuously differentiable functions has the difference property of $p$-th order, i.e. if a function $f\colon \mathbb{R}\to \mathbb{R}$ is such that $\Delta^p_h f\in C^p(\mathbb{R}\times \mathbb{R},\mathbb{R})$, where $\Delta^p_h f$ is the $p$-th iterate of the well-known difference operator $\Delta_h f(x):=f(x+h)-f(x)$, then there exists a polynomial function $\Gamma\!_{p-1}\colon \mathbb{R}\to \mathbb{R}$ of $(p-1)$-th order such that $f-\Gamma\!_{p-1}\in C^p(\mathbb{R},\mathbb{R})$. Moreover, some new equalities connected with the difference operator are also presented.

This article addresses the following question; 'how to approximate the spectrum of random bounded self-adjoint operators on separable Hilbert spaces'. This is an attempt to establish a link between the spectral theory of random operators and the rich theory of random matrices; including various notions of convergence. This study tries to develop a random version of the truncation method, which is useful in approximating spectrum of bounded self-adjoint operators. It is proved that the eigenvalue sequences of the truncations converge in distribution to the eigenvalues of the random bounded self-adjoint operator. The convergence of moments are also proved with some examples. In addition, the article discusses some new methods to predict the existence of spectral gaps between the bounds of essential spectrum. Some important open problems are also stated at the end.

Weighted EP Banach space operators and Banach algebra elements are characterized using different kinds of factorizations. The results presented extend well-known characterizations of (weighted) EP matrices, (weighted) EP Hilbert space operators and (weighted) EP $C^*$-algebra elements.

The purpose of this paper is to obtain some coincidence and periodic points results for generalized $F$-type contractions in a metric space endowed with a graph. Some examples are given to illustrate the new theory. Then, we apply our results to establishing the existence of solution for a certain type of nonlinear integral equation.

Using a `local' integral representation of a matrix connection of order $n$ corresponding to an interpolation function of the same order, for each integer $n$, we can describe an injective map from the class of matrix connections of order $n$ to the class of positive $n$-monotone functions on $(0,\infty)$ and the range of this corresponding covers the class of interpolation functions of order $2n$. In particular, the space of symmetric connections is isomorphic to the space of symmetric positive $n$-monotone functions. Moreover, we show that, for each $n$, the class of $n$-connections extremely contains that of $(n+2)$-connections.

In this paper we introduce Bessel multipliers, g-Bessel multipliers and Bessel fusion multipliers in Hilbert $C^\ast$--modules and we show that they share many useful properties with their corresponding notions in Hilbert and Banach spaces. We show that various properties of multipliers are closely related to their symbols and Bessel sequences, especially we consider multipliers when their Bessel sequences are modular Riesz bases and we see that in this case multipliers can be composed and inverted. We also study bounded below multipliers and generalize some of the results obtained for fusion frames in Hilbert spaces to Hilbert $C^\ast$--modules.

Let $A$ be a complex Banach algebra. If the spectrum of an invertible element $a\in A$ does not separate the plane, then $a$ admits a logarithm. We present two elementary proofs of this classical result which are independent of the holomorphic functional calculus. We also discuss the case of real Banach algebras. As applications, we obtain simple proofs that every invertible matrix over $\mathbb{C}$ has a logarithm and that every real matrix $M$ in $M_n(\mathbb{R})$ with $\det M>0$ is a product of two real exponential matrices.

In this paper, we explore approximate unitary equivalence of normaloid operators and classify several normaloid type operators including transaloid operators, polynomial-normaloid operators and von Neumann operators up to approximate unitary equivalence. As an application, we explore approximation of transaloid operators with closed numerical ranges. Among other things, it is proved that those transaloid operators with closed numerical ranges are norm dense in the class of transaloid operators.

Let $H$ be a compact subgroup of a locally compact group $G$. We consider the homogeneous space $G/H$ equipped with a strongly quasi-invariant Radon measure $\mu$. For $1\leq p\leq +\infty$, we introduce a norm decreasing linear map from $L^{p}(G)$ onto $L^{p}(G/H,\mu)$ and show that $L^{p}(G/H,\mu)$ may be identified with a quotient space of $L^{p}(G)$. Also, we prove that $L^{p}(G/H,\mu)$ is isometrically isomorphic to a closed subspace of $L^{p}(G)$. These help us study the structure of the classical Banach spaces constructed on a homogeneous space via those created on topological groups.

In this paper, the behavior for commutators and maximal commutators of a class of bilinear singular integral operators associated with non-smooth kernels on the products of Morrey space is studied. By some maximal operators and commutators, we proved that the commutators and maximal commutators of singular integral operators and ${\rm CMO}$ functions are bounded and compact.

The sharp order is a well known partial order defined on the set of complex matrices with index less or equal one. Following Šemrl's approach, Efimov extended this order to the set of those bounded Banach space operators $A$ for which the closure of the image and kernel are topologically complementary subspaces. In order to extend the sharp order to arbitrary ring $R$ (particulary to Rickart and Rickart $*$-rings) we use the notions of annihilators. The concept of the sharp order is extended to the set $\mathcal{I}_R$ of those elements for which left and right annihilators are respectively principal left and principal right ideals generated by the same idempotent. It is proved that the sharp order is a partial order relation on $\mathcal{I}_R$. Following the idea we also extend and discuss the recently introduced concept of core partial order.

In this paper, we generalize some determinantal inequalities for Hadamard product of $M$-matrices and inverse $M$-matrices obtained by Chen [Linear Algebra Appl. $\textbf{426}$ (2007), no. 2-3, 610--618.].

For a locally compact quantum group $\mathbb{G}$, the quantum group algebra $L^1(\mathbb{G})$ is operator amenable if and only if it has an operator bounded approximate diagonal. It is known that if $L^1(\mathbb{G})$ is operator biflat and has a bounded approximate identity then it is operator amenable. In this paper, we consider nets in $L^2(\mathbb{G})$ which suffice to show these two conditions and combine them to make an approximate diagonal of the form $\omega_{{W'}^*\xi\otimes\eta}$ where $W$ is the multiplicative unitary and $\xi\otimes\eta$ are simple tensors in $L^2(\mathbb{G})\otimes L^2(\mathbb{G})$. Indeed, if $L^1(\mathbb{G})$ and $L^1(\hat{\mathbb{G}})$ both have a bounded approximate identity and either of the corresponding nets in $L^2(\mathbb{G})$ satisfies a condition generalizing quasicentrality then this construction generates an operator bounded approximate diagonal. In the classical group case, this provides a new method for constructing approximate diagonals emphasizing the relation between the operator amenability of the group algebra $L^1(G)$ and the Fourier algebra $A(G)$.

The Hilbert space representation theory of the $q$-deformed quantum $*$-algebra $\mathcal{U}_q(su_{1,1})$ is studied using the inducing procedure. As a result we obtain four series of irreducible induced $*$-representations on Hilbert spaces. Furthermore, we show that there is a one-to-one correspondence between the induced series and the series of irreducible well-behaved $*$-representations of $\mathcal{U}_q(su_{1,1})$ computed by Burban and Klymik.

The issue of Ulam's type stability of an equation is understood in the following way: when a mapping which satisfies the equation approximately (in some sense), it is "close" to a solution of it. In this expository paper, we present a survey and a discussion of selected recent results concerning such stability of the equations of homomorphisms, focussing especially on some conditional versions of them.