‎‎‎‎This article deals with almost convergence of double sequences using a new generalization of fractional-order difference operator in modular spaces and application to the Korovkin-type approximation in the context of modular spaces for positive linear operators‎. ‎We then obtain several inclusion relations and present some examples‎, ‎include proper non-trivial extensions of the corresponding classical ones‎. ‎Further‎, ‎we extend our study to new modular forms of Korovkin-type approximation theorems‎. ‎Finally‎, ‎we give an example using bivariate Chlodowsky-Szász-Kantorovich-Charlier-type operators and outline possible further extensions and improvements‎, ‎in order to illustrate the effectiveness of the proposed methods‎.

‎For a constant $K\geq 1$‎, ‎let $\mathfrak{B}_K$ be the class of pairs $(X,(\mathbf e_n)_{n\in\omega})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(\mathbf e_n)_{n\in\omega}$ for $X$‎, ‎having the unconditional basic constant $K_u\le K$‎. ‎Such pairs are called $K$-based Banach spaces‎. ‎A based Banach space $X$ is rational if the unit ball of any finite-dimensional subspace spanned by finitely many basic vectors is a polyhedron whose vertices have rational coordinates in the Schauder basis of $X$‎.

Using the technique of Fraïssé theory‎, ‎we construct a rational $K$-based Banach space $\big(\mathbb U_K,(\mathbf e_n)_{n\in\omega}\big)$ which is $\mathfrak{RI}_K$-universal in the sense that each basis preserving isometry $f:\Lambda\to\mathbb U_K$ defined on a based subspace $\Lambda$ of a finite-dimensional rational $K$-based Banach space $A$ extends to a basis preserving isometry $\bar f:A\to\mathbb U_K$ of the based Banach space $A$‎. ‎We also prove that the $K$-based Banach space $\mathbb U_K$ is almost $\mathfrak{FI}_1$-universal in the sense that any base preserving‎ ‎$\varepsilon$-isometry $f:\Lambda\to\mathbb U_K$ defined on a based subspace $\Lambda$ of a finite-dimensional $1$-based Banach space $A$ extends to a base preserving $\varepsilon$-isometry $\bar f:A\to\mathbb U_K$ of the based Banach space $A$‎. ‎On the other hand‎, ‎we show that no almost $\mathfrak{FI}_K$-universal based Banach space exists for $K>1$‎.‎

The Banach space $\mathbb U_K$ is isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional Schauder basis‎, ‎constructed by Pełczyński in 1969‎.

‎‎Let $\mathcal{U}$ be a unitary system and let $\mathcal{B(U)}$ be the Bessel vector space for $\mathcal{U}$‎. ‎In this article‎, ‎we give a characterization of Bessel vector spaces and local commutant spaces at different complete frame vectors‎. ‎The relation between local commutant spaces at different complete frame vectors is investigated‎. ‎Moreover‎, ‎by introducing multiplication and adjoint on the Bessel vector space for a unital semigroup of unitary operators‎, ‎we give a $C^*$-algebra structure to $\mathcal{B(U)}$‎. ‎Then‎, ‎we construct some subsets of $K$-frame vectors that have a Banach space or Banach algebra structure‎. ‎Also‎, ‎as a consequence‎, ‎the set of complete frame vectors for different unitary systems contains Banach spaces or Banach algebras‎. ‎In the end‎, ‎we give several characterizations of $K$-frame generator multipliers and Parseval $K$-frame generator multipliers‎.

‎In this article‎, ‎the authors consider the Schrödinger type‎ ‎operator $L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$‎, ‎where the matrix $A$ is symmetric and satisfies‎ ‎the uniformly elliptic condition and the nonnegative potential‎ ‎$V$ belongs to the reverse Hölder class $RH_q(\mathbb{R}^n)$‎ ‎with $q\in(n/2,\,\infty)$‎. ‎Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$ be a variable exponent function‎ ‎satisfying the globally $\log$-Hölder continuous condition‎. ‎The authors introduce the variable Hardy space $H_L^{p(\cdot)}(\mathbb{R}^n)$ associated to $L$‎ ‎and establish its atomic characterization‎. ‎The atoms here are closer to the atoms of‎ ‎variable Hardy space $H^{p(\cdot)}(\mathbb{R}^n)$ in spirit‎, ‎which further implies that $H^{p(\cdot)}(\mathbb{R}^n)$ is continuously embedded in‎ ‎$H_L^{p(\cdot)}(\mathbb{R}^n)$‎.

‎ ‎In this paper‎, ‎we introduce the Itô-McShane integral and show that the classical Itô integral of an operator-valued stochastic process with respect to a Hilbert space-valued $Q$-Wiener process can be defined‎, ‎using the Itô-McShane integral‎.

In simple unital purely infinite $C^*$-algebra $A$‎, ‎Leen proved that any element in the identity component of the invertible group is‎ ‎a finite product of symmetries of $A$‎. ‎Revising Leen's factorization‎, ‎we show that a multiple of eight of such factors are $*$-symmetries of the form $1-2P_{i,j}(u)$‎, ‎where $P_{i,j}(u)$ are certain projections of the $C^*$-matrix algebra‎, ‎defined by Dye as‎ ‎\begin{equation*}‎ ‎P_{i,j}(u) = \frac{1}{2}(e_{i,i}+e_{j,j}‎ ‎+e_{i,1}ue_{1,j}+e_{j,1}u^*e_{1,i}),‎ ‎\end{equation*}‎ ‎for a given system of matrix units $\{e_{i,j}\}_{i,j=1}^n$ of $A$ and a unitary $u\in \mathcal{U}(A)$‎. ‎

‎In this paper‎, ‎we introduce new Hadamard type operators “with respect to another function”‎. ‎Some properties of the introduced operators are proved and some applications are discussed‎. ‎For our results‎, ‎some recent results related to Hadamard operators are deduced as some special cases‎.

‎The aim of this paper is to introduce a class of operators acting on a complex Hilbert space‎. ‎This class contains‎, ‎among others‎, ‎nonzero compact operators‎. ‎We give a characterization of this class in term of generalized numerical ranges and deduce that if $A$ is a compact operator‎, ‎then $ w(A)=\vert \lambda \vert $ with $ \lambda \in\mathit W(A) $‎, ‎where $ \mathit W(A)$ and $ w(A) $ are the numerical range and the numerical radius of $ A $‎, ‎respectively‎. ‎We will give some new necessary conditions for an operator to be compact‎. ‎We also show some light on the generalized numerical ranges of the elementary operators $\delta_{2,A,B}$ and $\mathcal{M}_{2,A,B}$‎.

‎The notion of multipolynomials was recently introduced and explored by T‎. ‎Velanga [Linear Multilinear Algebra‎. ‎66 (2018)‎, ‎no‎. ‎11‎, ‎2328-2348] as an attempt to encompass the theories of polynomials and multilinear operators‎. ‎In the present paper‎, ‎we push this subject further‎, ‎by proving Hardy-Littlewood inequalities for multipolynomials and‎, ‎en passant‎, ‎a variant of the Kahane-Salem-Zygmund inequality in this framework‎.

‎‎‎We develop new methods to study the zero product problem and the commutator of Toeplitz operators on the Fock space with harmonic symbols‎. ‎Our method gives us new results on the zero product problem and the commutator of Toeplitz operators on the Fock space‎. ‎We also extend some known result on the Bergman space setting to the Fock space‎.