On Polyanalytic Functions in Several Complex Variables

We give a characterisation of the polyanalytic type subspaces of the Hilbert spaces H , being the weighted L 2 function spaces on a connected simply connected domains D ⊂ C n . The typical examples considered in the paper are the unit ball B n and the whole space  C n . Our approach is based on the...

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Bibliographic Details
Published in:Complex analysis and operator theory Vol. 17; no. 6; p. 80
Main Author: Vasilevski, Nikolai
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.09.2023
Springer Nature B.V
Subjects:
Analysis
Commutation
Complex variables
Domains
Function space
Higher Dimensional Geometric Function Theory and Hypercomplex Analysis
Hilbert space
Mathematics
Mathematics and Statistics
Operator Theory
Operators (mathematics)
Subspaces
Poly-Bergman
Poly-Vekua
Extended Fock space construction
32A36
Primary 32A30
30H20
Poly-Fock
ISSN:1661-8254, 1661-8262
Online Access:Get full text
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Summary:We give a characterisation of the polyanalytic type subspaces of the Hilbert spaces H , being the weighted L 2 function spaces on a connected simply connected domains D ⊂ C n . The typical examples considered in the paper are the unit ball B n and the whole space  C n . Our approach is based on the use of the two tuples of operators a = ( a 1 , a 2 , … , a n ) and b = ( b 1 , b 2 , … , b n ) , which act invariantly in some linear space and satisfy therein the commutation relations a j , b ℓ = δ j , ℓ I , a j , a ℓ = 0 , b j , b ℓ = 0 , j , ℓ = 1 , 2 , . . . , n . We assume further that a common invariant domain of the above operators is a dense linear subspace in a Hilbert space H , and impose several additional conditions. The exact formulation is given in the Extended Fock space construction. Further we specify the obtained description to different concrete realizations of the above operators and Hilbert spaces H , illustrating a variety of possibilities that may occur in the characterisation of the polyanalytic type spaces in several complex variables.
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ISSN:1661-8254
1661-8262
DOI:10.1007/s11785-023-01386-0