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postgraduate thesis: Preservers of generalized numerical ranges
Title  Preservers of generalized numerical ranges 

Authors  
Advisors  Advisor(s):Chan, JT 
Issue Date  2013 
Publisher  The University of Hong Kong (Pokfulam, Hong Kong) 
Citation  Chan, K. [陳鋼]. (2013). Preservers of generalized numerical ranges. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5066218 
Abstract  Let B(H) denote the C^*algebra of all bounded linear operators on a complex Hilbert space H. For A ∈ B(H) and c = 〖(c1, . . . , cn)〗^t ∈ C^n with n being a positive integer such that n ≤ dim H, the cnumerical range and cnumerical radius of A are defined by
W_e (A)= {∑_(i=1)^n▒c_i 〈〖Ax〗_i, x_i 〉 : {x_1, …, x_n } is an orthonormal set in H}
and
W_C (A)={z :z ∈W_(c ) (A)}
respectively. When c = 〖(1, 0, . . . , 0)〗^t, Wc(A) reduces to the classical numerical
range W(A).
Preserver problems concern the characterization of maps between spaces of bounded linear operators that leave invariant certain functions, subsets, or relations etc. In this thesis, several preserver problems related to the numerical range or its generalizations were studied.
For A ∈ B(H), the diameter of its numerical range is
d_w(A) = sup{a  b : a, b ∈ W(A)}.
The first result in this thesis was a characterization of linear surjections on B(H) preserving the diameter of the numerical range, i.e., linear surjections T : B(H) → B(H) satisfying
d_w(T(A)) =d_w(A)
for all A ∈ B(H) were characterized.
Let Mn be the set of n × n complex matrices and Tn the set of upper triangular matrices in Mn. Suppose c = 〖(c1, . . . , cn)〗^t ∈ R^n. When wc(·) is a norm on Mn, mappings T on Mn (or Tn) satisfying
wc(T(A)  T(B)) = wc(A  B)
for all A,B were characterized.
Let V be either B(H) or the set of all selfadjoint operators in B(H). Suppose V^n is the set of ntuples of bounded operators Â = (A1, . . . ,An), with each Ai ∈ V. The joint numerical radius of Â is defined by
w(Â) = sup{(⟨A1x, x⟩, . . . , ⟨Anx, x⟩)∥ : x ∈ H, ∥x∥ = 1},
where ∥ · ∥ is the usual Euclidean norm on F^n with F = C or R. When H is infinitedimensional, surjective linear maps T : V^n→V^n satisfying
w(T(Â)) = w(Â)
for all Â ∈ V^n were characterized.
Another generalization of the numerical range is the DavisWielandt shell. For A ∈ B(H), its DavisWielandt shell is
DW(A) = {(⟨Ax, x⟩, ⟨Ax, Ax⟩): x ∈ H and∥x∥= 1}.
Define the DavisWielandt radius of A by
dw(A) = sup{(√(⟨Ax, x⟩ ^2 + ⟨Ax, Ax⟩ ^2) : x ∈ H and ∥x∥= 1}.
Its properties and relations with normaloid matrices were investigated. Surjective mappings T on B(H) satisfying
dw(T(A)  T(B))= dw(A  B)
for all A,B ∈ B(H) were also characterized.
A characterization of real linear surjective isometries on B(H) by Dang was used to prove the preserver result about the DavisWielandt radius. The result of Dang is proved by advanced techniques and is applicable on a more general setting than B(H). In this thesis, the characterization of surjective real linear isometries on B(H) was reproved using elementary operator theory techniques. 
Degree  Doctor of Philosophy 
Subject  Linear operators. Matrices. 
Dept/Program  Mathematics 
Persistent Identifier  http://hdl.handle.net/10722/191193 
HKU Library Item ID  b5066218 
DC Field  Value  Language 

dc.contributor.advisor  Chan, JT   
dc.contributor.author  Chan, Kong.   
dc.contributor.author  陳鋼.   
dc.date.accessioned  20130930T15:52:28Z   
dc.date.available  20130930T15:52:28Z   
dc.date.issued  2013   
dc.identifier.citation  Chan, K. [陳鋼]. (2013). Preservers of generalized numerical ranges. (Thesis). University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from http://dx.doi.org/10.5353/th_b5066218   
dc.identifier.uri  http://hdl.handle.net/10722/191193   
dc.description.abstract  Let B(H) denote the C^*algebra of all bounded linear operators on a complex Hilbert space H. For A ∈ B(H) and c = 〖(c1, . . . , cn)〗^t ∈ C^n with n being a positive integer such that n ≤ dim H, the cnumerical range and cnumerical radius of A are defined by W_e (A)= {∑_(i=1)^n▒c_i 〈〖Ax〗_i, x_i 〉 : {x_1, …, x_n } is an orthonormal set in H} and W_C (A)={z :z ∈W_(c ) (A)} respectively. When c = 〖(1, 0, . . . , 0)〗^t, Wc(A) reduces to the classical numerical range W(A). Preserver problems concern the characterization of maps between spaces of bounded linear operators that leave invariant certain functions, subsets, or relations etc. In this thesis, several preserver problems related to the numerical range or its generalizations were studied. For A ∈ B(H), the diameter of its numerical range is d_w(A) = sup{a  b : a, b ∈ W(A)}. The first result in this thesis was a characterization of linear surjections on B(H) preserving the diameter of the numerical range, i.e., linear surjections T : B(H) → B(H) satisfying d_w(T(A)) =d_w(A) for all A ∈ B(H) were characterized. Let Mn be the set of n × n complex matrices and Tn the set of upper triangular matrices in Mn. Suppose c = 〖(c1, . . . , cn)〗^t ∈ R^n. When wc(·) is a norm on Mn, mappings T on Mn (or Tn) satisfying wc(T(A)  T(B)) = wc(A  B) for all A,B were characterized. Let V be either B(H) or the set of all selfadjoint operators in B(H). Suppose V^n is the set of ntuples of bounded operators Â = (A1, . . . ,An), with each Ai ∈ V. The joint numerical radius of Â is defined by w(Â) = sup{(⟨A1x, x⟩, . . . , ⟨Anx, x⟩)∥ : x ∈ H, ∥x∥ = 1}, where ∥ · ∥ is the usual Euclidean norm on F^n with F = C or R. When H is infinitedimensional, surjective linear maps T : V^n→V^n satisfying w(T(Â)) = w(Â) for all Â ∈ V^n were characterized. Another generalization of the numerical range is the DavisWielandt shell. For A ∈ B(H), its DavisWielandt shell is DW(A) = {(⟨Ax, x⟩, ⟨Ax, Ax⟩): x ∈ H and∥x∥= 1}. Define the DavisWielandt radius of A by dw(A) = sup{(√(⟨Ax, x⟩ ^2 + ⟨Ax, Ax⟩ ^2) : x ∈ H and ∥x∥= 1}. Its properties and relations with normaloid matrices were investigated. Surjective mappings T on B(H) satisfying dw(T(A)  T(B))= dw(A  B) for all A,B ∈ B(H) were also characterized. A characterization of real linear surjective isometries on B(H) by Dang was used to prove the preserver result about the DavisWielandt radius. The result of Dang is proved by advanced techniques and is applicable on a more general setting than B(H). In this thesis, the characterization of surjective real linear isometries on B(H) was reproved using elementary operator theory techniques.   
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/B50662181   
dc.subject.lcsh  Linear operators.   
dc.subject.lcsh  Matrices.   
dc.title  Preservers of generalized numerical ranges   
dc.type  PG_Thesis   
dc.identifier.hkul  b5066218   
dc.description.thesisname  Doctor of Philosophy   
dc.description.thesislevel  Doctoral   
dc.description.thesisdiscipline  Mathematics   
dc.description.nature  published_or_final_version   
dc.identifier.doi  10.5353/th_b5066218   
dc.date.hkucongregation  2013   