arXiv subjects

Joseph A. Ball

Publications and source records attributed to Joseph A. Ball.

41 records · Page 3Linked to original sources

arXiv2006

Schur-class multipliers on the Fock space: de Branges-Rovnyak reproducing kernel spaces and transfer-function realizations

We introduce and study a Fock-space noncommutative analogue of reproducing kernel Hilbert spaces of de Branges-Rovnyak type. Results include: use of the de Branges-Rovnyak space ${\mathcal H}(K_{S})$ as the state space for the unique (up to unitary equivalence) observable, coisometric transfer-function realization of the Schur-class multiplier $S$, realization-theoretic characterization of inner Schur-class multipliers, and a calculus for obtaining a realization for an inner multiplier with prescribed left zero-structure. In contrast with the parallel theory for the Arveson space on the unit ball ${\mathbb B}^{d} \subset {\mathbb C}^{d}$ (which can be viewed as the symmetrized version of the Fock space used here), the results here are much more in line with the classical univariate case, with the extra ingredient of the existence of all results having both a ``left'' and a ``right'' version.

math.CA↗

arXiv2006

Transfer-function realization for multipliers of the Arveson space

An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers for the reproducing kernel Hilbert space ${\mathcal H}(k_{d})$ on the unit ball ${\mathbb B}^{d} \subset {\mathbb C}^{d}$, where $k_{d}$ is the positive kernel $k_{d}(λ, ζ) = 1/(1 - < λ, ζ>)$ on ${\mathbb B}^{d}$. We study this space from the point of view of realization theory and functional models of de Branges-Rovnyak type. We highlight features which depart from the classical univariate case: coisometric realizations have only partial uniqueness properties, the nonuniqueness can be described explicitly, and this description assumes a particularly concrete form in the functional-model context.

math.CA↗

arXiv2006

Schur-class multipliers on the Arveson space: de Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations

An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers $S$ for the reproducing kernel Hilbert space ${\mathcal H}(k_{d})$ on the unit ball ${\mathbb B}^{d} \subset {\mathbb C}^{d}$, where $k_{d}$ is the positive kernel $k_{d}(λ, ζ) = 1/(1 - < λ, ζ>)$ on ${\mathbb B}^{d}$. The reproducing kernel space ${\mathcal H}(K_{S})$ associated with the positive kernel $K_{S}(λ, ζ) = (I - S(λ) S(ζ)^{*}) \cdot k_{d}(λ, ζ)$ is a natural multivariable generalization of the classical de Branges-Rovnyak canonical model space. A special feature appearing in the multivariable case is that the space ${\mathcal H}(K_{S})$ in general may not be invariant under the adjoints $M_{λ_{j}}^{*}$ of the multiplication operators $M_{λ_{j}} \colon f(λ) \mapsto λ_{j} f(λ)$ on ${\mathcal H}(k_{d})$. We show that invariance of ${\mathcal H}(K_{S})$ under $M_{λ_{j}}^{*}$ for each $j = 1, ..., d$ is equivalent to the existence of a weakly coisometric realization for $S$ of the form $S(λ) = D + C (I - λ_{1}A_{1} ... - λ_{d} A_{d})^{-1}(λ_{1}B_{1} + ... + λ_{d} B_{d})$ such that the state operators $A_{1}, ..., A_{d}$ pairwise commute. We show that this special situation always occurs for the case of inner functions $S$ (where the associated multiplication operator $M_{S}$ is a partial isometry), and that inner multipliers are characterized by the existence of such a realization such that the state operators $A_{1}, >..., A_{d}$ satisfy an additional stability property.

math.CA↗

arXiv2005

Interpolation in the noncommutative Schur-Agler class

The class of Schur-Agler functions over a domain ${\mathcal D} \subset {\mathbb C}^{d}$ is defined as the class of holomorphic operator-valued functions on ${\mathcal D}$ for which a certain von Neumann inequality is satisfied when a commuting tuple of operators satisfying a certain polynomial norm inequality is plugged in for the variables. Such functions are alternatively characterized as those having a linear-fractional presentation which identifies them as transfer functions of a certain type of conservative structured multidimensional linear system. There now has been introduced a noncommutative version of the Schur-Agler class which consists of formal power series in noncommuting indeterminants satisfying a noncommutative version of the von Neumann inequality when a tuple of operators (not necessarily commuting) coming from a noncommutative operator ball are plugged in for the formal indeterminants. Formal power series in this noncommutative Schur-Agler class in turn are characterized as those having a certain linear-fractional presentation in noncommuting variables identifying them as transfer functions of a recently introduced class of conservative structure multidimensional linear systems having evolution along a free semigroup rather than along an integer lattice. The purpose of this paper is to extend the previously developed interpolation theory for the commutative Schur-Agler class to this noncommutative setting.

math.FA↗

arXiv1999

Zero-pole interpolation for matrix meromorphic functions on a compact Riemann surface, and a matrix Fay trisecant identity

This paper presents a new approach to constructing a meromorphic bundle map between flat vector bundles over a compact Riemann surface having a prescribed Weil divisor (i.e., having prescribed zeros and poles with directional as well as multiplicity information included in the vector case). This new formalism unifies the earlier approach of Ball-Clancey (in the setting of trivial bundles over an abstract Riemann surface) with an earlier approach of the authors (where the Riemann surface was assumed to be the normalizing Riemann surface for an algebraic curve embedded in ${\bold C}^2$ with determinantal representation, and the vector bundles were assumed to be presented as the kernels of linear matrix pencils). The main tool is a version of the Cauchy kernel appropriate for flat vector bundles over the Riemann surface. Our formula for the interpolating bundle map (in the special case of a single zero and a single pole) can be viewed as a generalization of the Fay trisecant identity from the usual line bundle case to the vector bundle case in terms of Cauchy kernels. In particular we obtain a new proof of the Fay trisecant identity.

alg-geom↗