We consider nets $\left(T_j\right)$ of operators acting on complex functions, and we investigate the algebraic and the topological structure of the set $\{f:T_j(\left|f{|}^2\right)-|T_{j}f{|}^2\to 0\}$. Our results extend and improve some known results from the literature, which are connected with Korovkin’s theorem. Applications to Abel–Poisson-type operators and Bernstein-type operators are given.

We propose a new iterative algorithm to compute the symmetric solution of the matrix equations $AXB=E$ and $CXD=F$. The greatest advantage of this new algorithm is higher speed and lower computational cost at each step compared with existing numerical algorithms. We state the solutions of these matrix equations as the intersection point of some closed convex sets, and then we use the alternating projection method to solve them. Finally, we use some numerical examples to show that the new algorithm is feasible and effective.

Let $X,Y$ be Banach spaces, and let $T$, $\delta T:X\to Y$ be bounded linear operators. Put $\overline{T}=T+\delta T$. In this article, utilizing the gap between closed subspaces and the perturbation bounds of metric projections, we first present some error estimates of the upper bound of $\Vert {\overline{T}}^M-T^M\Vert$ in $L^p$ ($1<p<+\infty$) spaces. Then, by using the concept of strong uniqueness and modulus of convexity, we further investigate the corresponding perturbation bound $\Vert {\overline{T}}^M-T^M\Vert$ in uniformly convex Banach spaces.

In this article, we introduce the notion of a scattered locally $C^{*}$-algebra and we give the conditions for a locally $C^{*}$-algebra to be scattered. Given an action $\alpha$ of a locally compact group $G$ on a scattered locally $C^{*}$-algebra $A\left[{\tau }_{\Gamma }\right]$, it is natural to ask under what conditions the crossed product $A\left[{\tau }_{\Gamma }\right]{\times }_{\alpha }G$ is also scattered. We obtain some results concerning this question.

In a previous article, we proved the equivalence of six conditions on a continuous function $f$ on an interval. These conditions determine a subset of the set of operator-convex functions whose elements are called strongly operator-convex. Two of the six conditions involve operator-algebraic semicontinuity theory, as given by Akemann and Pedersen, and the other four conditions do not involve operator algebras at all. Two of these conditions are operator inequalities, one is a global condition on $f$, and the fourth is an integral representation of $f$, stronger than the usual integral representation for operator-convex functions. The purpose of this article is to make the equivalence of these four conditions accessible to people who do not know operator algebra theory as well as to operator algebraists who do not know the semicontinuity theory. A treatment of other operator inequalities characterizing strong operator convexity is included.

We construct topological bases in spaces of Whitney functions on Cantor sets, which were introduced by the first author. By means of suitable individual extensions of basis elements, we construct a linear continuous extension operator, when it exists for the corresponding space. In general, elements of the basis are restrictions of polynomials to certain subsets. In the case of small sets, we can present strict polynomial bases as well.

In this article, the authors first give a Littlewood–Paley characterization for inhomogeneous Lipschitz spaces of variable order with the help of inhomogeneous Calderón identity and almost-orthogonality estimates. As applications, the boundedness of inhomogeneous Calderón–Zygmund singular integral operators of order $(ϵ,\sigma )$ on these spaces has been presented. Finally, we note that a class of pseudodifferential operators $T_a\in \mathcal{O}p{S}_{1,1}^0$ are continuous on the inhomogeneous Lipschitz spaces of variable order as a corollary. We may observe that those operators are not, in general, continuous in $L^2$.

The purpose of this paper is to give a new atomic decomposition for variable Hardy spaces via the discrete Littlewood–Paley–Stein theory. As an application of this decomposition, we assume that $T$ is a linear operator bounded on $L^q$ and $H^{p(\cdot )}$, and we thus obtain that $T$ can be extended to a bounded operator from $H^{p(\cdot )}$ to $L^{p(\cdot )}$.

We study several properties of the modulus of order bounded disjointness-preserving operators. We show that, if $T$ is an order bounded disjointness-preserving operator, then $T$ and $\left|T\right|$ have the same compactness property for several types of compactness. Finally, we characterize Banach lattices having $b$-$\mathit{AM}$-compact (resp., $\mathit{AM}$-compact) operators defined between them as having a modulus that is $b$-$\mathit{AM}$-compact (resp., $\mathit{AM}$-compact).

We study the problem of proximal split feasibility of two objective convex functions in Hilbert spaces. We prove that, under suitable conditions, certain strong convergence theorems of the Halpern-type algorithm present solutions to the proximal split feasibility problem. Finally, we provide some related applications as well as numerical experiments.

We introduce the notion of the $p$-Schur property ($1\le p\le \infty$) as a generalization of the Schur property of Banach spaces, and then we present a number of basic properties and some examples. We also study its relation with some geometric properties of Banach spaces, such as the Gelfand–Phillips property. Moreover, we verify some necessary and sufficient conditions for the $p$-Schur property of some closed subspaces of operator spaces.

In this paper, we investigate convergence and divergence of partial sums with respect to the $2$-dimensional Walsh system on the martingale Hardy spaces. In particular, we find some conditions for the modulus of continuity which provide convergence of partial sums of Walsh–Fourier series. We also show that these conditions are in a sense necessary and sufficient.