On Polyanalytic Functions in Several Complex Variables

Abstract

AbstractWe give a characterisation of the polyanalytic type subspaces of the Hilbert spaces $$\mathcal {H}$$ H , being the weighted $$L_2$$ L 2 function spaces on a connected simply connected domains $$D \subset \mathbb {C}^n$$ D ⊂ C n . The typical examples considered in the paper are the unit ball $$\mathbb {B}^n$$ B n and the whole space $$\mathbb {C}^n$$ C n . Our approach is based on the use of the two tuples of operators $$\begin{aligned} \mathbf {\mathfrak {a}}= (\mathfrak {a}_1, \ \mathfrak {a}_2, \ \ldots , \ \mathfrak {a}_n) \quad \textrm{and} \quad \mathbf {\mathfrak {b}}= (\mathfrak {b}_1, \ \mathfrak {b}_2, \ \ldots , \ \mathfrak {b}_n), \end{aligned}$$ 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 $$\begin{aligned} \left[ \mathfrak {a}_j, \mathfrak {b}_{\ell }\right] = \delta _{j,\ell }I, \quad \left[ \mathfrak {a}_j, \mathfrak {a}_{\ell }\right] = 0, \quad \left[ \mathfrak {b}_j, \mathfrak {b}_{\ell }\right] = 0, \quad j,\ell = 1,2,...,n. \end{aligned}$$ 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 $$\mathcal {H}$$ 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 $$\mathcal {H}$$ H , illustrating a variety of possibilities that may occur in the characterisation of the polyanalytic type spaces in several complex variables.