<p>We provide a number of conditions which characterize the closed range weighted composition–differentiation operator <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2024_1650_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$D_{(u,\psi ,n)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>ψ</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> on Fock spaces. We, in particular, prove <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2024_1650_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="53" /> </InlineMediaObject> <EquationSource Format="TEX">$D_{(u,\psi ,n)}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo>,</mo> <mi>ψ</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </msub> </math></EquationSource> </InlineEquation> has a closed range if and only if it is either trivial or surjective, and the latter happens if and only if <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11785_2024_1650_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="105" /> </InlineMediaObject> <EquationSource Format="TEX">$\psi (z)= az+b$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ψ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>a</mi> <mi>z</mi> <mo>+</mo> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation> and <i>a</i> belongs to the unit circle.</p>

On the Closed Range Weighted Composition–Differentiation Operator

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2024年12月31日

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Tesfa Mengestie

摘要

We provide a number of conditions which characterize the closed range weighted composition–differentiation operator $D_{(u,\psi ,n)}$ $D_{(u, ψ, n)}$ on Fock spaces. We, in particular, prove $D_{(u,\psi ,n)}$ $D_{(u, ψ, n)}$ has a closed range if and only if it is either trivial or surjective, and the latter happens if and only if $\psi (z)= az+b$ $ψ (z) = a z + b$ and a belongs to the unit circle.