Banach Journal of Mathematical Analysis

A field-theoretic operator model and Cowen–Douglas class

Björn Gustafsson and Mihai Putinar

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In resonance with a recent geometric framework proposed by Douglas and Yang, we develop a functional model for certain linear bounded operators with rank-one self-commutator acting on a Hilbert space. By taking advantage of the refined existing theory of the principal function of a hyponormal operator, we transfer the whole action outside the spectrum, on the resolvent of the underlying operator, localized at a distinguished vector. The whole construction turns out to rely on an elementary algebra body involving analytic multipliers and Cauchy transforms. We propose a natural field theory interpretation of the resulting resolvent functional model.

Article information

Banach J. Math. Anal., Volume 13, Number 2 (2019), 338-358.

Received: 27 June 2018
Accepted: 25 November 2018
First available in Project Euclid: 28 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47B20: Subnormal operators, hyponormal operators, etc.
Secondary: 30A31 76C05

hyponormal operator exponential transform Cauchy transform ideal fluid flow


Gustafsson, Björn; Putinar, Mihai. A field-theoretic operator model and Cowen–Douglas class. Banach J. Math. Anal. 13 (2019), no. 2, 338--358. doi:10.1215/17358787-2018-0041.

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  • [1] K. F. Clancey and B. L. Wadhwa, Local spectra of seminormal operators, Trans. Amer. Math. Soc. 280, no. 1 (1983), 415–428.
  • [2] M. J. Cowen and R. G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978), no. 3–4, 187–261.
  • [3] R. G. Douglas and R. Yang, “Hermitian geometry on resolvent set” in Operator Theory, Operator Algebras, and Matrix Theory, Oper. Theory Adv. Appl. 267, Birkhäuser, Cham, 2018, 167–183.
  • [4] P. A. Fuhrmann, Functional models in linear algebra, Linear Algebra Appl. 162/164 (1992), 107–151.
  • [5] B. Gustafsson and M. Putinar, Hyponormal Quantization of Planar Domains: Exponential Transform in Dimension Two, Lecture Notes in Math. 2199, Springer, Cham, 2017.
  • [6] B. Gustafsson and M. Putinar, Line bundles defined by the Schwarz function, Anal. Math. Phys. 8 (2018), no. 2, 171–183.
  • [7] M. Martin and M. Putinar, Lectures on Hyponormal Operators, Oper. Theory Adv. Appl. 39, Birkhäuser, Basel, 1989.
  • [8] J. D. Pincus, D. Xia, and J. B. Xia, The analytic model of a hyponormal operator with rank one self-commutator, Integral Equations Operator Theory 7 (1984), no. 4, 516–535.
  • [9] G.-C. Rota, On models for linear operators, Comm. Pure Appl. Math. 13 (1960), 469–472.