Protecting points from operator pencils (original) (raw)

Some spectral sets of linear operators pencils on non-archimedean Banach spaces

Bulletin of the Transilvania University of Brasov. Series III: Mathematics and Computer Science

In this paper, we define the notions of trace pseudo-spectrum, ε−determinant spectrum, and ε−trace of bounded linear operator pencils on non-Archimedean Banach spaces. Many results are proved about trace pseudo-spectrum, ε−determinant spectrum, and ε−trace of bounded linear operator pencils on non-Archimedean Banach spaces. Examples are given to support our work.

On the spectrum of linear operator pencils

Matematičnì studìï, 2019

We consider a linear operator pencil L(λ) = A − λB, λ ∈ C, where A and B are bounded operators on Hilbert space. The purpose of this paper is to study the conditions under which the spectrum of L(.) is the whole complex plane or empty. This leads to some criteria for the spectrum to be bounded.

On a conjecture about spectral sets

Topology and Its Applications, 2004

Let R be a commutative ring with identity. We denote by Spec(R) the set of prime ideals of R. Call a partial ordered set spectral if it is order isomorphic to (Spec(R), ⊆) for some R. A longstanding open question about spectral sets (since 1976), is that of Lewis and Ohm [Canad. J. Math. 28 (1976) 820, Question 3.4]: "If (X, ) is an ordered disjoint union of the posets (X λ , λ ), λ ∈ Λ, and if (X, ) is spectral, then are the (X λ , λ ) also spectral?".

On Real and Complex Spectra in some Real C*-Algebras and Applications

Zeitschrift für Analysis und ihre Anwendungen, 1999

A real extension A of a complex C-algebra A by some element rn which has a number of special properties is proposed. These properties allow us to introduce some suitable operations of addition, multiplication and involution on A. After then we are able to study Moore-Penrose invertibility in A. Because this notion strongly depends on the element m, we study under what conditions different elements m produce just the same involution on A. It is shown that the set of all additive continuous operators £aa(fl) acting in a complex Hilbert space fl possesses unique involution only (in the sense defined below). In addition, we consider some properties of the real and complex spectra of elements belonging to A, and show that whenever an operator sequence {A,} C £aa(fl) is weakly asymptotically Moore-Penrose invertible, then the real spectrum of AA can be split in two special parts. This property has been earlier known for sequences of linear operators.

Isolated Points of Spectrum for Quasi - * -Class A Operators

Let T be a Quasi - * -class A operator on a complex Hilbert space H if T * (|T 2 | − |T * | 2 )T ≥ 0. In this paper, we prove that if E is the Riesz idempotent for a non-zero isolated point λ of the spectrum of T ∈ Quasi - *class A operator, then E is self-adjoint and EH = ker(T − λ) = ker (T − λ) * .

SP ] 6 S ep 2 01 8 On universal realizability of spectra . ∗

2018

A list Λ = {λ1, λ2, . . . , λn} of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. The list Λ is said to be universally realizable (UR) if it is the spectrum of a nonnegative matrix for each possible Jordan canonical form allowed by Λ. It is well known that an n×n nonnegative matrix A is co-spectral to a nonnegative matrix B with constant row sums. In this paper, we extend the co-spectrality between A and B to a similarity between A and B, when the Perron eigenvalue is simple. We also show that if ǫ ≥ 0 and Λ = {λ1, λ2, . . . , λn} is UR, then {λ1 + ǫ, λ2, . . . , λn} is also UR. We give counter-examples for the cases: Λ = {λ1, λ2, . . . , λn} is UR implies {λ1 + ǫ, λ2 − ǫ, λ3, . . . , λn} is UR, and Λ1,Λ2 are UR implies Λ1 ∪ Λ2 is UR.

Note on spectra of non-selfadjoint operators over dynamical systems

We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.