Connection and curvature on bundles of Bergman and Hardy spaces

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F20%3A00524631" target="_blank" >RIV/67985840:_____/20:00524631 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.25537/dm.2020v25.189-217" target="_blank" >http://dx.doi.org/10.25537/dm.2020v25.189-217</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.25537/dm.2020v25.189-217" target="_blank" >10.25537/dm.2020v25.189-217</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Connection and curvature on bundles of Bergman and Hardy spaces

Popis výsledku v původním jazyce

We consider a complex domain D×V in the space Cm×Cn and a family of weighted Bergman spaces on V defined by a weight e-kϕ(z,w) for a pluri-subharmonic function ϕ(z,w) with a quantization parameter k. The weighted Bergman spaces define an infinite dimensional Hermitian vector bundle over the domain D. We consider the natural covariant differentiation ∇Z on the sections, namely the unitary Chern connections preserving the Bergman norm. We prove a Dixmier trace formula for the curvature of the unitary connection and we find the asymptotic expansion for the curvatures R(k) (Z,Z) for large k and for the induced connection [∇(k)Z,T(k)f] on Toeplitz operators Tf. In the special case when the domain D is the Siegel domain and the weighted Bergman spaces are the Fock spaces we find the exact formula for [∇(k)Z,T(k)f] as Toeplitz operators. This generalizes earlier work of J. E. Andersen in [Commun. Math. Phys. 255, No. 3, 727--745]. Finally, we also determine the formulas for the curvature and for the induced connection in the general case of D×V replaced by a general strictly pseudoconvex domain V⊂Cm×Cn fibered over a domain D⊂Cm. The case when the Bergman space is replaced by the Hardy space on the boundary of the domain is likewise discussed.

Název v anglickém jazyce

Connection and curvature on bundles of Bergman and Hardy spaces

Popis výsledku anglicky

We consider a complex domain D×V in the space Cm×Cn and a family of weighted Bergman spaces on V defined by a weight e-kϕ(z,w) for a pluri-subharmonic function ϕ(z,w) with a quantization parameter k. The weighted Bergman spaces define an infinite dimensional Hermitian vector bundle over the domain D. We consider the natural covariant differentiation ∇Z on the sections, namely the unitary Chern connections preserving the Bergman norm. We prove a Dixmier trace formula for the curvature of the unitary connection and we find the asymptotic expansion for the curvatures R(k) (Z,Z) for large k and for the induced connection [∇(k)Z,T(k)f] on Toeplitz operators Tf. In the special case when the domain D is the Siegel domain and the weighted Bergman spaces are the Fock spaces we find the exact formula for [∇(k)Z,T(k)f] as Toeplitz operators. This generalizes earlier work of J. E. Andersen in [Commun. Math. Phys. 255, No. 3, 727--745]. Finally, we also determine the formulas for the curvature and for the induced connection in the general case of D×V replaced by a general strictly pseudoconvex domain V⊂Cm×Cn fibered over a domain D⊂Cm. The case when the Bergman space is replaced by the Hardy space on the boundary of the domain is likewise discussed.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2020

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Documenta Mathematica

ISSN

1431-0643

e-ISSN

—

Svazek periodika

25

Číslo periodika v rámci svazku

June

Stát vydavatele periodika

DE - Spolková republika Německo

Počet stran výsledku

29

Strana od-do

189-217

Kód UT WoS článku

000592702600007

EID výsledku v databázi Scopus

2-s2.0-85103664228