L. A. Coburn

BMO in the Bergman Metric on Bounded Symmetric Domains

For bounded symmetric domains Ω in Cn, a notion of “bounded mean oscillation” in terms of the Bergman metric is introduced. It is shown that for ƒ in L2(Ω, dv), ƒ is in BMO(Ω) if and only if the densely-defined operator [Mƒ, P] ≡ MƒP − PMƒ on L2(Ω, dv) is bounded (here, Mƒ is “multiplication by ƒ” and P is the Bergman projection with range the Bergman subspace H2(Ω, dv) = La2(Ω, dv) of holomorphic functions in L2(Ω, dv)). An analogous characterization of compactness for [Mƒ, P] is provided by functions of “vanishing mean oscillation at the boundary of Ω”.

Toeplitz operators and quantum mechanics

Abstract Toeplitz operators on the Segal-Bargmann spaces of Gaussian measure square-integrable entire functions on complex n-space C n are studied. The C∗-algebra generated by the Weyl form of the canonical commutation relations consists precisely of the uniform limits of almost-periodic Toeplitz operators. The question of “which Toeplitz operators admit a symbol calculus modulo the compact operators” is raised and sufficient conditions are given for such a calculus. These conditions involve a notion of “slow oscillation at infinity.”

FUNCTION THEORY ON CARTAN DOMAINS AND THE BEREZIN-TOEPLITZ SYMBOL CALCULUS*

Cn with dv(z) normalized Lebesgue measure on 9. We let L = L2(Q, dv) be the usual space of Lebesgue square-integrable complex-valued functions on 9. By H2 = H2(Q, dv) we denote the Bergman subspace of L2 consisting of holomorphic functions. The orthogonal projection operator from L2 onto H2 is denoted by P. Forf in L-(Q), the space of essentially bounded, measurable functions on 9, we will consider the multiplication operator

A Lipschitz estimate for Berezin's operator calculus

F. A. Berezin introduced a general symbol calculus for linear operators on reproducing kernel Hilbert spaces. For the particular Hilbert space of Gaussian square-integrable entire functions on complex n-space, C n , we obtain Lipschitz estimates for the Berezin symbols of arbitrary bounded operators. Additional properties of the Berezin symbol and extensions to more general reproducing kernel Hilbert spaces are discussed.

BMO on the Bergman spaces of the classical domains

Let Ü be a bounded symmetric (Cartan) domain with its Harish-Chandra realization in C n [T]. For dv the usual Euclidean volume measure on C n = R n , normalized so that v(Q) = 1, we consider the Hubert space of squareintegrable complex-valued functions L = L(Q,dv) and the Bergman subspace H = H(Q) of holomorphic functions in L. The self-adjoint projection from L onto H is denoted by P. For ƒ, g in L, we consider the multiplication operator Mf on L given by Mjg = f g and the Hankel operator Hf on L given by H f = (I — P)MfP. For ƒ in L, these operators are only densely defined and may be unbounded. The commutator [Mf,P] = MfP — PMf is densely defined on L and may also be unbounded. From the equations

Berezin transform and Weyl-type unitary operators on the Bergman space

For D the open complex unit disc with normalized area measure, we consider the Bergman space La(D) of square-integrable holomorphic functions on D. Induced by the group Aut(D) of biholomorphic automorphisms of D, there is a standard family of Weyl-type unitary operators on La(D). For all bounded operators X on La(D), the Berezin transform X is a smooth, bounded function on D. The range of the mapping Ber: X → X is invariant under Aut(D). The “mixing properties” of the elements of Aut(D) are visible in the Berezin transforms of the induced unitary operators. Computations involving these operators show that there is no real number M > 0 with M‖ X‖∞ ≥ ‖X‖ for all bounded operators X and are used to check other possible properties of X. Extensions to other domains are discussed.

The Koecher Norm and Toeplitz Operators in Several Variables

In this note, I sketch portions of the proofs of several results recently obtained jointly with C. Berger and A. Koranyi [2]. One of the main technical results of that paper has been simplified by use of a classical fact about “harmonic projection” of polynomials.