A definition of frames in Krein spaces is stated and a complete characterization is given by comparing them to frames in the associated Hilbert space. The basic tools of frame theory are described in the formalism of Krein spaces. It is shown how to transfer a frame for Hilbert spaces to Krein spaces given by a $W$-metric, where the Gram operator $W$ is not necessarily regular and possibly unbounded.

For any $n \times n$ nonnegative matrix $A$, and any norm $\|.\|$ on $\mathbb{R}^n$, $\eta_{\|.\|}(A)$ is defined as $ \sup\ \{\frac{\|A \otimes x\|}{\|x\|} :\ x\in \mathbb{R}_+^n \ , \ x\neq 0\}.$ Let $P(\lambda)$ be a matrix polynomial in the max algebra. In this paper, we introduce $\eta_{\|.\|}[P(\lambda)]$, as a generalization of the matrix norm $\eta_{\|.\|}(.)$, and we investigate some algebraic properties of this notion. We also study some properties of the maximum circuit geometric mean of the companion matrix of $P(\lambda)$ and the relationship between this concept and the matrices $P(1)$ and coefficients of $P(\lambda)$. Some properties of $\eta_{\|.\|}(\Psi)$, for a bounded set of max matrix polynomials $\Psi$, are also investigated.

In this work, given a unital Banach algebra $\mathcal{A}$ and $a\in \mathcal{A}$ such that $a$ has a Moore--Penrose inverse $a^\dagger$, it will be characterized when $aa^\dagger- a^\dagger a$ is invertible. A particular subset of this class of objects will also be studied. In addition, perturbations of this class of elements will be studied. Finally, the Banach space operator case will be also considered.

In this paper, we prove that if $X$ is a separable space and $X^{*}$ is a locally $2$-uniform convex space, then for any $\varepsilon \in (0,1)$, there exist sequences $\{x_{n,1}^*\}_{n = 1}^\infty $ and $\{x_{n,2}^*\}_{n = 1}^\infty $ of strongly extreme points such that \[ \mathop \cup \limits_{n = 1}^\infty \left\{B(x_{n,1}^*,1 - \frac{1}{8}\varepsilon) \cup B(x_{n,2}^*,1 - \frac{1}{8}\varepsilon ) \cup B(\frac{{x_{n,1}^* + x_{n,2}^*}}{2},\varepsilon )\right\} \] is a ball-covering of $X^{*}$. Moreover, we also prove that if (1) $X$ is a separable space; (2) $X$ is a locally $2$-uniform convex space; (3) $X$ is a uniformly nonsquare space, then there exists a sequence $\{x_n\}_{n = 1}^\infty $ of strongly extreme points such that $\mathop \cup \limits_{n = 1}^\infty B(x_n,{r_n})$ is a ball-covering of $X$.

In this paper, we study disjoint, strongly disjoint and weakly disjoint $g$-frames in Hilbert spaces and we provide necessary and sufficient conditions for disjointness, strongly disjointness and weakly disjointness of $g$-frames. Also, by using the orthogonal projections in Hilbert spaces, we prove that dual $g$-frames for a Hilbert space can be dilated to a $g$-Riesz basis for some larger Hilbert space and its dual $g$-Riesz basis.

In this paper, in relation with interpolation functions we study some generalized Powers-Størmer's type inequalities and monotonicity inequality of indefinite type which generalizes a result of Ando.

In the article, the boundedness of vector-valued sublinear operators in Herz--Morrey spaces with variable exponents $M\dot{K}^{\alpha(\cdot),\lambda}_{q,p(\cdot)}(\mathbb{R}^{n})$ are obtained. Then Herz--Morrey type Besov and Triebel-Lizorkin spaces with variable exponents are introduced. Finally, we prove the equivalent quasi-norms on these spaces by Peetre's maximal operators.

We give a direct definition of Morita equivalence for Hilbert $C^{*}$ -modules, by introducing an explicit list of axioms for an imprimitivity bimodule. We show that Hilbert $C^{*}$-modules with unit vectors, over Morita equivalent unital $C^{*}$-algebras, are Morita equivalent, and Morita equivalence is an equivalence relation in the category of left Hilbert $C^{*} $-modules with unit vectors.

In this paper some inequalities involving quasi-arithmetic means for a continuous field of self-adjoint operators, a field of positive linear mappings and continuous strictly monotone functions are refined. These refined converses are presented by using the Mond-Pečarić method improvement. Obtained results are applied to refine selected inequalities with power functions.

In this paper, two new classes of subalgebras of $L^{1}\left( G\right) $ generated by Lorentz--Karamata spaces are investigated and some fundamental properties of these spaces are examined. Also, the multipliers space (homomorphisms space) on $L^{1}\left( G\right) \cap L\left( p,q;b\right) \left( G\right) $ spaces are characterized.

In this paper we describe the multiplier algebra of a perfect complete locally $m$-convex algebra with an approximate identity and with complete Arens-Michael normed factors.

For an almost radial and typical weight $v$ and any weight $w$, we characterize the continuity, compactness and we estimate the essential norm of weighted composition operators $u C_{\varphi}$, acting from the weighted Banach spaces of analytic functions $H_{v}^{\infty}$ into $H_{w}^{\infty}$, in terms of the quotients of the $w$-norm of the product of $u$ with $\varphi^n$ and the $v$-norm of the $n$th power of the identity function on $\Bbb D$, where $u: \mathbb{D} \to \mathbb{C}$ and $\varphi: \mathbb{D} \to \mathbb{D}$ are analytic. As a consequence, we estimate the essential norm of composition operators $C_\varphi$ (in terms of $\varphi^n$) acting on $\mu$-Bloch spaces, for very general weights $\mu$. We also characterize continuity and compactness of weighted composition operators $uC_\varphi$ acting on $\log$-Bloch space.

In this paper, we study the concept of harmonic functionals for certain Banach algebras such as generalized Fourier algebras. For a non-zero character $\phi$ on Banach algebra ${\mathcal A}$, we also characterize the concept of $\phi$-amenability in terms of harmonic functionals. Finally, for a locally compact group $G$ we investigate the space $H_{\sigma, x}$ of $\sigma$-harmonic functionals in the dual of generalized Fourier algebra $A_p(G)$. The main result states that $G$ is first countable if and only if $\sigma$ is adapted if and only if $H_{\sigma, x}={\Bbb C}\phi_x$.

Some operator inequalities for positive linear maps are presented. These inequalities improve and generalize the corresponding results due to Fu and He [Linear Multilinear Algebra, doi: 10.1080/03081087.2014.880432.].

In this paper, we study an algebra $\mathcal{A}$ consisting of all arithmetic functions, and corresponding dynamical systems acting on $% \mathcal{A}$ determined by a fixed prime $p$. Starting from free probabilistic models on $\mathcal{A}$ determined by $p$, we construct certain group dynamical systems induced by the additive group $\Bbb{R}$ of all real numbers. We investigate the basic properties and free-probabilistic data of such dynamical systems by constructing corresponding crossed product algebras.

The weights $W_{\alpha}(x)=\mathrm{exp}(-|x|^{\alpha})$ $(\alpha>1)$ form a subclass of Freud weights on the real line. Primarily from a functional analytic angle, we investigate the subspace of $L_{2}(\mathbb{R}, W_{\alpha}^{2} dx)$ consisting of those elements that can be rapidly approximated by polynomials. This subspace has a natural Fr\'echet topology, in which it is isomorphic to the space of rapidly decreasing sequences. We show that it consists of smooth functions and obtain concrete results on its topology. For $\alpha=2$, there is a complete and elementary description of this topological vector space in terms of the Schwartz functions.

The operator-product $ABA$ appears in many mathematical contexts, such as in algebraic Riccati equation, in operator entropy and in operator-mean theory. The purpose of the present paper is to investigate a reasonable analogue of $ABA$ when the positive linear operators $A$ and $B$ are convex functionals. As consequence, the square of a convex functional extending $A^2$ is provided as well.

Let $X$ and $Y$ be topological spaces and $f : X \rightarrow Y$ be a continuous function. We are interested in finding points of $Y$ at which $f$ is open or closed. We will show that under certain conditions, the set of points of openness or closedness of $f$ in $Y$ , i.e. points of $Y$ at which $f$ is open (resp. closed) is a $G_{\delta}$ subset of $Y$. We will extend some results of S. Levi, R. Engelking and I. A. Vaĭnšteĭn.

An operator $T$ on a complex Hilbert space $\mathcal{H}$ is called skew symmetric if $T$ can be represented as a skew symmetric matrix relative to some orthonormal basis for $\mathcal{H}$. We first give a canonical decomposition for general skew symmetric operators. Based on this decomposition, we provide a classification of skew symmetric weighted shifts.

In this paper we provide two representation theorems for two relevant classes of operators from any $p$-convex order continuous Banach lattice with weak unit into a Banach space: the class of continuous operators and the class of cone absolutely summing operators. We prove that they can be characterized as spaces of vector measures with finite $p$-semivariation and $p$-variation, respectively, with respect to a fixed vector measure. We give in this way a technique for representing operators as integrals with respect to vector measures.