Spectrum localization of a perturbed operator in a strip and applications

Let \(A\) and \(\tilde{A}\) be bounded operators in a Hilbert space. We consider the following problem: let the spectrum of \(A\) lie in some strip. In what strip the spectrum of \(\tilde{A}\) lies if \(A\) and \(\tilde{A}\) are "close"? Applications of the obtained results to integral operators and matrices are also discussed. In addition, we apply our perturbation results to approximate the spectral strip of a Hilbert-Schmidt operator by the spectral strips of finite matrices.

Simultaneous approximation of Birkhoff interpolation and the associated sharp inequalities

This paper gives a kind of sharp simultaneous approximation error estimation of Birkhoff interpolation [Formula: see text], [Formula: see text] where [Formula: see text] and [Formula: see text] is the Birkhoff interpolation based on [Formula: see text] pairs of numbers [Formula: see text] with its P[Formula: see text]lya interpolation matrix to be regular. First, based on the integral remainder formula of Birkhoff interpolation, we refer the computation of [Formula: see text] to the norm of an integral operator. Second, we refer the values of [Formula: see text] and [Formula: see text] to two explicit integral expressions and the value of [Formula: see text] to the computation of the maximum eigenvalue of a Hilbert–Schmidt operator. At the same time, we give the corresponding sharp Wirtinger inequality [Formula: see text] and sharp Picone inequality [Formula: see text].

Some consequences of quasicentral approximate units modulo Hilbert-Schmidt class

Conjecture 4 of Voiculescu implies that almost normal operators must satisfy a Fuglede-Putnam theorem, namely [T∗, X] is a Hilbert-Schmidt operator whenever [T, X] is in the same class for an arbitrary operator X. In this note, a partial answer to this question is given, namely when X ∈ 𝓒4, the Fuglede-Putnam theorem holds.

Homogeneous robin boundary conditions and discrete spectrum of fractional eigenvalue problem

We discuss a fractional eigenvalue problem with the fractional Sturm-Liouville operator mixing the left and right derivatives of order in the range (1/2, 1], subject to a variant of Robin boundary conditions. The considered differential fractional Sturm-Liouville problem (FSLP) is equivalent to an integral eigenvalue problem on the respective subspace of continuous functions. By applying the properties of the explicitly calculated integral Hilbert-Schmidt operator, we prove the existence of a purely atomic real spectrum for both eigenvalue problems. The orthogonal eigenfunctions’ systems coincide and constitute a basis in the corresponding weighted Hilbert space. An analogous result is obtained for the reflected fractional Sturm-Liouville problem.

Fuglede–Putnam type theorems via the generalized second Aluthge transform

Let [Formula: see text] and [Formula: see text] be the polar decompositions of [Formula: see text] and [Formula: see text]. A pair [Formula: see text] is said to have the FP-property if [Formula: see text] implies [Formula: see text] for any [Formula: see text]. Let [Formula: see text] denote the generalized second Aluthge transform of a bounded linear operator [Formula: see text] such that [Formula: see text] is the polar decomposition of [Formula: see text] where [Formula: see text] denotes the first Aluthge transform of operator [Formula: see text]. We show that (i) if [Formula: see text] is class [Formula: see text] and [Formula: see text] is invertible class [Formula: see text] operators with [Formula: see text] such that [Formula: see text] for some Hilbert Schmidt operator [Formula: see text], then [Formula: see text]; (ii) if [Formula: see text] for any [Formula: see text], then [Formula: see text] for any [Formula: see text], furthermore, if [Formula: see text] is invertible, then [Formula: see text]. Finally, if [Formula: see text] and [Formula: see text] and [Formula: see text] is an operator such that [Formula: see text], then we prove that [Formula: see text] for any [Formula: see text] such that [Formula: see text].

Resolvents of functions of operators with Hilbert-Schmidt hermitian components

Let H be a separable Hilbert space with the unit operator I. We derive a sharp norm estimate for the operator function (?I-f(A))-1 (? ? C), where A is a bounded linear operator in H whose Hermitian component (A- A*)/2i is a Hilbert-Schmidt operator and f(z) is a function holomorphic on the convex hull of the spectrum of A. Here A* is the operator adjoint to A. Applications of the obtained estimate to perturbations of operator equations, whose coefficients are operator functions and localization of spectra are also discussed.

A BOUND FOR SIMILARITY CONDITION NUMBERS OF UNBOUNDED OPERATORS WITH HILBERT–SCHMIDT HERMITIAN COMPONENTS

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ be a linear unbounded operator in a Hilbert space. It is assumed that the resolvent of $H$ is a compact operator and $H-H^*$ is a Hilbert–Schmidt operator. Various integro-differential operators satisfy these conditions. It is shown that $H$ is similar to a normal operator and a sharp bound for the condition number is suggested. We also discuss applications of that bound to spectrum perturbations and operator functions.