My research interests include operator theory, C*-algebras, and quantum mechanics from the viewpoint of deformations of C*-algebras. The operators on which I focus are usually of Toeplitz-type and act on the square- integrable holomorphic functions on phase-space. I am interested in C*-algebras of these operators with various interesting "symbols" and the relation of such algebras to algebras of pseudo-differential operators which have been studied classically.

In the past several years, I have become interested in the structure of the Berezin symbol calculus of general operators on Bergman reproducing kernel Hilbert spaces. This calculus serves as a model for "quantization" and has been the object of considerable attention since it was introduced by Berezin in the 1970's. The paper (with Bo Li) "Directional derivative estimates for Berezin's operator calculus," appears in the Proceedings of the AMS 136 (2008) pp. 641-649. This paper is a sequel to my papers "Sharp Berezin Lipschitz estimates" ( Proceedings of the AMS 135 (2007) pp. 1163-1168) and "A Lipschitz estimate for Berezin's operator calculus" (Proceedings of the AMS 133 (2005) pp. 127-131). These papers show that the Berezin symbols of general bounded operators must satisfy certain severe Bloch-type growth limitations which were not previously known. In his Ph.D thesis (degree granted in August, 2008), Bo Li extended these results further, to obtain Bloch-type estimates on the higher derivatives of general Berezin symbols.

More recently, in ``Toeplitz operators with BMO symbols on the Segal-Bargmann space" (joint work with Joshua Isralowitz and Bo Li), we have shown (Transactions of the AMS 363 (2011) pp. 3015-3030 ) that, under certain (BMO) regularity conditions on the symbol, boundedness and compactness of Toeplitz operators on the Segal-Bargmann space (of Gaussian square-integrable entire functions on complex n-dimensional space) are completely determined by the Berezin symbols of these operators. Another paper (written jointly with Wolfram Bauer and Joshua Isralowitz), "Heat flow, BMO and the compactness of Toeplitz operators" (Journal of Functional Analysis 259 (2010) pp. 57-78), shows that, under the same (BMO) condition on the symbol f, the Toeplitz operator T_{f} is compact if and only if (a) the "heat transform" f^{(t_0)} vanishes at infinity for some t_0 > 0 or (b) f^{(t)} vanishes at infinity for all t > 0. This result is remarkable as a purely function-theoretic result about the interaction of BMO and the heat flow.

In ``Berezin transform and Weyl-type unitary operators on the Bergman space," Proceedings of the AMS, 140 (2012) pp. 3445-3451, I have examined the detailed structure of the Berezin transform, Ber, and shown, among other things, that, for the classical Bergman space of the open unit disc, D, as well as for the Segal-Bargmann space on C^{n}, range(Ber) is a non-closed linear subspace of the supremum norm Banach space BC(D) of bounded continuous functions on D or of BC(C^{n}). The analysis uses explicit calculation of the Berezin transforms of unitary operators which arise from the biholomorphic automorphisms of the underlying complex spaces D and C^{n}.In the paper``Heat flow, weighted Bergman spaces, and real-analytic Lipschitz approximation," Journal fur die reine und angewandte Mathematik (Crelle), 703 (2015) pp. 225-246, Wolfram Bauer and I have shown that for the Bergman metric on bounded symmetric domains (BSD), real-analytic Lipschitz functions are uniformly dense in the space of all uniformly continuous functions. Our analysis relies upon the fact that uniformly continuous functions on BSD's are in BMO.

In the paper ``Toeplitz operators with uniformly continuous symbols," Integral equations and operator theory, 83 (2015) pp. 25-34, Wolfram Bauer and I have shown, as a consequence of the Crelle paper and other recent results, that for f uniformly continuous with respect to the Bergman metric on any BSD, the Toeplitz operator T_{f} is bounded iff f is bounded and T_{f} is compact iff f vanishes at the boundary.

In the paper ``Uniformly continuous functions and quantization on the Fock space," Bol. Soc. Mat. Mex., 22 (2016) pp. 669-677, Wolfram Bauer and I have shown that, for all bounded uniformly continuous functions f and g on C^{n}, (*) lim || T_{f} T_{g} - T_{fg}||_{t} = 0 as t goes to 0 in the usual scale of Gaussian probability measures d\mu_{t} on C^{n}. More precisely, for t > 0, d\mu_{t}(z) = (4 \pi t)^{-n} \exp{-|z|^{2} / 4t} dv(z), L^{2}_{t} = L^{2}( C^{n}, d\mu_{t}): the Toeplitz operator T_{f} acts on H^{2}_{t}, the closed subspace of entire functions in L^{2}_{t}, by T_{f} h = P_{t} (f h), where P_{t} is the orthogonal projection from L^{2}_{t} onto H^{2}_{t} . This result fails for some rapidly oscillating bounded functions f, g.

In``Toeplitz quantization on Fock space," Journal of Functional Analysis, 274 (2018) pp. 3531-3551, Wolfram Bauer, Raffael Hagger and I have extended the ``quantization" result (*) of the previous paper to f, g general uniformly continuous functions or bounded functions of vanishing mean oscillation (VMO \cap L^{\infty}. The last algebra contains all conjugate-closed subalgebras of L^{\infty} with property (*). For f^{(t)}(a) = \int f( a + z) d\mu_{t}(z) the heat transform of f at time t > 0, we show that, for all bounded measurable f, \lim_{t \rightarrow 0}f^{(t)}(a) = f(a) except on a set of measure zero so that \lim_{t \rightarrow 0}||T_{f}||_{t} = ||f||_{\infty}. This is, evidently, the best extension of previously known results. We also provide a counterexample to (*) with f, g bounded and real analytic.

My survey article ``Fock space, the Heisenberg group, heat flow and Toeplitz operators," has appeared (Chapter 1, pp. 1 - 15) in the Handbook of analytic operator theory, edited by Kehe Zhu, ISBN 9781138486416, CRC Press, 2019.

In ``Positivity, complex FIOs, and Toeplitz operators," Pure and Applied Analysis, 1 (2019) pp. 327-357, Michael Hitrik, Johannes Sjoestrand and I have used a characterization of complex linear canonical transformations that are positive with respect to a pair of strictly plurisubharmonic quadratic weights to show that the boundedness of a class of Toeplitz operators on the Bargmann space having symbols which are exponentials of quadratic forms in z_{k}, \bar z_{j} is implied by the boundedness of their Weyl symbols. There is a follow-on paper "Weyl symbols and boundedness of Toeplitz operators", arXiv 1907/.06132, where, with Francis White, we extended the previous results to Toeplitz symbols which are exponentials of general degree two polynomials. This work is part of a program, initiated with Charles Berger in our 1994 paper in the American Journal of Mathematics, which suggests that boundedness of Toeplitz operators on the Bargmann space may be equivalent to the boundedness of their Weyl symbols.

updated on 7/23/2019

Send mail to lcoburn@buffalo.edu