# Generalized Fock spaces, interpolation, multipliers, circle geometry.

Jaak Peetre; Sundaram Thangavelu; Nils-Olof Wallin

Revista Matemática Iberoamericana (1996)

- Volume: 12, Issue: 1, page 63-110
- ISSN: 0213-2230

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topPeetre, Jaak, Thangavelu, Sundaram, and Wallin, Nils-Olof. "Generalized Fock spaces, interpolation, multipliers, circle geometry.." Revista Matemática Iberoamericana 12.1 (1996): 63-110. <http://eudml.org/doc/39513>.

@article{Peetre1996,

abstract = {By a (generalized) Fock space we understand a Hilbert space of entire analytic functions in the complex plane C which are square integrable with respect to a weight of the type e-Q(z), where Q(z) is a quadratic form such that tr Q > 0. Each such space is in a natural way associated with an (oriented) circle C in C. We consider the problem of interpolation between two Fock spaces. If C0 and C1 are the corresponding circles, one is led to consider the pencil of circles generated by C0 and C1. If H is the one parameter Lie group of Moebius transformations leaving invariant the circles in the pencil, we consider its complexification Hc, which permutes these circles and with the aid of which we can construct the Calderón curve giving the complex interpolation. Similarly, real interpolation leads to a multiplier problem for the transformation that diagonalizes all the operators in Hc. It turns out that the result is rather sensitive to the nature of the pencil, and we obtain nearly complete results for elliptic and parabolic pencils only.},

author = {Peetre, Jaak, Thangavelu, Sundaram, Wallin, Nils-Olof},

journal = {Revista Matemática Iberoamericana},

keywords = {Interpolación; Operadores de Fock; Espacios de Fock; Espacios de Hilbert; Calderón curve; Fock space; Hilbert space of entire analytic functions; interpolation between two Fock spaces; pencil of circles; one parameter Lie group of Moebius transformations; multiplier problem; elliptic and parabolic pencils},

language = {eng},

number = {1},

pages = {63-110},

title = {Generalized Fock spaces, interpolation, multipliers, circle geometry.},

url = {http://eudml.org/doc/39513},

volume = {12},

year = {1996},

}

TY - JOUR

AU - Peetre, Jaak

AU - Thangavelu, Sundaram

AU - Wallin, Nils-Olof

TI - Generalized Fock spaces, interpolation, multipliers, circle geometry.

JO - Revista Matemática Iberoamericana

PY - 1996

VL - 12

IS - 1

SP - 63

EP - 110

AB - By a (generalized) Fock space we understand a Hilbert space of entire analytic functions in the complex plane C which are square integrable with respect to a weight of the type e-Q(z), where Q(z) is a quadratic form such that tr Q > 0. Each such space is in a natural way associated with an (oriented) circle C in C. We consider the problem of interpolation between two Fock spaces. If C0 and C1 are the corresponding circles, one is led to consider the pencil of circles generated by C0 and C1. If H is the one parameter Lie group of Moebius transformations leaving invariant the circles in the pencil, we consider its complexification Hc, which permutes these circles and with the aid of which we can construct the Calderón curve giving the complex interpolation. Similarly, real interpolation leads to a multiplier problem for the transformation that diagonalizes all the operators in Hc. It turns out that the result is rather sensitive to the nature of the pencil, and we obtain nearly complete results for elliptic and parabolic pencils only.

LA - eng

KW - Interpolación; Operadores de Fock; Espacios de Fock; Espacios de Hilbert; Calderón curve; Fock space; Hilbert space of entire analytic functions; interpolation between two Fock spaces; pencil of circles; one parameter Lie group of Moebius transformations; multiplier problem; elliptic and parabolic pencils

UR - http://eudml.org/doc/39513

ER -

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