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Search: MSC category 15A60 ( Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] )

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1. CJM 2008 (vol 60 pp. 520)

Chen, Chang-Pao; Huang, Hao-Wei; Shen, Chun-Yen
Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences
Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\|A\|_{E, F}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces: $\ell_p$, $d(w,p)$, and $\ell_p(w)$. The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour.

Keywords:norms of matrices, normed Riesz spaces, weighted mean matrices, Nörlund mean matrices, summability matrices, matrices with row decreasing
Categories:15A60, 40G05, 47A30, 47B37, 46B42

2. 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

3. CJM 2003 (vol 55 pp. 91)

Choi, Man-Duen; Li, Chi-Kwong; Poon, Yiu-Tung
Some Convexity Features Associated with Unitary Orbits
Let $\mathcal{H}_n$ be the real linear space of $n\times n$ complex Hermitian matrices. The unitary (similarity) orbit $\mathcal{U} (C)$ of $C \in \mathcal{H}_n$ is the collection of all matrices unitarily similar to $C$. We characterize those $C \in \mathcal{H}_n$ such that every matrix in the convex hull of $\mathcal{U}(C)$ can be written as the average of two matrices in $\mathcal{U}(C)$. The result is used to study spectral properties of submatrices of matrices in $\mathcal{U}(C)$, the convexity of images of $\mathcal{U} (C)$ under linear transformations, and some related questions concerning the joint $C$-numerical range of Hermitian matrices. Analogous results on real symmetric matrices are also discussed.

Keywords:Hermitian matrix, unitary orbit, eigenvalue, joint numerical range
Categories:15A60, 15A42

4. CJM 2000 (vol 52 pp. 141)

Li, Chi-Kwong; Tam, Tin-Yau
Numerical Ranges Arising from Simple Lie Algebras
A unified formulation is given to various generalizations of the classical numerical range including the $c$-numerical range, congruence numerical range, $q$-numerical range and von Neumann range. Attention is given to those cases having connections with classical simple real Lie algebras. Convexity and inclusion relation involving those generalized numerical ranges are investigated. The underlying geometry is emphasized.

Keywords:numerical range, convexity, inclusion relation
Categories:15A60, 17B20

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