Preservers of Generalized Numerical Ranges

This dissertation, "Preservers of Generalized Numerical Ranges" by Kong, Chan, 陳鋼, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Let B(H) denote the C DEGREES*-algebra of all bounded linear operators on a complex Hilbert space H. For A ∈ B(H) and c = 〖(c1, . . ., cn)〗 DEGREESt ∈ C DEGREESn with n being a positive integer such that n W_e (A)= {∑_(i=1) DEGREESn▒c_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)〗 DEGREESt, 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)〗 DEGREESt ∈ R DEGREESn. 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 DEGREESn is the set of n-tuples of bounded operators A = (A1, . . ., An), with each Ai ∈ V. The joint numerical radius of A is defined by w(A) = sup{(⟨A1x, x⟩, . . ., ⟨Anx, x⟩)∥ x ∈ H, ∥x∥ = 1}, where ∥ - ∥ is the usual Euclidean norm on F DEGREESn with F = C or R. When H is infinite-dimensional, surjective linear maps T: V DEGREESn→V DEGREESn satisfying w(T(A)) = w(A) for all A ∈ V DEGREESn 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⟩ DEGREES2 + ⟨Ax, Ax⟩ DEGREES2): 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. DOI: 10.5353/th_b5066218 Subjects: Linear operators Matrices