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• The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
1

#### The joint numerical range and the joint essential numerical range

Lam, Tsz-mang., 林梓萌. January 2013 (has links)
﻿Let B(H) denote the algebra of bounded linear operators on a complex Hilbert space H. The (classical) numerical range of T ∈ B(H) is the set W(T) = {〈T x; x〉: x ∈ H; ‖x‖ = 1} Writing T= T_1 + iT_2 for self-adjoint T_1, T_2 ∈ B(H), W(T) can be identified with the set {(〈T_1 x, x〉,〈T_2 x, x〉) : x ∈ H, ‖x‖ = 1}. This leads to the notion of the joint numerical range of T= 〖(T〗_1, T_2, …, T_n) ∈ 〖B(H)〗^n. It is defined by W(T) = {(〈T_1 x, x〉,〈T_2 x, x〉, …, 〈T_n x, x〉) : x ∈ H, ‖x‖ = 1}. The joint numerical range has been studied extensively in order to understand the joint behaviour of operators. Let K(H) be the set of all compact operators on a Hilbert space H. The essential numerical range of T ∈ B(H) is defined by W_(e ) (T)=∩{W(T+K) :K∈K(H) }. The joint essential numerical range of T= 〖(T〗_1, T_2, …, T_n) ∈〖 B(H)〗^n is defined analogously by W_(e ) (T)=∩{ /W(T+K) :K∈〖K(H)〗^n }. These notions have been generalized to operators on a Banach space. In Chapter 1 of this thesis, the joint spatial essential numerical range were introduced. Also the notions of the joint algebraic numerical range V(T) and the joint algebraic essential numerical range Ve(T) were reviewed. Basic properties of these sets were given. In 2010, Müller proved that each n-tuple of operators T on a separable Hilbert space has a compact perturbation T + K so that We(T) = W(T + K). In Chapter 2, it was shown that any n-tuple T of operators on lp has a compact perturbation T +K so that Ve(T) = V (T +K), provided that Ve(T) has an interior point. A key step was to find for each n-tuple of operators on lp a compact perturbation and a sequence of finite-dimensional subspaces with respect to which it is block 3 diagonal. This idea was inspired by a similar construction of Chui, Legg, Smith and Ward in 1979. Let H and L be separable Hilbert spaces and consider the operator D_AB=A⨂I_L⨂B on the tensor product space H ⨂▒L. In 1987 Magajna proved that W_(e ) (D_AB )=co[W_(e ) (A)- /(W(B)))∪/W(A) - W_(e ) (B))] by considering quasidiagonal operators. An alternative proof of the equality was given in Chapter 3 using block 3 diagonal operators. The maximal numerical range and the essential maximal numerical range of T ∈ B(H) were introduced by Stampi in 1970 and Fong in 1979 respectively. In 1993, Khan extended the notions to the joint essential maximal numerical range. However the set may be empty for some T ∈ B(H). In Chapter 4, the kth joint essential maximal numerical range, spatial maximal numerical range and algebraic numerical range were introduced. It was shown that kth joint essential maximal numerical range is non-empty and convex. Also, it was shown that the kth joint algebraic maximal numerical range is the convex hull of the kth joint spatial maximal numerical range. This extends the corresponding result of Fong. / published_or_final_version / Mathematics / Master / Master of Philosophy
2

#### On the q-numerical range

published_or_final_version / Mathematics / Master / Master of Philosophy
3

#### Plotting generalized numerical ranges

published_or_final_version / Mathematics / Master / Master of Philosophy
4

#### On the q-numerical range /

Lau, Hing-sang. January 1999 (has links)
Thesis (M. Phil.)--University of Hong Kong, 1998. / Includes bibliographical references (leaves 81-87).
5

#### Plotting generalized numerical ranges /

Cheng, Kam-muk. January 1998 (has links)
Thesis (M. Phil.)--University of Hong Kong, 1999. / Includes bibliographical references (leaves 56-59).
6

#### The L-numerical range : liear image of joint unitary orbit of hermitian matrices /

Chan, Yiu-Kwong. January 2002 (has links)
Thesis (M. Phil.)--University of Hong Kong, 2002. / Includes bibliographical references (leaves 53-56).
7

#### The L-numerical range: liear image of joint unitary orbit of hermitian matrices

published_or_final_version / Mathematics / Master / Master of Philosophy
8

#### On numerical range and its application to Banach algebra.

Sims, Brailey January 1972 (has links)
The spatial numerical range of an operator on a normed linear space and the algebra numerical range of an element of a unital Banach algebra, as developed by G. Lumer and F. F. Bonsall, are considered and the theory of such numerical ranges applied to Banach algebra. The first part of the thesis is largely expository as in it we introduce the basic results on numerical ranges. For an element of a unital Banach algebra, the question of approximating its spectrum by numerical ranges has been considered by F. F. Bonsall and J. Duncan. We give an alternative proof that the convex hull of the spectrum of an element may be approximated by its numerical range defined with respect to equivalent renormings of the algebra. In the particular case of operators on a Hilbert space, this leads to a sharper version of a result by J. P. Williams. An element is hermitian if it has real numerical range. Such an element is characterized in terms of the linear subspace spanned by the unit, the element and its square. This is used to characterize Banach*–algebras in which every self–adjoint element is hermitian. From this an elementary proof that such algebras are B*-algebras in an equivalent norm is given. As indicated by T. W. Palmer, a formula of L. Harris is then used to show that the equivalent renorming is unnecessary, thus giving a simple proof of Palmer's characterization of B*-algebras among Banach algebras. The closure properties of the spatial numerical range are studied. A construction of B. Berberian is extended to normed linear spaces, however because the numerical range need not be convex, the result obtained is weaker than that of Berberian for Hilbert spaces. A Hilbert space or an Lp-space, for p between one and infinity, is seen to be finite dimensional if and only if all the compact operators have closed spatial numerical range. The spatial numerical range of a compact operator, on a Hilbert space or an Lp-space, for p between one and infinity, is shown to contain all the non–zero extreme points of its closure. So, for a compact operator on a Hilbert space the spatial numerical range is closed if and only if it contains the origin, thus answering a question of P. R. Halmos. Operators that attain their numerical radius are also considered. A result of D. Hilbert is extended to a class of Banach spaces. In a Hilbert space the hermitian operators, which attain their numerical radius, are shown to be dense among all the hermitian operators. This leads to a stronger form of a result by J. Lindenstrauss in the spatial case of operators on a Hilbert space. / PhD Doctorate
9

#### On the Numerical Range of Compact Operators

Dabkowski, Montserrat 01 June 2022 (has links) (PDF)
One of the many characterizations of compact operators is as linear operators whichcan be closely approximated by bounded finite rank operators (theorem 25). It iswell known that the numerical range of a bounded operator on a finite dimensionalHilbert space is closed (theorem 54). In this thesis we explore how close to beingclosed the numerical range of a compact operator is (theorem 56). We also describehow limited the difference between the closure and the numerical range of a compactoperator can be (theorem 58). To aid in our exploration of the numerical range ofa compact operator we spend some time examining its spectra, as the spectrum of abounded operator is closely tied to its numerical range (theorem 45). Throughout,we use the forward shift operator and the diagonal operator (example 1) to illustratethe exceptional behavior of compact operators.
10