We establish asymptotic formulas for polynomials that are orthogonal over the unit disk with respect to a weight of the form |w(z)|2, where w(z) is a polynomial without zeros on the unit circle |z| = 1. The formulas put in evidence a strong connection between the behavior of the polynomials and the reproducing kernel of an associated weighted Bergman space, which produces interesting new featur...

Dedicated to my " father " Leonard Gross, and to the memory of my " grandfather " Irving Segal. Contents 1. Introduction 1 2. Basics of holomorphic function spaces 2 3. Examples of holomorphic function spaces 7 4. A special property of the Segal-Bargmann and weighted Bergman spaces 12 5. Canonical commutation relations 16 6. The Segal-Bargmann transform 21 7. Quantum mechanics and quantization ...

We define the weighted Bergman space b p β (S T) consisting of temperature functions on the cylinder S T = S 1 × (0,T) and belonging to L p (Ω T ,t β dxdt), where Ω T = (0, 2) × (0,T). For β > −1 we construct a family of bounded projections of L p (Ω T ,t β dxdt) onto b p β (S T). We use this to get, for 1 < p < ∞ and 1/p +1/p = 1, a duality b p β (S T) * = b p β (S T), where β depends on p and β.

For α > −1, let A2α be the corresponding weighted Bergman space of the unit ball in C. For a bounded measurable function f , let Tf be the Toeplitz operator with symbol f on A 2 α. This paper describes all the functions f for which Tf commutes with a given Tg, where g(z) = z1 1 · · · z Ln n for strictly positive integers L1, . . . , Ln, or g(z) = |z1| s1 · · · |zn| nh(|z|) for non-negative real...

Improving upon a recent result of L. Coburn and J. Xia, we show that for any bounded linear operator T on the Segal-Bargmann space, the Berezin transform of T is a function whose partial derivatives of all orders are bounded. Similarly, if T is a bounded operator on any one of the usual weighted Bergman spaces on a bounded symmetric domain, then the appropriately defined “invariant derivatives”...

Let be weighted Bergman space on a bounded symmetric domain . It has analytic continuation in the weight and for in the so-called Wallach set still forms unitary irreducible (projective) representations of . We give the irreducible decomposition of the tensor product of the representation for any two unitary weights and we find the highest weight vectors of the irreducible components. We find a...

A family of recently discovered commutative C∗-algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines a set of symbols consisting of functions which are constant on the corresponding cycles, the orthogonal trajectories to lines forming a pencil. The C∗-algebra generated by Toeplitz operators with such symbols turns out t...