On linear maps preserving spectrum, spectral radius and essential spectral radius

Other Title(s)

حول الدوال الخطية الحافظة للطيف، لنصف قطر الطيف و لنصف قطر الطيف الأساسي

Dissertant

al-Sirr, Huda Abd al-Karim Diyab

Thesis advisor

Asad, Yusuf Asad

Comitee Members

Safi, Ziyad Rizq
al-Mabhuh, Ahmad Abd al-Rauf

University

Islamic University

Faculty

Faculty of Science

Department

Department of Mathematics

University Country

Palestine (Gaza Strip)

Degree

Master

Degree Date

2016

English Abstract

In this thesis, we focus our study on a part of the linear maps on algebras of operators that preserve certain properties of operators.

We study linear maps preserving spectrum.

Let X and Y be Banach spaces.

We show that a spectrum preserving surjective linear map ϕ from B(X) to B(Y) is either of the form ϕ(T) = ATA-1 for an isomorphism A of X onto Y or the form ϕ(T) = BT×B-1 for an isomorphism B of X/ onto Y.

After this, we study linear maps preserving the spectral radius.

Let X be a complex Banach space.

We show that if ϕ: B(X) ⟶ B(X) is a surjective linear map such that T and ϕ(T) have the same spectral radius for every T ∈ B(X), then ϕ = cθ where θ is either an algebra-automorphism or an antiautomorphism of B(X) and c is a complex constant such that |c|=1.

Finally, we characterize linear maps from B(H) onto itself that preserve the essential spectral radius.

Where B(H) is the algebra of all bounded linear operators on an infinite dimensional complex Hilbert space H.

Main Subjects

Mathematics

No. of Pages

65

Table of Contents

• APA
• MLA
• AMA

Language

English

Data Type

Arab Theses

Record ID

BIM-724356