Connection and curvature on bundles on Bergman and Hardy spaces

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F47813059%3A19610%2F20%3AA0000067" target="_blank" >RIV/47813059:19610/20:A0000067 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.elibm.org/article/10012026" target="_blank" >https://www.elibm.org/article/10012026</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 on Bergman and Hardy spaces

Popis výsledku v původním jazyce

We consider a complex domain D x V in the space C-m x C-n and a family of weighted Bergman spaces on V defined by a weight e(-k phi(z , w)) for a pluri-subharmonic function phi(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 del(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 [del((k))(Z), T-f((k))] on Toeplitz operators T-f. 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 [del((k))(Z), T-f((k))] as Toeplitz operators. This generalizes earlier work of J.E. Andersen in Comm. Math. Phys. 255 (2005), 727-745. Finally, we also determine the formulas for the curvature and for the induced connection in the general case of D x V replaced by a general strictly pseudoconvex domain V subset of C-m x C-n fibered over a domain D subset of C-m. 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 on Bergman and Hardy spaces

Popis výsledku anglicky

We consider a complex domain D x V in the space C-m x C-n and a family of weighted Bergman spaces on V defined by a weight e(-k phi(z , w)) for a pluri-subharmonic function phi(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 del(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 [del((k))(Z), T-f((k))] on Toeplitz operators T-f. 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 [del((k))(Z), T-f((k))] as Toeplitz operators. This generalizes earlier work of J.E. Andersen in Comm. Math. Phys. 255 (2005), 727-745. Finally, we also determine the formulas for the curvature and for the induced connection in the general case of D x V replaced by a general strictly pseudoconvex domain V subset of C-m x C-n fibered over a domain D subset of C-m. 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

<a href="/cs/project/GA16-25995S" target="_blank" >GA16-25995S: Teorie funkcí a teorie operátorů v Bergmanových prostorech a jejich aplikace II</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

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

February

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