Let $\lambda =({\lambda }_n{)}_{n\ge 1}$ be a nondecreasing sequence of positive numbers tending to infinity such that ${\lambda }_1=1$ and ${\lambda }_{n+1}\le {\lambda }_n+1$ for all $n$, and let $I_n=[n-{\lambda }_n+1,n]$ for $n=1,2,\dots$. Then for any given nonzero sequence $\mu$, we define by ${\Delta }^{+}\left(\mu \right)$ the operator that generalizes the operator of the first difference and is defined by ${\Delta }^{+}\left(\mu \right)x_k={\mu }_k(x_k-x_{k+1})$. In this article, for any given integer $r\ge 1$, we deal with the ${\lambda }^{+r}\left(\mu \right)$ -statistical convergence that generalizes in a certain sense the well-known ${\lambda }_E^r$-statistical convergence. The main results consist in determining sets of sequences $\chi$ and ${\chi }^{\prime }$ of the form $s_{\xi }^0$ satisfying $\chi \subset [V,\lambda {]}_0({\Delta }^{+r}\left(\mu \right))\subset {\chi }^{\prime }$ and sets $\kappa$ and ${\kappa }^{\prime }$ of the form $s_{\xi }$ satisfying $\kappa \le [V,\lambda {]}_{\infty }({\lambda }^{+r}\left(\mu \right))\le {\kappa }^{\prime }$. This study is justified since the infinite matrix associated with the operator ${\Delta }^{+r}\left(\mu \right)$ cannot be explicitly calculated for all $r$.

By using some geometric properties and nested sequence of balls, we prove seven necessary and sufficient conditions such that a point $x$ in the unit sphere of Banach space $X$ is a nearly rotund point of the unit ball of the bidual space. For any closed convex set $C\subset X$ and $x\in X\setminus C$ with $P_C\left(x\right)\ne \varnothing$, we give a series of characterizations such that $C$ is approximatively compact or approximatively weakly compact for $x$ by using three types of nearly convex points. Furthermore, making use of an S point, we present a characterization such that the convex subset $C$ is approximatively compact for some $x$ in $X\setminus C$. We also establish a relationship between nested sequence of balls and the approximate compactness of the closed convex subset $C$ for some $x\in X\setminus C$.

Let $B\mathcal{(H)}$, $K\mathcal{(H)}$ and $T\mathcal{(H)}$ be the set of all bounded linear operators, compact operators, and trace-class operators on the Hilbert space $\mathcal{H}$. The cone of all completely positive maps from $K\mathcal{(H)}$ into $T\mathcal{(K)}$ and all normal completely positive maps from $B\mathcal{(K)}$ into $T\mathcal{(H)}$ is denoted by $\mathit{CP}(K\mathcal{(H)},T\mathcal{(K)})$ and $\mathit{NCP}(B\mathcal{(K)},T\mathcal{(H)})$, respectively. In this note, the order structures of the positive cones $\mathit{CP}(K\mathcal{(H)},T\mathcal{(K)})$ and $\mathit{NCP}(B\mathcal{(K)},T\mathcal{(H)})$ are investigated. First, we show that $\mathit{CP}(K\mathcal{(H)},T\mathcal{(K)})$, $\mathit{NCP}(B\mathcal{(K)},T\mathcal{(H)})$, and $T{\mathcal{(K\otimes H)}}^{+}$ are cone-isomorphic. Then we give the operator sum representation for the map $\Phi \in \mathit{CP}(K\mathcal{(H)},T\mathcal{(K)})$.

Due to recent advances in the theory of ideals of Lipschitz mappings, we introduce $(p,\sigma )$-absolutely Lipschitz mappings as an interpolating class between Lipschitz mappings and Lipschitz absolutely $p$-summing mappings. Among other results, we prove a factorization theorem that provides a reformulation to the one given by Farmer and Johnson for Lipschitz absolutely $p$-summing mappings.

In this article we suggest a vector bundle description for multiplier modules of vector bundles over noncompact spaces. We prove that the isomorphism classes of multiplier modules are dependent on the isomorphism classes of their underlying modules. This gives a way to evaluate the set of extensions of Hilbert modules in topological terms in simple cases.

An operator $T$ on a complex Hilbert space $\mathcal{H}$ is said to be complex symmetric if $T$ can be represented as a symmetric matrix relative to some orthonormal basis for $\mathcal{H}$. In this article we explore the stability of complex symmetry under the condition of similarity. It is proved that the similarity orbit of an operator $T$ is included in the class of complex symmetric operators if and only if $T$ is an algebraic operator of degree at most $2$.

Right $\phi$-contractibility and right character contractibility of Banach algebras have been introduced and investigated. Here, we introduce and generalize these concepts for Fréchet algebras. We then verify available results about right $\phi$-contractibility and right character contractibility of Banach algebras for Fréchet algebras. Moreover, we provide related results about Segal–Fréchet algebras.

In this article, we obtain new additive results on the generalized Drazin inverse of a sum of two elements in a Banach algebra. Applying these additive results, we also give explicit formulas for the generalized Drazin inverse of a block matrix in a Banach algebra.

The present article deals with the modified forms of the Baskakov and Szász basis functions. We introduce a Durrmeyer-type operator having the basis functions in summation and integration due to Stancu (1970) and Pǎltǎnea (2008). We obtain some approximation results, which include the Voronovskaja-type asymptotic formula, local approximation, error estimation in terms of the modulus of continuity, and weighted approximation. Also, the rate of convergence for functions with derivatives of bounded variation is established. Furthermore, the convergence of these operators to certain functions is shown by illustrative graphics using MAPLE algorithms.

In 2001, Renaud obtained a Grüss type operator inequality involving the usual trace functional. In this article, we give a refinement of that result, and we answer positively Renaud’s open problem.

It is shown that, for an operator algebra $A$, the operator system $S=A+A^{\ast }$ in the $C^{\ast }$-algebra $C^{\ast }\left(S\right)$, and any representation $\rho$ of $C^{\ast }\left(S\right)$ on a Hilbert space $\mathcal{H}$, the restriction ${\rho }_{{|}_S}$ has a unique extension property if and only if the Hilbert module $\mathcal{H}$ over $A$ is both orthogonally projective and orthogonally injective. As a corollary we deduce that, when $S$ is separable, the hyperrigidity of $S$ is equivalent to the Hilbert modules over $A$ being both orthogonally projective and orthogonally injective.

Suppose that $X$, $Y$ are positive random variables and $m$ is a numerical (commutative) mean. We prove that the inequality $\mathrm{E}\left(m\right(X,Y\left)\right)\le m\left(\mathrm{E}\right(X),\mathrm{E}(Y\left)\right)$ holds if and only if the mean is generated by a concave function. With due changes we also prove that the same inequality holds for all operator means in the Kubo–Ando setting. The case of the harmonic mean was proved by C. R. Rao and B. L. S. Prakasa Rao.