Related papers: Phillips symmetric operators and their extensions

We prove that if T is an operator on an infinite-dimensional Hilbert space whose spectrum and essential spectrum are both connected and whose Fredholm index is only 0 or 1, then the only nontrivial norm-stable invariant subspaces of T are…

Supersymmetric (SUSY) transformation operators corresponding to complex factorization constants are analyzed as operators acting in the Hilbert space of functions square integrable on the positive semiaxis. Obtained results are applied to…

A commuting triple of Hilbert space operators $(A,S,P)$ is said to be a \textit{$\mathbb{P}$-contraction} if the closed pentablock $\overline{\mathbb P}$ is a spectral set for $(A,S,P)$, where \[ \mathbb{P}:=\left\{(a_{21}, \mbox{tr}(A_0),…

In this paper it is proved that each densely defined $J$-skew-symmetric operator (or each $J$-isometric operator with $\overline{D(A)}=\overline{R(A)}=H$) in a Hilbert space $H$ has a $J$-skew-self-adjoint (respectively $J$-unitary)…

A Hilbert space operator $S\in\B$ is $n$-quasi left $m$-invertible (resp., left $m$-invertible) by $T\in\B$, $m,n \geq 1$ some integers, if $S^{*n}p(S,T)S^n=0$ (resp., $p(S,T)=0$), where…

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…

We consider the indefinite Sturm-Liouville differential expression \[\mathfrak{a}(f) := - \frac{1}{w}\left( \frac{1}{r} f' \right)',\] where $\mathfrak{a}$ is defined on a finite or infinite open interval $I$ with $0\in I$ and the…

The issue of general covariance of spinors and related objects is reconsidered. Given an oriented manifold $M$, to each spin structure $\sigma$ and Riemannian metric $g$ there is associated a space $S_{\sigma, g}$ of spinor fields on $M$…

Given a densely defined and closed operator $A$ acting on a complex Hilbert space $\mathcal{H}$, we establish a one-to-one correspondence between its closed extensions and subspaces $\mathfrak{M}\subset\mathcal{D}(A^*)$, that are closed…

Consider the set of unitary operators on a complex separable Hilbert space $\hilh$, denoted as $\mathcal{U}(\hilh)$. Consider $1<p<\infty$. We establish that a function $f$ defined on the unit circle $\cir$ is $n$ times continuously…

The theory of symmetric, non-selfadjoint operators has several deep applications to the complex function theory of certain reproducing kernel Hilbert spaces of analytic functions, as well as to the study of ordinary differential operators…

We ask the question whether for a given unitary representation $U$ the associated operator $\rho_{U}\in\operatorname{Mor}(U,U^{c\, c})$ has spectrum invariant under inversion -- in this case we say that $\rho_{U}$ has symmetric eigenvalues.…

Let $\mathcal{H}$ be a complex, separable Hilbert space and $\mathcal{B}(\mathcal{H})$ denote the algebra of all bounded linear operators acting on $\mathcal{H}$. Given a unitarily-invariant norm $\| \cdot \|_u$ on…

Let T:=[T_1,..., T_n] be an n-tuple of operators on a Hilbert space such that T is a completely non-coisometric row contraction. We establish the existence of a "one-to-one" correspondence between the joint invariant subspaces under…

We consider in a Hilbert space a self-adjoint operator H and a family Phi=(Phi_1,...,Phi_d) of mutually commuting self-adjoint operators. Under some regularity properties of H with respect to Phi, we propose two new formulae for a time…

If a differential operator $D$ on a smooth Hermitian vector bundle $S$ over a compact manifold $M$ is symmetric, it is essentially self-adjoint and so admits the use of functional calculus. If $D$ is also elliptic, then the Hilbert space of…

Let $L_0$ be a closed symmetric positive definite operator with nonzero defect indices $n_\pm(L_0)$ in a separable Hilbert space ${\mathscr H}$. It determines a family of dynamical systems $\alpha^T$, $T>0$, of the form \begin{align*} &…

We establish a spectral duality for certain unbounded operators in Hilbert space. The class of operators includes discrete graph Laplacians arising from infinite weighted graphs. The problem in this context is to establish a practical…

Let $(G,+)$ be a topological abelian group with a neutral element $e$ and let $\mu : G\longrightarrow\mathbb{C}$ be a continuous character of $G$. Let $(\mathcal{H}, \langle \cdot,\cdot \rangle)$ be a complex Hilbert space and let…

In this note, under a certain assumption on an affine space of operators, which admit embedded eigenvalues, it is shown that the singular part of the spectral shift function of any pair of operators from this space is an integer-valued…