AN UNBOUNDED OPERATOR WITH SPECTRUM IN A STRIP AND MATRIX DIFFERENTIAL OPERATORS

Michael Gil’

doi:10.1017/s0004972721000241

Abstract

AbstractLet A and $\tilde A$ be unbounded linear operators on a Hilbert space. We consider the following problem. Let the spectrum of A lie in some horizontal strip. In which strip does the spectrum of $\tilde A$ lie, if A and $\tilde A$ are sufficiently ‘close’? We derive a sharp bound for the strip containing the spectrum of $\tilde A$ , assuming that $\tilde A-A$ is a bounded operator and A has a bounded Hermitian component. We also discuss applications of our results to regular matrix differential operators.

Highlights

Let A and Abe unbounded linear operators on a Hilbert space
In which strip does the spectrum of Alie if ωst(A) is known and Aand A are sufficiently ‘close’? We discuss applications of our results to matrix differential operators
The strip-type operators form a wide class of unbounded operators in a Banach space

Summary

Introduction

Let A and Abe unbounded linear operators on a Hilbert space. We consider the following problem. Following [10, Section 4.1], we will say that an operator A on H is a strip-type operator of height ω (in short, A ∈ Strip(ω)) if σ(A) ⊂ Hω and sup|Im z|≥ω′ Rz(A) < ∞ for all ω′ > ω. Ωst(A) := inf{ω ≥ 0 : A ∈ Strip(ω)} is called the spectral height of A. Let A and Abe strip-type operators on H. We discuss applications of our results to matrix differential operators.

Results

Conclusion