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We show that the Hardy space on the unit disk is the only nontrivial irreducible reproducing kernel Hilbert space which satisfies the complete Nevanlinna-Pick property and hyponormality of all multiplication operators. [ABSTRACT FROM AUTHOR]

We study the isomorphism problem for the multiplier algebras of irreducible complete Pick kernels. These are precisely the restrictions MV of the multiplier algebra M of Drury-Arveson space to a holomorphic subvariety V of the unit ball Bd. We find that MV is completely isometrically isomorphic to MW if and only if W is the image of V under a biholomorphic automorphism of the ball. In this case, the isomorphism is unitarily implemented. This is then strengthened to show that when d < ∞ every isometric isomorphism is completely isometric. The problem of characterizing when two such algebras are (algebraically) isomorphic is also studied. When V and W are each a finite union of irreducible varieties and a discrete variety, when d < ∞, an isomorphism between MV and MW determines a biholomorphism (with multiplier coordinates) between the varieties; and the isomorphism is composition with this function. These maps are automatically weak-* continuous. We present a number of examples showing that the converse fails in several ways. We discuss several special cases in which the converse does hold-- particularly, smooth curves and Blaschke sequences. We also discuss the norm closed algebras associated to a variety, and point out some of the differences. [ABSTRACT FROM AUTHOR]

We introduce a reproducing kernel structure for Hilbert spaces of functions where differences of point evaluations are bounded. The associated reproducing kernels are characterized in terms of conditionally negative functions. [ABSTRACT FROM AUTHOR]

Various statements on the distribution of eigenvalues of random matrices are obtained by considering the limiting behavior of the reproducing kernels of a certain naturally associated sequence of orthogonal polynomials. We establish a universal limiting behavior of this type in the case when the underlying measure does not have finite moments. In this case the orthogonal polynomials are replaced by a nested family of de Branges spaces of entire functions. [ABSTRACT FROM AUTHOR]

A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions includes the Szegö and the Bergman kernel on the symmetrized polydisc. [ABSTRACT FROM AUTHOR]

We investigate the Pick problem for the polydisk and unit ball using dual algebra techniques. Some factorization results for Bergman spaces are used to describe a Pick theorem for any bounded region in Cd. [ABSTRACT FROM AUTHOR]