Related papers: On the norm closure problem for complex symmetric …

We study a specific family of symmetric norms on the algebra $\mathcal B(\mathcal H)$ of operators on a separable infinite-dimensional Hilbert space. With respect to each symmetric norm in this family the identity operator fails to attain…

We show that the set of $2$-isometries on an infinite-dimensional Hilbert space is not closed in the strong operator topology. In fact, we prove that its SOT-closure coincides with the set of all expansive operators.

The class of absolutely norming operators on complex Hilbert spaces of arbitrary dimensions was introduced in [6] and a spectral characterization theorem for these operators was established in [11]. In this paper we extend the concept of…

This paper aims to study reducible and irreducible approximation in the set $\textsl{CSO}$ of all complex symmetric operators on a separable, complex Hilbert space $\mathcal H$. When ${\rm dim} \mathcal H=\infty$, it is proved that both…

We give a simple, straightforward proof of the non-hypercyclicity of an arbitrary (bounded or not) normal operator $A$ in a complex Hilbert space as well as of the collection $\left\{e^{tA}\right\}_{t\ge 0}$ of its exponentials, which,…

We describe the norm-closures of the set $\mathfrak{C}_{\mathfrak{E}}$ of commutators of idempotent operators and the set $\mathfrak{E} - \mathfrak{E}$ of differences of idempotent operators acting on a finite-dimensional complex Hilbert…

We investigate $\rho$-orthogonality and its local symmetry in the space of bounded linear operators. A characterization of Hilbert space operators with symmetric numerical range is established in terms of $\rho$-orthogonality. Further, we…

In this paper we consider a problem of the similarity of complex symmetric operators to perturbations of restrictions of normal operators. For a subclass of cyclic complex symmetric operators in a finite-dimensional Hilbert space we prove…

Given a complex, separable Hilbert space $\mathcal{H}$, we characterize those operators for which $\| P T (I-P) \| = \| (I-P) T P \|$ for all orthogonal projections $P$ on $\mathcal{H}$. When $\mathcal{H}$ is finite-dimensional, we also…

In this paper we study the theory of operators on complex Hilbert spaces, which achieve the norm in the unit sphere. We prove important results concerning the characterization of the AN operators, see Definition 1.2. The class of AN…

Suppose $L(H)$ is the space of all bounded linear operators on a complex Hilbert space $H.$ This article deals with the problem of characterizing the extreme contractions of $L(H)$ with respect to the numerical radius norm on $L(H).$ In…

We establish a spectral characterization theorem for the operators on complex Hilbert spaces of arbitrary dimensions that attain their norm on every closed subspace. The class of these operators is not closed under addition. Nevertheless,…

An operator $T \in B(\h)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:\h\to\h$ so that $T = CT^*C$. We provide a concrete description of all complex symmetric partial isometries. In particular, we prove…

We introduce a very simple, exactly solvable PT-symmetric non-Hermitian model with real spectrum, and derive a closed formula for the metric operator which relates the problem to a Hermitian one.

Functional analysis, especially the theory of Hilbert spaces and of operators on these, form an important area in mathematics. We formalized the Isabelle/HOL library Complex_Bounded_Operators containing a large amount of theorems about…

We establish the convergence of pseudospectra in Hausdorff distance for closed operators acting in different Hilbert spaces and converging in the generalised norm resolvent sense. As an assumption, we exclude the case that the limiting…

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a…

We show that (for the weak operator topology) the set of unitary operators on a separable infinite-dimensional Hilbert space is residual in the set of all contractions. The analogous result holds for isometries and the strong operator…

Let $\mathcal{H}$ be a complex, separable Hilbert space, and set $\mathfrak{c}($NIL$_2)=\{ MN - NM : N, M \in \mathcal{B}(\mathcal{H}), M^2 = 0 = N^2 \}$. When $\dim\, \mathcal{H}$ is finite, we characterise the set $\mathfrak{c}($NIL$_2)$…

Let $\mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal…