Related papers: The Taylor joint spectrum and restriction to hyper…

Let $\mathcal{H}$ be a right quaternionic Hilbert space and let $T$ be a quaternionic normal operator with the domain $\mathcal{D}(T) \subset \mathcal{H}$. Then for a fixed unit imaginary quaternion $m$, there exists a Hilbert basis…

Let $T$ be an absolutely continuous polynomially bounded operator, and let $\theta$ be a singular inner function. It is shown that if $\theta(T)$ is invertible and some additional conditions are fulfilled, then $T$ has nontrivial…

This paper explores the Invariant Subspace Problem in operator theory and functional analysis, examining its applications in various branches of mathematics and physics. The problem addresses the existence of invariant subspaces for bounded…

A closed densely defined operator $ T $ on a Hilbert space $ \mathcal{H} $ is callled $M$-hyponormal if $\mathcal{D}(T) \subset \mathcal{D}(T^{*}) $ and there exists $ M > 0 $ for which $ \parallel(T-zI)^{*}x \parallel \leq M…

Let $T:D(T)\rightarrow H_2$ be a densely defined closed operator with domain $D(T)\subset H_1$. We say $T$ to be absolutely minimum attaining if for every closed subspace $M$ of $H_1$, the restriction operator $T|_M:D(T)\cap M\rightarrow…

Joint spectra of tuples of operators are subsets in complex projective space. The corresponding tuple of operators can be viewed as an infinite dimensional analog of a determinantal representation of the joint spectrum. We investigate the…

Let $\mathbf{T} \equiv (T_1,\cdots,T_n)$ be a commuting $n$-tuple of operators on a Hilbert space $\mathcal{H}$, and let $T_i \equiv V_i P \; (1 \le i \le n)$ be its canonical joint polar decomposition (i.e.,…

A commuting tuple of Hilbert space operators $(T_1, \dotsc, T_n)$ is said to be an \textit{$\mathbb{A}_r^n$-contraction} if the closure of the polyannulus \[ \mathbb A_r^n=\left\{(z_1, \dotsc, z_n) \ : \ r<|z_i|<1, \ 1 \leq i \leq n…

We extend the concept of Lifshits--Krein spectral shift function associated with a pair of self-adjoint operators to the case of pairs of admissible operators that are similar to self-adjoint operators. Our main result is the following. Let…

Let $A\colon H\rightarrow H$ be a normal operator on an infinite-dimensional separable Hilbert space $H$ and let $S\subseteq H$ be a finite subset such that $\{A^nx\}_{n\geq 0,\,x\in S}$ can be rescaled to form a frame for $H$. That is,…

We consider the following question: Let $\mathcal{A}$ be an abelian self-adjoint algebra of bounded operators on a Hilbert space $\mathcal{H}$. Assume that $\mathcal{A}$ is invariant under conjugation by a unitary operator $U$, i.e., $U^*…

Closed operators in Hilbert space defined by a non-self-adjoint resolution of the identity $\{X(\lambda)\}_{\lambda\in {\mb R}}$, whose adjoints constitute also a resolution of the identity, are studied . In particular, it is shown that a…

Let $H$ be a separable Hilbert space, $A_c:\mathcal D_c\subset H\to H$ a densely defined unbounded operator, bounded from below, let $\mathcal D_{\min}$ be the domain of the closure of $A_c$ and $\mathcal D_{\max}$ that of the adjoint.…

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 investigate the relation between the spectrum of matrix (or operator) polynomials and the Taylor spectrum of its coefficients. We prove that the polynomial of commuting matrices is singular, i.e. its spectrum is the whole complex plane,…

In this work, we focus on the multiplicity of singular spectrum for operators of the form $A^\omega=A+\sum_{n}\omega_n C_n$ on a separable Hilbert space $\mathcal{H}$, for a self-adjoint operator $A$ and a countable collection $\{C_n\}_{n}$…

For a bounded right linear operators $A$, in a right quaternionic Hilbert space $V_{\mathbb{H}}^{R}$, following the complex formalism, we study the Berberian extension $A^\circ$, which is an extension of $A$ in a right quaternionic Hilbert…

A new geometric proof of the spectral theorem for unbounded self-adjoint operators A in a Hilbert space H is given based on a splitting of A in positive and negative parts A+ and A-. For both operators A+ and A- the spectral family can be…

Let $U$ be a unitary operator defined on some infinite-dimensional complex Hilbert space ${\cal H}$. Under some suitable regularity assumptions, it is known that a local positive commutation relation between $U$ and an auxiliary…

We give new inequalities for $A$-operator seminorm and $A$-numerical radius of semi-Hilbertian space operators and show that the inequalities obtained here generalize and improve on the existing ones. Considering a complex Hilbert space…