Berezin Transform in Polynomial Bergman Spaces
2010 (English)In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 63, no 12, 1533-1584 p.Article in journal (Refereed) Published
Fix a smooth weight function Q in the plane, subject to a growth condition from below Let K-m,K-n denote the reproducing kernel for the Hilbert space of analytic polynomials of degree at most n - 1 of finite L-2-norm with respect to the measure e-(mQ) dA Here dA is normalized area measure, and m is a positive real scaling parameter The (polynomial) Berezin measure dB(m,n)(< z0 >) (z) = K-m,K-n(z(0).z(0))(-1) vertical bar K-m,K-n(z.z(0))vertical bar(2)e(-mQ(z)) dA(z) for the point z(0) is a probability measure that defines the (polynomial) Berezin transform B-m,B-n f(z(0)) = integral(C) f dB(m,n)(< z0 >) for continuous f is an element of L-infinity (C). We analyze the semiclassical limit of the Berezin measure (and transform) as m -> +infinity while n = m tau + o(1), where tau is fixed, positive, and real We find that the Berezin measure for z(0) converges weak-star to the unit point mass at the point z(0) provided that Delta Q(z(0)) > 0 and that z(0) is contained in the interior of a compact set f(tau). defined as the coincidence set for an obstacle problem. As a refinement, we show that the appropriate local blowup of the Berezin measure converges to the standardized Gaussian measure in the plane For points z(0) is an element of C\f(tau), the Berezin measure cannot converge to the point mass at z(0) In the model case Q(z) = vertical bar z vertical bar(2), when f(tau) is a closed disk, we find that the Berezin measure instead converges to harmonic measure at z(0) relative to C\f(tau) Our results have applications to the study of the cigenvalues of random normal matrices The auxiliary results include weighted L-2-estimates for the equation partial derivative u = f when f is a suitable test function and the solution u is restricted by a polynomial growth bound at infinity.
Place, publisher, year, edition, pages
2010. Vol. 63, no 12, 1533-1584 p.
IdentifiersURN: urn:nbn:se:kth:diva-26281ISI: 000283336000001ScopusID: 2-s2.0-77958567873OAI: oai:DiVA.org:kth-26281DiVA: diva2:386045
QC 201101122011-01-122010-11-212012-03-20Bibliographically approved