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Search: MSC category 15A45 ( Miscellaneous inequalities involving matrices )

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1. CMB 2012 (vol 57 pp. 25)

Bourin, Jean-Christophe; Harada, Tetsuo; Lee, Eun-Young
 Subadditivity Inequalities for Compact Operators Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings. Keywords:concave or convex function, Hilbert space, unitary orbits, compact operators, compressions, matrix inequalitiesCategories:47A63, 15A45

2. CMB 2008 (vol 51 pp. 86)

Nakazato, Hiroshi; Bebiano, Natália; Providência, Jo\ ao da
 The Numerical Range of 2-Dimensional Krein Space Operators The tracial numerical range of operators on a $2$-dimensional Krein space is investigated. Results in the vein of those obtained in the context of Hilbert spaces are obtained. Keywords:numerical range, generalized numerical range, indefinite inner product spaceCategories:15A60, 15A63, 15A45

3. CMB 2006 (vol 49 pp. 281)

Ragnarsson, Carl Johan; Suen, Wesley Wai; Wagner, David G.
 Correction to a Theorem on Total Positivity A well-known theorem states that if $f(z)$ generates a PF$_r$ sequence then $1/f(-z)$ generates a PF$_r$ sequence. We give two counterexamples which show that this is not true, and give a correct version of the theorem. In the infinite limit the result is sound: if $f(z)$ generates a PF sequence then $1/f(-z)$ generates a PF sequence. Keywords:total positivity, Toeplitz matrix, PÃ³lya frequency sequence, skew Schur functionCategories:15A48, 15A45, 15A57, 05E05

4. CMB 2003 (vol 46 pp. 332)

Đoković, Dragomir Z.; Tam, Tin-Yau
 Some Questions about Semisimple Lie Groups Originating in Matrix Theory We generalize the well-known result that a square traceless complex matrix is unitarily similar to a matrix with zero diagonal to arbitrary connected semisimple complex Lie groups $G$ and their Lie algebras $\mathfrak{g}$ under the action of a maximal compact subgroup $K$ of $G$. We also introduce a natural partial order on $\mathfrak{g}$: $x\le y$ if $f(K\cdot x) \subseteq f(K\cdot y)$ for all $f\in \mathfrak{g}^*$, the complex dual of $\mathfrak{g}$. This partial order is $K$-invariant and induces a partial order on the orbit space $\mathfrak{g}/K$. We prove that, under some restrictions on $\mathfrak{g}$, the set $f(K\cdot x)$ is star-shaped with respect to the origin. Categories:15A45, 20G20, 22E60