§3.3. H p spaces again §3.4. #°°(ID>) as a multiplier algebra §3.5. Inner functions 45 §3.6. Historical notes Chapter 4. P 2 (/_i) 49 §4.1. Other spaces with H°°(TD>) as the multiplier algebra 49 §4.2. Vector-valued-P 2 (/u) spaces Chapter 5. Pick Redux §5.1. Necessity of positivity of the Pick matrix §5.2. The Szego kernel has the Pick property §5.3. The Caratheodory problem §5.4. Uniqueness of the Szego kernel §5.5. Historical notes Chapter 6. Qualitative Properties of the Solution of the Pick Problem in H°°(B) §6.1. A formula for the solution §6.2. The realization formula for H°°(B) §6.3. Another formula for the solution §6.4. The Nevanlinna problem Chapter 7. Characterizing Kernels with the Complete Pick Property §7.1. Characterization of the complete Pick property §7.2. Another characterization of the complete Pick property §7.3. Holomorphic spaces with the complete Pick property §7.4. The Sobolev space §7.5. The M sxt Pick property §7.6. Historical notes Contents xi Chapter 8. The Universal Pick Kernel §8.1. The universal kernel 97 §8.2. The realization formula for the universal kernel §8.3. Qualitative properties of solutions of the Pick problem for complete Pick kernels §8.4. The Toeplitz-corona theorem §8.5. Beurling theorems §8.6. Holomorphic complete Pick spaces §8.7. The Nevanlinna problem §8.8. Uniqueness of kernels with the Pick property §8.9. Historical notes Chapter 9.

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IntroductionIn this paper, we shall be looking at a special class of bordered (algebraic) varieties that are contained in the bidisk D 2 in C 2. Condition (0.2) means that the variety exits the bidisk through the distinguished boundary of the bidisk, the torus. We shall use OV to denote the set given by (0.2): topologically, it is the boundary of V within Zp, the zero set of p, rather than in all of C 2. We shall always assume that p is chosen to be minimal, i.e. so that no irreducible component of Zp is disjoint from D 2 and so that p has no repeated irreducible factors.Why should one single out distinguished varieties from other bordered varieties?One of the most important results in operator theory is T. Andb's inequality

i ¢ on a Hilbert space whose inner product is defined in terms of periodic distributions and we relate this model theory for the case when m = 2 to a disconjugacy theory for a subclass of Toeplitz operators of the type studied by Boutet de Monvel and Guilliman, classical function theoretic ideas on the Dirichlet space, and the theory of nonstationary stochastic processes. This is presented in a series of three papers. In this first paper, we concentrate on a model for these T.

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