Connection and curvature on bundles of Bergman and Hardy spaces

Result code in 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>
Result on the web
<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>

Result language
angličtina
Original language name
Connection and curvature on bundles of Bergman and Hardy spaces
Original language description
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.
Czech name
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Czech description
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Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics

Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2020
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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