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Search: MSC category 47B49 ( Transformers, preservers (operators on spaces of operators) )

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1. CMB 2004 (vol 47 pp. 369)

Fošner, Ajda; Šemrl, Peter
Spectrally Bounded Linear Maps on ${\cal B}(X)$
We characterize surjective linear maps on ${\cal B}(X)$ that are spectrally bounded and spectrally bounded below.

Keywords:spectrally bounded linear map.
Category:47B49

2. CMB 2003 (vol 46 pp. 216)

Li, Chi-Kwong; Rodman, Leiba; Šemrl, Peter
Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range
Let $H$ be a complex Hilbert space, and $\HH$ be the real linear space of bounded selfadjoint operators on $H$. We study linear maps $\phi\colon \HH \to \HH$ leaving invariant various properties such as invertibility, positive definiteness, numerical range, {\it etc}. The maps $\phi$ are not assumed {\it a priori\/} continuous. It is shown that under an appropriate surjective or injective assumption $\phi$ has the form $X \mapsto \xi TXT^*$ or $X \mapsto \xi TX^tT^*$, for a suitable invertible or unitary $T$ and $\xi\in\{1, -1\}$, where $X^t$ stands for the transpose of $X$ relative to some orthonormal basis. Examples are given to show that the surjective or injective assumption cannot be relaxed. The results are extended to complex linear maps on the algebra of bounded linear operators on $H$. Similar results are proved for the (real) linear space of (selfadjoint) operators of the form $\alpha I+K$, where $\alpha$ is a scalar and $K$ is compact.

Keywords:linear map, selfadjoint operator, invertible, positive definite, numerical range
Categories:47B15, 47B49

3. CMB 2001 (vol 44 pp. 270)

Cheung, Wai-Shun; Li, Chi-Kwong
Linear Operators Preserving Generalized Numerical Ranges and Radii on Certain Triangular Algebras of Matrices
Let $c = (c_1, \dots, c_n)$ be such that $c_1 \ge \cdots \ge c_n$. The $c$-numerical range of an $n \times n$ matrix $A$ is defined by $$ W_c(A) = \Bigl\{ \sum_{j=1}^n c_j (Ax_j,x_j) : \{x_1, \dots, x_n\} \text{ an orthonormal basis for } \IC^n \Bigr\}, $$ and the $c$-numerical radius of $A$ is defined by $r_c (A) = \max \{|z| : z \in W_c (A)\}$. We determine the structure of those linear operators $\phi$ on algebras of block triangular matrices, satisfying $$ W_c \bigl( \phi(A) \bigr) = W_c (A) \text{ for all } A \quad \text{or} \quad r_c \bigl( \phi(A) \bigr) = r_c (A) \text{ for all } A. $$

Keywords:linear operator, numerical range (radius), block triangular matrices
Categories:15A04, 15A60, 47B49

4. CMB 1998 (vol 41 pp. 434)

Mascioni, Vania; Molnár, Lajos
Linear maps on factors which preserve the extreme points of the unit ball
The aim of this paper is to characterize those linear maps from a von~Neumann factor $\A$ into itself which preserve the extreme points of the unit ball of $\A$. For example, we show that if $\A$ is infinite, then every such linear preserver can be written as a fixed unitary operator times either a unital $\ast$-homomorphism or a unital $\ast$-antihomomorphism.

Categories:47B49, 47D25

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