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Unbounded Operators on the Segal–Bargmann Space

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The Bergman Kernel and Related Topics (HSSCV 2022)

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 447))

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Abstract

In this paper we analyse the domains of differential and multiplication operators on the Segal–Bargmann space. We consider the basic estimate for the \(\partial \)- complex, remark that this estimate is closely related to the uncertainty principle in quantum mechanics and compute the Bergman kernel of the graph norm. It is shown that the set of all functions u in the Segal–Bargmann \(A^2(\mathbb C, e^{-|z|^2})\) such that the multiplication with a polynomial p is norm bounded gives a relatively compact subset of the Segal–Bargmann space. In the following section we give a survey of recent results on the \(\partial \)-complex on weighted Bergman spaces on Hermitian manifolds, analysing metrics which produce a similar duality between differentiation and multiplication as in the Segal–Bargmann space. Finally we study the basic estimates and the corresponding questions of compactness for the generalized \(\partial \)-complex.

F. Haslinger—Supported by the FWF-grant P 36884

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Authors and Affiliations

  1. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090, Wien, Austria

    Friedrich Haslinger

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  1. Friedrich Haslinger
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Correspondence to Friedrich Haslinger .

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Editors and Affiliations

  1. Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan

    Kengo Hirachi

  2. Graduate School of Mathematics, Nagoya University, Nagoya, Japan

    Takeo Ohsawa

  3. Graduate School of Mathematical Sciences, The University of Tokyo, Tokyo, Japan

    Shigeharu Takayama

  4. Faculty of Mathematics, Kyushu University, Fukuoka, Japan

    Joe Kamimoto

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Haslinger, F. (2024). Unbounded Operators on the Segal–Bargmann Space. In: Hirachi, K., Ohsawa, T., Takayama, S., Kamimoto, J. (eds) The Bergman Kernel and Related Topics. HSSCV 2022. Springer Proceedings in Mathematics & Statistics, vol 447. Springer, Singapore. https://doi.org/10.1007/978-981-99-9506-6_6

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