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Search: MSC category 15A04 ( Linear transformations, semilinear transformations )

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1. CJM 2010 (vol 62 pp. 758)

Dolinar, Gregor; Kuzma, Bojan
General Preservers of Quasi-Commutativity
Let ${ M}_n$ be the algebra of all $n \times n$ matrices over $\mathbb{C}$. We say that $A, B \in { M}_n$ quasi-commute if there exists a nonzero $\xi \in \mathbb{C}$ such that $AB = \xi BA$. In the paper we classify bijective not necessarily linear maps $\Phi \colon M_n \to M_n$ which preserve quasi-commutativity in both directions.

Keywords:general preservers, matrix algebra, quasi-commutativity
Categories:15A04, 15A27, 06A99

2. CJM 2008 (vol 60 pp. 1050)

Huang, Wen-ling; Semrl, Peter \v
Adjacency Preserving Maps on Hermitian Matrices
Hua's fundamental theorem of the geometry of hermitian matrices characterizes bijective maps on the space of all $n\times n$ hermitian matrices preserving adjacency in both directions. The problem of possible improvements has been open for a while. There are three natural problems here. Do we need the bijectivity assumption? Can we replace the assumption of preserving adjacency in both directions by the weaker assumption of preserving adjacency in one direction only? Can we obtain such a characterization for maps acting between the spaces of hermitian matrices of different sizes? We answer all three questions for the complex hermitian matrices, thus obtaining the optimal structural result for adjacency preserving maps on hermitian matrices over the complex field.

Keywords:rank, adjacency preserving map, hermitian matrix, geometry of matrices
Categories:15A03, 15A04, 15A57, 15A99

3. CJM 2004 (vol 56 pp. 134)

Li, Chi-Kwong; Sourour, Ahmed Ramzi
Linear Operators on Matrix Algebras that Preserve the Numerical Range, Numerical Radius or the States
Every norm $\nu$ on $\mathbf{C}^n$ induces two norm numerical ranges on the algebra $M_n$ of all $n\times n$ complex matrices, the spatial numerical range $$ W(A)= \{x^*Ay : x, y \in \mathbf{C}^n,\nu^D(x) = \nu(y) = x^*y = 1\}, $$ where $\nu^D$ is the norm dual to $\nu$, and the algebra numerical range $$ V(A) = \{ f(A) : f \in \mathcal{S} \}, $$ where $\mathcal{S}$ is the set of states on the normed algebra $M_n$ under the operator norm induced by $\nu$. For a symmetric norm $\nu$, we identify all linear maps on $M_n$ that preserve either one of the two norm numerical ranges or the set of states or vector states. We also identify the numerical radius isometries, {\it i.e.}, linear maps that preserve the (one) numerical radius induced by either numerical range. In particular, it is shown that if $\nu$ is not the $\ell_1$, $\ell_2$, or $\ell_\infty$ norms, then the linear maps that preserve either numerical range or either set of states are ``inner'', {\it i.e.}, of the form $A\mapsto Q^*AQ$, where $Q$ is a product of a diagonal unitary matrix and a permutation matrix and the numerical radius isometries are unimodular scalar multiples of such inner maps. For the $\ell_1$ and the $\ell_\infty$ norms, the results are quite different.

Keywords:Numerical range, numerical radius, state, isometry
Categories:15A60, 15A04, 47A12, 47A30

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