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postgraduate thesis: The joint numerical range and the joint essential numerical range
Title  The joint numerical range and the joint essential numerical range 

Authors  
Advisors  Advisor(s):Chan, JT 
Issue Date  2013 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Lam, T. [林梓萌]. (2013). The joint numerical range and the joint essential numerical range. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4985885 
Abstract  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 selfadjoint 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 ntuple 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 ntuple 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 ntuple of operators on lp a compact perturbation and a sequence of finitedimensional 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 nonempty 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. 
Degree  Master of Philosophy 
Subject  Numerical range. 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/181884 
HKU Library Item ID  b4985885 
DC Field  Value  Language 

dc.contributor.advisor  Chan, JT   
dc.contributor.author  Lam, Tszmang.   
dc.contributor.author  林梓萌.   
dc.date.accessioned  20130320T06:29:50Z   
dc.date.available  20130320T06:29:50Z   
dc.date.issued  2013   
dc.identifier.citation  Lam, T. [林梓萌]. (2013). The joint numerical range and the joint essential numerical range. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b4985885   
dc.identifier.uri  http://hdl.handle.net/10722/181884   
dc.description.abstract  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 selfadjoint 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 ntuple 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 ntuple 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 ntuple of operators on lp a compact perturbation and a sequence of finitedimensional 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 nonempty 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.   
dc.language  eng   
dc.publisher  The University of Hong Kong (Pokfulam, Hong Kong)   
dc.relation.ispartof  HKU Theses Online (HKUTO)   
dc.rights  The author retains all proprietary rights, (such as patent rights) and the right to use in future works.   
dc.rights  This work is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License.   
dc.source.uri  http://hub.hku.hk/bib/B49858853   
dc.subject.lcsh  Numerical range.   
dc.title  The joint numerical range and the joint essential numerical range   
dc.type  PG_Thesis   
dc.identifier.hkul  b4985885   
dc.description.thesisname  Master of Philosophy   
dc.description.thesislevel  Master   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b4985885   
dc.date.hkucongregation  2013   