Lewis Coburn's Home Page
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 \(\mathbb{C}^{n}\), 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(\(\mathbb{C}^{n}\)). 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 \(\mathbb{C}^{n}\).
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 \(\mathbb{C}^{n}\), (*)
\(\lim || T_{f} T_{g} - T_{fg}||_{t} = 0\) as \(t\) goes to 0 in
the usual scale of Gaussian probability measures \(d\mu_{t}\) on
\(\mathbb{C}^{n}\). More precisely, for \(t\) > 0, \(d\mu_{t}(z) =
(4 \pi t)^{-n} \exp{-|z|^{2} / 4t}
dv(z)\), \(L^{2}_{t} = L^{2}( \mathbb{C}^{n},
d\mu_{t})\): 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", Mathematical
Research Letters, 28 (2021) pp. 681-696, 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.
Most recently, Michael Hitrik, Johannes Sjoestrand and I have considered the problem of composing Toeplitz operators on Bargmann space (see arXiv 2205.08649 ). I have also revisited the problem of approximation by Lipschitz functions (see arXiv: 2104.13153).
updated on 7/25/2022
Send mail to lcoburn@buffalo.edu