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• W4384342801 abstract "Abstract We give a characterisation of the polyanalytic type subspaces of the Hilbert spaces $$mathcal {H}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>H</mml:mi> </mml:math> , being the weighted $$L_2$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> function spaces on a connected simply connected domains $$D subset mathbb {C}^n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> . The typical examples considered in the paper are the unit ball $$mathbb {B}^n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> and the whole space $$mathbb {C}^n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:math> . Our approach is based on the use of the two tuples of operators $$begin{aligned} mathbf {mathfrak {a}}= (mathfrak {a}_1, mathfrak {a}_2, ldots , mathfrak {a}_n) quad textrm{and} quad mathbf {mathfrak {b}}= (mathfrak {b}_1, mathfrak {b}_2, ldots , mathfrak {b}_n), end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mspace /> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mspace /> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace /> <mml:mtext>and</mml:mtext> <mml:mspace /> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mspace /> <mml:msub> <mml:mi>b</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mo>…</mml:mo> <mml:mo>,</mml:mo> <mml:mspace /> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> which act invariantly in some linear space and satisfy therein the commutation relations $$begin{aligned} left[ mathfrak {a}_j, mathfrak {b}_{ell }right] = delta _{j,ell }I, quad left[ mathfrak {a}_j, mathfrak {a}_{ell }right] = 0, quad left[ mathfrak {b}_j, mathfrak {b}_{ell }right] = 0, quad j,ell = 1,2,...,n. end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mfenced> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:mfenced> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>δ</mml:mi> <mml:mrow> <mml:mi>j</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:msub> <mml:mi>I</mml:mi> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mfenced> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mfenced> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>ℓ</mml:mi> </mml:msub> </mml:mfenced> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>j</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>.</mml:mo> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>.</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> We assume further that a common invariant domain of the above operators is a dense linear subspace in a Hilbert space $$mathcal {H}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>H</mml:mi> </mml:math> , 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 $$mathcal {H}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>H</mml:mi> </mml:math> , illustrating a variety of possibilities that may occur in the characterisation of the polyanalytic type spaces in several complex variables." @default.
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• W4384342801 date "2023-07-14" @default.
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• W4384342801 title "On Polyanalytic Functions in Several Complex Variables" @default.
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• W4384342801 doi "https://doi.org/10.1007/s11785-023-01386-0" @default.
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