Preservers of generalized numerical ranges

Preservers of generalized numerical ranges

陳鋼(Kong Chan)

香港大學/Doctor of Philosophy/Mathematics/博士(2013年)

並列摘要

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 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.

並列關鍵字

Linear operators. ； Matrices.