Spherical Tuples of Hilbert Space Operators

We introduce and study a class of operator tuples in complex Hilbert spaces, which we call spherical tuples. In particular, we characterize spherical multi-shifts, and more generally, multiplication tuples on RKHS. We further use these characterizations to describe various spectral parts including the Taylor spectrum. We also find a criterion for the Schatten $S_p$-class membership of cross-commutators of spherical $m$-shifts. We show, in particular, that cross-commutators of non-compact spherical $m$-shifts cannot belong to $S_p$ for $p \le m$. We specialize our results to some well-studied classes of multi-shifts. We prove that the cross-commutators of a spherical joint $m$-shift, which is a $q$-isometry or a $2$-expansion, belongs to $S_p$ if and only if $p > m$. We further give an example of a spherical jointly hyponormal $2$-shift, for which the cross-commutators are compact but not in $S_p$ for any $p <\infty$.Comment: a version close to final on

Operator monotone functions and L\"owner functions of several variables

We prove generalizations of L\"owner's results on matrix monotone functions to several variables. We give a characterization of when a function of $d$ variables is locally monotone on $d$-tuples of commuting self-adjoint $n$-by-$n$ matrices. We prove a generalization to several variables of Nevanlinna's theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone

Independence in computable algebra

We give a sufficient condition for an algebraic structure to have a computable presentation with a computable basis and a computable presentation with no computable basis. We apply the condition to differentially closed, real closed, and difference closed fields with the relevant notions of independence. To cover these classes of structures we introduce a new technique of safe extensions that was not necessary for the previously known results of this kind. We will then apply our techniques to derive new corollaries on the number of computable presentations of these structures. The condition also implies classical and new results on vector spaces, algebraically closed fields, torsion-free abelian groups and Archimedean ordered abelian groups.Comment: 24 page

Free biholomorphic functions and operator model theory,II

This is a continuation of the paper entitled "Free biholomorphic functions and operator model theory", in our attempt to transfer the free analogue of Nagy-Foias theory from the unit ball [B(\cH)^n]_1 to other noncommutative domains and varieties in B(\cH)^n, using appropriate maps. The present paper treats the completely non-coisometric case and the case of noncommutative varieties.Comment: 37 page