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1. CMB 2012 (vol 57 pp. 25)
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 inequalities Categories:47A63, 15A45 |
2. CMB 2011 (vol 56 pp. 92)
On Perturbations of Continuous Maps We give sufficient conditions for the following problem: given a
topological space $X$, a metric space $Y$, a subspace $Z$ of $Y$, and
a continuous map $f$ from $X$ to $Y$, is it possible, by applying to
$f$ an arbitrarily small perturbation, to ensure that $f(X)$ does not
meet $Z$? We also give a relative variant: if $f(X')$ does not meet
$Z$ for a certain subset $X'\subset X$, then we may keep $f$ unchanged
on $X'$. We also develop a variant for continuous sections of
fibrations and discuss some applications to matrix perturbation
theory.
Keywords:perturbation theory, general topology, applications to operator algebras / matrix perturbation theory Category:54F45 |
3. CMB 2009 (vol 52 pp. 95)
Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory In the work presented below the classical subject of orthogonal
polynomials on the unit
circle is discussed in the matrix setting. An explicit matrix
representation of the matrix valued orthogonal polynomials in terms of
the moments of the measure is presented. Classical recurrence
relations are revisited using the matrix representation of the
polynomials. The matrix expressions for the kernel polynomials and the
Christoffel--Darboux formulas are presented for the first time.
Keywords:Matrix valued orthogonal polynomials, unit circle, Schur complements, recurrence relations, kernel polynomials, Christoffel-Darboux Category:42C99 |
4. CMB 2009 (vol 52 pp. 145)
$2$-Clean Rings A ring $R$ is said to be $n$-clean if every
element can be written as a sum of an idempotent and $n$ units.
The class of these rings contains clean rings and $n$-good rings
in which each element is a sum of $n$ units. In this paper, we
show that for any ring $R$, the endomorphism ring of a free
$R$-module of rank at least 2 is $2$-clean and that the ring $B(R)$
of all $\omega\times \omega$ row and column-finite matrices over
any ring $R$ is $2$-clean. Finally, the group ring $RC_{n}$ is
considered where $R$ is a local ring.
Keywords:$2$-clean rings, $2$-good rings, free modules, row and column-finite matrix rings, group rings Categories:16D70, 16D40, 16S50 |
5. CMB 2006 (vol 49 pp. 281)
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 function Categories:15A48, 15A45, 15A57, 05E05 |
6. CMB 2000 (vol 43 pp. 145)
On the 2-Parallel Versions of Links In this paper, we show that the absolute value of the signature of
the $2$-parallel version of a link is less than or equal to the
nullity of it and show that the signature, nullity, and Minkowski
units of the $2$-parallel version of a certain class of links are
always equal to $0$, $2$, and $1$ respectively.
Keywords:braid, Goeritz matrix, Minkowski unit, nullity, signature, 2-parallel version Category:57M25 |