Related papers: Norm attaining operators which satisfy a Bollob\'a…

Let Q and P be the position and momentum operators of a particle in one dimension. It is shown that all compact operators can be approximated in norm by linear combinations of the basic resolvents (aQ + bP - i r)^{-1} for real constants…

An operator $T$ acting on a Hilbert space is called $(\alpha ,\beta)$-normal ($0\leq \alpha \leq 1\leq \beta $) if \begin{equation*} \alpha ^{2}T^{\ast }T\leq TT^{\ast}\leq \beta ^{2}T^{\ast}T. \end{equation*} In this paper we establish…

A criterion on the similarity of a (bounded, linear) operator $T$ on a (complex, separable) Hilbert space $\mathcal H$ in terms of shift-type invariant subspaces of $T$ to a contraction of class $C_{\cdot 0}$ with finite unequal defects is…

We obtain a complete characterization of the norm attainment set of a bounded linear functional on a normed space, in terms of a semi-inner-product defined on the space. Motivated by this result, we further apply the concept of…

We study the Schr\"{o}dinger operators on a non-compact star graph with the Coulomb-type potentials having singularities at the vertex. The convergence of regularized Hamiltonians $H_\varepsilon$ with cut-off Coulomb potentials coupled with…

Let X be a proper metric space, which has finite asymptotic dimension in the sense of Gromov (or more generally, straight finite decomposition complexity of Dranishnikov and Zarichnyi). New descriptions are provided of the Roe algebra of X:…

The notion of a regular operator with compact supports between function spaces is introduced. On that base we obtain a characterization of absolute extensors for zero-dimensional spaces in terms of regular extension operators having compact…

In this paper, first we study surjective isometries (not necessarily linear) between completely regular subspaces $A$ and $B$ of $C_0(X,E)$ and $C_0(Y,F)$ where $X$ and $Y$ are locally compact Hausdorff spaces and $E$ and $F$ are normed…

Given an operator $T:U_X(\Sigma)\to Y$ or ${T:U(\Sigma)\to Y$, one may consider the net of conditional expectation operators $(T_\pi)$ directed by refinement of the partitions $\pi$. It has been shown previously that $(T_\pi)$ does not…

Let $A$ be a $C^{*}$ algebra and $T: A\rightarrow A$ be a linear map which satisfies the functional equation $\begin{cases}T(x)T(y)=T^{2}(xy)\\T(x^{*})=T(x)^{*} \end{cases}$ We prove that under each of the following conditions, $T$ must be…

Consider a complex Hilbert space $\left(\mathcal{H}, \langle \cdot, \cdot \rangle\right)$ equipped with a positive bounded linear operator $A$ on $\mathcal{H}$. This induces a semi-norm $\|\cdot\|_A$ through the semi-inner product $\langle…

We study idempotent, model, and Toeplitz operators that attain the norm. Notably, we prove that if $\mathcal{Q}$ is a backward shift invariant subspace of the Hardy space $H^2(\mathbb{D})$, then the model operator $S_{\mathcal{Q}}$ attains…

We study approximate Birkhoff-James orthogonality of bounded linear operators defined between normed linear spaces $\mathbb{X}$ and $\mathbb{Y}.$ As an application of the results obtained, we characterize smoothness of a bounded linear…

For any Ritt operator T:L^{p}(\Omega) --> L^{p}(\Omega), for any positive real number \alpha, and for any x in L^{p}, we consider the square functions |x |_{T,\alpha} = \Bigl| \Bigl(\sum_{k=1}^{\infty} k^{2\alpha -1}\bigl…

Given a contraction A on a Hilbert space H, an operator T on H is said to be A-invariant if <Tx,x>=<TAx,Ax> for every x in H such that ||Ax||=||x||. In the special case in which both defect indices of A are equal to 1, we show that every…

Given a bounded normal operator $A$ in a Hilbert space and a fixed vector $x$, we elaborate on the problem of finding necessary and sufficient conditions under which $(A^kx)_{k\in\mathbb N}$ constitutes a Bessel sequence. We provide a…

Projection operators arise naturally as one-particle density operators associated to Slater determinants in fields such as quantum mechanics and the study of determinantal processes. In the context of the semiclassical approximation of…

Let $(X,+,d)$ be an Abelian metric group and $A\subset X$. We investigate the spectre of a set $A$, defined as the set of all elements $z\in X$ such that for every $x\in A$ either $x+z \in A$ or $x-z \in A$. We consider the corresponding to…

Let $(\mathbb{X} , d, \mu )$ be a proper metric measure space and let $\Omega \subset \mathbb{X}$ be a bounded domain. For each $x\in \Omega$, we choose a radius $0< \varrho (x) \leq \mathrm{dist}(x, \partial \Omega ) $ and let $B_x$ be the…

We study expansions of Hilbert spaces with a bounded normal operator $T$. We axiomatize this theory in a natural language and identify all of its completions. We prove the definability of the adjoint $T^*$ and prove quantifier elimination…