These notes give an introduction to the theory of reproducing kernel Hilbert spaces and their multipliers. We begin with the material that is contained in Aronszajn's classic paper on the subject. We take a somewhat algebraic view of some of his results and discuss them in the context of pull-back and push-out constructions. We then prove Schoenberg's results on negative definite functions and his characterization of metric spaces that can be embedded isometrically in Hilbert spaces. Following this we study multipliers of reproducing kernel Hilbert spaces and use them to give classic proofs of Pick's and Nevanlinna's theorems. In later chapters we consider generalizations of this theory to the vector-valued setting.

We obtain necessary and sufficient conditions for Nevanlinna-Pick interpolation on the unit disk with the additional restriction that all analytic interpolating functions satisfy f ′ (0) = 0. Alternatively, these results can be interpreted as interpolation results for H ∞ (V ), where V is the intersection of the bidisk with an algebraic variety. We use an analysis of C*-envelopes to show that these same conditions do not suffice for matrix interpolation.2000 Mathematics Subject Classification. Primary 47A57; Secondary 30E05, 46E22.

We study the Nevanlinna-Pick problem for a class of subalgebras of H ∞ . This class includes algebras of analytic functions on embedded disks, the algebras of finite codimension in H ∞ and the algebra of bounded analytic functions on a multiply connected domain. Our approach uses a distance formula that generalizes Sarason's [23] work. We also investigate the difference between scalar-valued and matrix-valued interpolation through the use of C * -envelopes. (2000). Primary 47A57; Secondary 46E22, 30E05. Mathematics Subject ClassificationKeywords. Nevanlinna-Pick interpolation, distance formulae, reproducing kernel Hilbert space. n i,j=1 is positive (semidefinite).

We generalize Abrahamse's interpolation theorem from the setting of a multiply connected domain to that of a more general Riemann surface. Our main result provides the scalarvalued interpolation theorem for the fixed-point subalgebra of H ∞ associated to the action of a Fuchsian group. We rely on two results from a paper of Forelli. This allows us to prove the interpolation result using duality techniques that parallel Sarason's approach to the interpolation problem for H ∞ [Donald Sarason, Generalized interpolation in H ∞ , Trans. Amer. Math. Soc. 127 (1967) 179-203, MR0208383. [26]]. In this process we prove a more general distance formula, very much like Nehari's theorem, and obtain relations between the kernel function for the character automorphic Hardy spaces and the Szegö kernel for the disk. Finally, we examine our interpolation results in the context of the two simplest examples of Fuchsian groups acting on the disk.

Abstract. We study the question of whether or not contractive representations of logmodular algebras are completely contractive. We prove that a 2-contractive representation of a logmodular algebra extends to a positive map on the enveloping C * -algebra, which we show generalizes a result of Foias and Suciu on uniform logmodular algebras. Our proof uses non-commutative operator space generalizations of classical results on 2-summing maps and semispectral measures. We establish some matrix factorization results for uniform logmodular algebras.