Linear Operator Pencils

1990 ◽  
pp. 49-58 ◽  
Author(s):  
Israel Gohberg ◽  
Seymour Goldberg ◽  
Marinus A. Kaashoek
Keyword(s):  
Linear Operator ◽  
Operator Pencils

scholarly journals Essential numerical ranges for linear operator pencils

2019 ◽  
Vol 40 (4) ◽  
pp. 2256-2308
Author(s):  
Sabine Bögli ◽  
Marco Marletta
Keyword(s):  
Linear Operator ◽  
Numerical Range ◽  
Projection Methods ◽  
Operator Pencil ◽  
Operator Matrices ◽  
New Concepts ◽  
Block Operator ◽  
Block Operator Matrices ◽  
Spectral Pollution ◽  
Operator Pencils

Abstract We introduce concepts of essential numerical range for the linear operator pencil $\lambda \mapsto A-\lambda B$. In contrast to the operator essential numerical range, the pencil essential numerical ranges are, in general, neither convex nor even connected. The new concepts allow us to describe the set of spectral pollution when approximating the operator pencil by projection and truncation methods. Moreover, by transforming the operator eigenvalue problem $Tx=\lambda x$ into the pencil problem $BTx=\lambda Bx$ for suitable choices of $B$, we can obtain nonconvex spectral enclosures for $T$ and, in the study of truncation and projection methods, confine spectral pollution to smaller sets than with hitherto known concepts. We apply the results to various block operator matrices. In particular, Theorem 4.12 presents substantial improvements over previously known results for Dirac operators while Theorem 4.5 excludes spectral pollution for a class of nonselfadjoint Schrödinger operators which has not been possible to treat with existing methods.

The condition pseudospectrum of a operator pencil

2020 ◽  
pp. 2150057
Author(s):  
Aymen Ammar ◽  
Aref Jeribi ◽  
Kamel Mahfoudhi
Keyword(s):  
Linear Operator ◽  
Operator Pencil ◽  
Linear Operators ◽  
Essential Condition ◽  
The Stability ◽  
Operator Pencils

In this paper, we establish some properties concerning the condition pseudospectrum of the linear operator pencils [Formula: see text] when, [Formula: see text] is not necessarily invertible. Also, we give some results related to the generalized condition pseudospectra and the generalized essential condition pseudospectra of linear operators. We start by studying the stability of these condition pseudospectra and some characterization.

Definitizable Operators on a Krein Space

1995 ◽  
Vol 38 (4) ◽  
pp. 496-506 ◽  
Author(s):  
Petr Zizler
Keyword(s):  
Linear Operator ◽  
Selfadjoint Operator ◽  
Inner Product ◽  
Indefinite Inner Product ◽  
Bounded Linear ◽  
Positive Type ◽  
Approximate Point Spectrum ◽  
Definitizable Operators ◽  
Pontrjagin Space ◽  
Operator Pencils

AbstractLet A be a bounded linear operator on a Hilbert space H. Assume that A is selfadjoint in the indefinite inner product defined by a selfadjoint, bounded, invertible linear operator G on H; [x,y] := (Gx,y). In the first part of the paper we define two orders of neutrality for the pair (G, A) and a connection is made with the "types" of numbers in the point and approximate point spectrum of A. The main results of the paper are in the second part and they deal with strong and uniform definitizability of a bounded selfadjoint operator on a Pontrjagin space. They state:A) Let A be a bounded strongly definitizable operator on a Pontrjagin space ΠK, then A is uniformly definitizable.B) A bounded selfadjoint operator A on a Pontrjagin space ΠK is uniformly definitizable if and only if all the eigenvalues of A are of definite type and all the nonisolated eigenvalues of A are of positive type.Some applications to the theory of linear selfadjoint operator pencils are given.

scholarly journals On the spectrum of linear operator pencils

2019 ◽  
Vol 52 (2) ◽  
Author(s):  
F. Bouchelaghem ◽  
M. Benharrat
Keyword(s):  
Linear Operator ◽  
Operator Pencils
Sign in / Sign up

Export Citation Format

Share Document