Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {X}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">X</mi> </math></EquationSource> </InlineEquation> be a certain Banach space of analytic functions on the open unit disk <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {D}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">D</mi> </math></EquationSource> </InlineEquation> in the complex plane. Let p be an analytic polynomial and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$M_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> denote the operator of multiplication by p. Under certain conditions on p, we characterize the structure of the operator S such that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$M_pS=SM_p$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>M</mi> <mi>p</mi> </msub> <mi>S</mi> <mo>=</mo> <mi>S</mi> <msub> <mi>M</mi> <mi>p</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. We determine the commutant of direct sum of certain multiplication operators on direct sum of certain spaces of functions. Assume that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\psi$</EquationSource> <EquationSource Format="MATHML"><math> <mi>ψ</mi> </math></EquationSource> </InlineEquation> is an analytic automorphism of the unit disk, we also characterize the commutant of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$M_{p(\psi )}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mi>ψ</mi> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> under certain conditions on p.

On the Commutant of Certain Multiplication Operators on Certain Banach Spaces of Functions

期刊论文
发表日期：

2025年6月26日

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Mahshid Yarmohammadi,
Bahram Khani Robati

摘要

Let $\mathcal {X}$ $X$ be a certain Banach space of analytic functions on the open unit disk $\mathbb {D}$ $D$ in the complex plane. Let p be an analytic polynomial and let $M_p$ $M_{p}$ denote the operator of multiplication by p. Under certain conditions on p, we characterize the structure of the operator S such that $M_pS=SM_p$ $M_{p} S = S M_{p}$ . We determine the commutant of direct sum of certain multiplication operators on direct sum of certain spaces of functions. Assume that $\psi$ $ψ$ is an analytic automorphism of the unit disk, we also characterize the commutant of $M_{p(\psi )}$ $M_{p (ψ)}$ under certain conditions on p.