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postgraduate thesis: Preservers of generalized numerical ranges

TitlePreservers of generalized numerical ranges
Authors
Advisors
Advisor(s):Chan, JT
Issue Date2013
PublisherThe 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
AbstractLet 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 c-numerical range and c-numerical 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 self-adjoint operators in B(H). Suppose V^n is the set of n-tuples of bounded operators  = (A1, . . . ,An), with each Ai ∈ V. The joint numerical radius of  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 infinite-dimensional, surjective linear maps T : V^n→V^n satisfying w(T(Â)) = w(Â) for all  ∈ V^n were characterized. Another generalization of the numerical range is the Davis-Wielandt shell. For A ∈ B(H), its Davis-Wielandt shell is DW(A) = {(⟨Ax, x⟩, ⟨Ax, Ax⟩): x ∈ H and∥x∥= 1}. Define the Davis-Wielandt 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 Davis-Wielandt 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 re-proved using elementary operator theory techniques.
DegreeDoctor of Philosophy
SubjectLinear operators.
Matrices.
Dept/ProgramMathematics
Persistent Identifierhttp://hdl.handle.net/10722/191193

 

DC FieldValueLanguage
dc.contributor.advisorChan, JT-
dc.contributor.authorChan, Kong.-
dc.contributor.author陳鋼.-
dc.date.accessioned2013-09-30T15:52:28Z-
dc.date.available2013-09-30T15:52:28Z-
dc.date.issued2013-
dc.identifier.citationChan, 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.urihttp://hdl.handle.net/10722/191193-
dc.description.abstractLet 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 c-numerical range and c-numerical 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 self-adjoint operators in B(H). Suppose V^n is the set of n-tuples of bounded operators  = (A1, . . . ,An), with each Ai ∈ V. The joint numerical radius of  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 infinite-dimensional, surjective linear maps T : V^n→V^n satisfying w(T(Â)) = w(Â) for all  ∈ V^n were characterized. Another generalization of the numerical range is the Davis-Wielandt shell. For A ∈ B(H), its Davis-Wielandt shell is DW(A) = {(⟨Ax, x⟩, ⟨Ax, Ax⟩): x ∈ H and∥x∥= 1}. Define the Davis-Wielandt 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 Davis-Wielandt 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 re-proved using elementary operator theory techniques.-
dc.languageeng-
dc.publisherThe University of Hong Kong (Pokfulam, Hong Kong)-
dc.relation.ispartofHKU Theses Online (HKUTO)-
dc.rightsThe author retains all proprietary rights, (such as patent rights) and the right to use in future works.-
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.source.urihttp://hub.hku.hk/bib/B50662181-
dc.subject.lcshLinear operators.-
dc.subject.lcshMatrices.-
dc.titlePreservers of generalized numerical ranges-
dc.typePG_Thesis-
dc.identifier.hkulb5066218-
dc.description.thesisnameDoctor of Philosophy-
dc.description.thesislevelDoctoral-
dc.description.thesisdisciplineMathematics-
dc.description.naturepublished_or_final_version-
dc.identifier.doi10.5353/th_b5066218-
dc.date.hkucongregation2013-

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