<p>We consider the commutativity problem for the Berezin transform on weighted Fock spaces. Given a real number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$m>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, for every <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, we denote by <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$B_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> the Berezin transform associated to the measure <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mu _{\alpha ,m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> with density proportional to <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$e^{-\alpha |z|^m}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msup> <mrow> <mi>α</mi> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> </mrow> <mi>m</mi> </msup> </mrow> </msup> </math></EquationSource> </InlineEquation> with respect to the Lebesgue measure on the complex plane and normalized so that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mu _{\alpha ,m}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>μ</mi> <mrow> <mi>α</mi> <mo>,</mo> <mi>m</mi> </mrow> </msub> </math></EquationSource> </InlineEquation>. We show that the commutativity relation <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$B_{\alpha }B_{\beta }f=B_{\beta }B_{\alpha }f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>B</mi> <mi>α</mi> </msub> <msub> <mi>B</mi> <mi>β</mi> </msub> <mi>f</mi> <mo>=</mo> <msub> <mi>B</mi> <mi>β</mi> </msub> <msub> <mi>B</mi> <mi>α</mi> </msub> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> holds for all <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$f\in L^{\infty }(\mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mi>∞</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\alpha ,\beta > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$m=2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

On the commutativity of the Berezin transform

期刊论文
发表日期：

2026年3月23日

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Alexander Borichev,
Gérard Fantolini,
El-Hassan Youssfi

摘要

We consider the commutativity problem for the Berezin transform on weighted Fock spaces. Given a real number $m>0$ $m > 0$ , for every $\alpha >0$ $α > 0$ , we denote by $B_{\alpha }$ $B_{α}$ the Berezin transform associated to the measure $\mu _{\alpha ,m}$ $μ_{α, m}$ with density proportional to $e^{-\alpha |z|^m}$ $e^{- {α | z |}^{m}}$ with respect to the Lebesgue measure on the complex plane and normalized so that $\mu _{\alpha ,m}$ $μ_{α, m}$ . We show that the commutativity relation $B_{\alpha }B_{\beta }f=B_{\beta }B_{\alpha }f$ $B_{α} B_{β} f = B_{β} B_{α} f$ holds for all $f\in L^{\infty }(\mathbb {C})$ $f \in L^{\infty} (C)$ and $\alpha ,\beta > 0$ $α, β > 0$ if and only if $m=2$ $m = 2$ .