Linear Maps on Operator Spaces Preserving Spectrum and Numerical Range
Let us denote by H a complex Hilbert space. We will consider the following operator spaces:
- B(H) is the complex linear space of all bounded operators on H,
- H(H) is the real linear space of all self-adjoint bounded operators on H,
- B_{K}(H) is the complex linear space of all operators on H which have the form α id_{H} + K, where α is a scalar coefficient and K is a compact operator,
- H_{K}(H) = B_{K}(H) cap H(H) (it is a real linear space).
We denote by σ(X) the spectrum of an operator X in B(H), and define the numerical range of X by W(X) = {<Xv,v> : v ∈ H, <v,v> = 1}, where <⋅,-> is the inner product on H.
The main goal of the talk is to present a proof (based on the original paper) of the following theorem [1].
Theorem 1. [Chi-Kwong Li, Rodman, v Semrl]
Let V be one of the above operator spaces and let ε: V→V be a surjective linear map. Then the following conditions are equivalent
(1) ε maps the set of self-adjoint elements into itself, and σ(ε(X)) = σ(X) for all self-adjoint X in V,
(2) ε is continuous (in the norm topology), maps the set of self-adjoint elements into itself, and is such that σ(ε(X)) = σ(X) for all orthogonal projections X in V,
(3) W(ε(X)) = W(X) for all X in V,
(4) W(ε(X)) = W(X) for all X in V,
(5) there is a unitary operator U in B(H) such that varphi is of the form X ↦ UXU* or X ↦ UX^{t}U*, where X^{t} is the transpose of X with respect to some fixed orthonormal basis in H.
At the end of the talk we will present a generalization of the theorem [2].
The bibliography:
[1] Chi-Kwong Li, L. Rodman, P. v Semrl, Linear Maps on Selfadjoint Operators. Preserving Invertibility, Positive Definiteness, Numerical Range, Canad. Math. Bull. 46 (2): 216-228 (2003).
[2] S. Clark, Chi-Kwong Li, J. Mahle, L. Rodman, Linear Preservers of Higher Rank Numerical Ranges and Radii, Linear Multilinear Algebra 57 (5): 503-521 (2009).