Bulletin of Mathematical Sciences

In this paper, Geronimus transformations for matrix orthogonal polynomials in the real line are studied. The orthogonality is understood in a broad sense, and is given in terms of a nondegenerate continuous sesquilinear form, which in turn is determined by a quasi-definite matrix of bivariate generalized functions with a well-defined support. The discussion of the orthogonality for such a sesquilinear form includes, among others, matrix Hankel cases with linear functionals, general matrix Sobolev orthogonality and discrete orthogonal polynomials with an infinite support. The results are mainly concerned with the derivation of Christoffel-type formulas, which allow to express the perturbed matrix biorthogonal polynomials and its norms in terms of the original ones. The basic tool is the Gauss–Borel factorization of the Gram matrix, and particular attention is paid to the non-associative character, in general, of the product of semi-infinite matrices. The Geronimus transformation in which a right multiplication by the inverse of a matrix polynomial and an addition of adequate masses are performed, is considered. The resolvent matrix and connection formulas are given. Two different methods are developed. A spectral one, based on the spectral properties of the perturbing polynomial, and constructed in terms of the second kind functions. This approach requires the perturbing matrix polynomial to have a nonsingular leading term. Then, using spectral techniques and spectral jets, Christoffel–Geronimus formulas for the transformed polynomials and norms are presented. For this type of transformations, the paper also proposes an alternative method, which does not require of spectral techniques, that is valid also for singular leading coefficients. When the leading term is nonsingular, a comparison of between both methods is presented. The nonspectral method is applied to unimodular Christoffel perturbations, and a simple example for a degree one massless Geronimus perturbation is given.

Consider a function $F(X,Y)$ of pairs of positive matrices with values in the positive matrices such that whenever $X$ and $Y$ commute $F(X,Y)=X^p{Y}^q.$ Our first main result gives conditions on $F$ such that $\mathrm{\text{Tr}}[X \mathrm{\text{log}}(F(Z,Y))]\le \mathrm{\text{Tr}}[X(p \mathrm{\text{log}} X+q \mathrm{\text{log}} Y)]$ for all $X,Y,Z$ such that $\mathrm{\text{Tr}}Z=\mathrm{\text{Tr}}X$. (Note that $Z$ is absent from the right side of the inequality.) We give several examples of functions $F$ to which the theorem applies. Our theorem allows us to give simple proofs of the well-known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables $X,Y,Z$ instead of just $X,Y$ alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy $D(X\parallel Y)=\mathrm{\text{Tr}}[X(\mathrm{\text{log}} X- \mathrm{\text{log}} Y])$, and two others, the Donald relative entropy $D_D(X\parallel Y)$, and the Belavkin–Stasewski relative entropy $D_{\mathrm{\text{BS}}}(X\parallel Y)$. They are known to satisfy $D_D(X\parallel Y)\le D(X\parallel Y)\le D_{\mathrm{\text{BS}}}(X\parallel Y)$. We prove that the Donald relative entropy provides the sharp upper bound, independent of $Z$ on $\mathrm{\text{Tr}}[X \mathrm{\text{log}}(F(Z,Y))]$ in a number of cases in which $F(Z,Y)$ is homogeneous of degree $1$ in $Z$ and $-1$ in $Y$. We also investigate the Legendre transforms in $X$ of $D_D(X\parallel Y)$ and $D_{\mathrm{\text{BS}}}(X\parallel Y)$, and show how our results for these lead to new refinements of the Golden–Thompson inequality.

Given a group-word $w$ and a group $G$, the verbal subgroup $w(G)$ is the one generated by all $w$-values in $G$. The word $w$ is said to be concise if $w(G)$ is finite whenever the set of $w$-values in $G$ is finite. In 1960s, Hall asked whether every word is concise but later Ivanov answered this question in the negative. On the other hand, Hall’s question remains wide open in the class of residually finite groups. In the present paper we show that various generalizations of the Engel word are concise in residually finite groups.

The paper is the first part of a program devoted to the study of the behavior of operator-valued multipliers in Morrey spaces. Embedding theorems and uniform separability properties involving $E$-valued Morrey spaces are proved. As a consequence, maximal regularity for solutions of infinite systems of anisotropic elliptic partial differential equations are established.

We study the bilinear Weyl product acting on quasi-Banach modulation spaces. We find sufficient conditions for continuity of the Weyl product and we derive necessary conditions. The results extend known results for Banach modulation spaces.