In this article, we establish some upper bounds for numerical radius inequalities, including those of $2\times 2$ operator matrices and their off-diagonal parts. Among other inequalities, it is shown that if , then $${\omega }^r\left(T\right)\le 2^{r-2}\Vert f^{2r}\left(\right|X\left|\right)+g^{2r}\left(\right|Y^{\ast }\left|\right){\Vert }^{\frac{1}{2}}\Vert f^{2r}\left(\right|Y\left|\right)+g^{2r}\left(\right|X^{\ast }\left|\right){\Vert }^{\frac{1}{2}}$$ and $${\omega }^r\left(T\right)\le 2^{r-2}\Vert f^{2r}\left(\right|X\left|\right)+f^{2r}\left(\right|Y^{\ast }\left|\right){\Vert }^{\frac{1}{2}}\Vert g^{2r}\left(\right|Y\left|\right)+g^{2r}\left(\right|X^{\ast }\left|\right){\Vert }^{\frac{1}{2}},$$ where $X,Y$ are bounded linear operators on a Hilbert space $\mathcal{H}$, $r\ge 1$, and $f$, $g$ are nonnegative continuous functions on $[0,\infty )$ satisfying the relation $f\left(t\right)g\left(t\right)=t$ ($t\in [0,\infty )$). Moreover, we present some inequalities involving the generalized Euclidean operator radius of operators $T_1,\dots ,T_n$.

In this article we use the atomic decomposition of a Herz-type Hardy space of variable smoothness and integrability to obtain the boundedness of the central Calderón–Zygmund operators on Herz-type Hardy spaces with variable smoothness and integrability.

In this article, we give equivalent conditions for the hypercyclicity of bilateral operator-weighted shifts on $L^2\left(\mathcal{K}\right)$ with weight sequence $\{A_n{\}}_{n=-\infty }^{\infty }$ of positive invertible diagonal operators on a separable complex Hilbert space $\mathcal{K}$, as well as for hereditarily hypercyclicity and supercyclicity.

In this article, we investigate the surjective linear isometries between the differentiable function spaces $C_0^p(X,E)$ and $C_0^q(Y,F)$, where $X$, $Y$ are open subsets of $\mathbb{R}$ and $E$, $F$ are strictly convex Banach spaces with dimension greater than $1$. We show that such isometries can be written as weighted composition operators.

The main concern of this article is the perturbation problem for outer inverses of linear bounded operators in Banach spaces. We consider the following perturbed problem. Let $T\in B(X,Y)$ with an outer inverse $T^{\left\{2\right\}}\in B(Y,X)$ and $\delta T\in B(X,Y)$ with $\Vert \delta T{T}^{\left\{2\right\}}\Vert <1$. What condition on the small perturbation $\delta T$ can guarantee that the simplest possible expression $B=T^{\left\{2\right\}}(I+\delta T{T}^{\left\{2\right\}}{)}^{-1}$ is a generalized inverse, Moore–Penrose inverse, group inverse, or Drazin inverse of $T+\delta T$? In this article, we give a complete solution to this problem. Since the generalized inverse, Moore–Penrose inverse, group inverse, and Drazin inverse are outer inverses, our results extend and improve many previous results in this area.

We axiomatize and study the matrices of type $H\in M_N\left(A\right)$ having unitary entries $H_{ij}\in U\left(A\right)$ and whose rows and columns are subject to orthogonality-type conditions. Here $A$ can be any $C^{\ast }$-algebra, for instance $A=\mathbb{C}$, where we obtain the usual complex Hadamard matrices, or $A=C\left(X\right)$, where we obtain the continuous families of complex Hadamard matrices. Our formalism allows the construction of a quantum permutation group $G\subset S_N^{+}$, whose structure and computation are discussed here.

In this article, we prove that the tensor product of two hyperrigid operator systems is hyperrigid in the spatial tensor product of $C^{\ast }$-algebras. We deduce this by establishing that the unique extension property for unital completely positive maps on operator systems carry over to tensor products such maps defined on the tensor product operator systems. Hopenwasser’s result about the tensor product of boundary representations follows as a special case. We also provide examples to illustrate the hyperrigidity property of tensor products of operator systems.

Wiener-type variation spaces, also known as ${BV}_p$-spaces ($1\le p<\infty$), are complete normed linear spaces. A function $g$ is called a multiplier from ${BV}_p$ to ${BV}_q$ if the pointwise multiplication $fg$ belongs to ${BV}_q$ for each $f\in {BV}_p$. In this article, we characterize the multipliers from ${BV}_p$ to ${BV}_q$ for the cases $1\le q<p$ and $1\le p\le q$.

In this article, we consider some Jensen-type inequalities for Lipschitzian maps between Banach spaces and functions defined by power series. We obtain as applications some inequalities of Levinson type for Lipschitzian maps. Applications for functions of norms in Banach spaces are provided as well.

We give sharp conditions for boundedness of Hausdorff operators on certain modulation and Wiener amalgam spaces.

In this article, we give a boundedness criterion for Cauchy singular integral operators in generalized weighted grand Lebesgue spaces. We establish a necessary and sufficient condition for the couple of weights and curves ensuring boundedness of integral operators generated by the Cauchy singular integral defined on a rectifiable curve. We characterize both weak and strong type weighted inequalities. Similar problems for Calderón–Zygmund singular integrals defined on measured quasimetric space and for maximal functions defined on curves are treated. Finally, as an application, we establish existence and uniqueness, and we exhibit the explicit solution to a boundary value problem for analytic functions in the class of Cauchy-type integrals with densities in weighted grand Lebesgue spaces.

In this article, we completely describe the perturbation class, the commuting perturbation class, and the topological interior of the class of all bounded linear algebraic operators. As applications, we also focus on the stability of the essential ascent spectrum and the essential descent spectrum under finite-rank perturbations.