<p>A weighted composition operator on a reproducing kernel Hilbert space is given by a composition, followed by a multiplication. We study unitary and co-isometric weighted composition operators on unitarily invariant spaces on the Euclidean unit ball <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2496_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb B_d$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">B</mi> <mi>d</mi> </msub> </math></EquationSource> </InlineEquation>. We establish a dichotomy between the spaces <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2496_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {H}_\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">H</mi> <mi>γ</mi> </msub> </math></EquationSource> </InlineEquation> with reproducing kernel <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2496_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="102" /> </InlineMediaObject> <EquationSource Format="TEX">$(1 - \langle z,w \rangle )^{-\gamma }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>-</mo> <mrow> <mo stretchy="false">⟨</mo> <mi>z</mi> <mo>,</mo> <mi>w</mi> <mo stretchy="false">⟩</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mi>γ</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2496_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$\gamma > 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and all other spaces. Whereas the former admit many unitary weighted composition operators, the latter only admit trivial ones. This extends results of Martín, Mas and Vukotić from the disc to the ball. Some of our results continue to hold when <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="25_2025_2496_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="52" /> </InlineMediaObject> <EquationSource Format="TEX">$d = \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>.</p>

Weighted Composition Operators on Hilbert Function Spaces on the Ball

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2025年8月12日

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Michael Hartz,
Maximilian Tornes

摘要

A weighted composition operator on a reproducing kernel Hilbert space is given by a composition, followed by a multiplication. We study unitary and co-isometric weighted composition operators on unitarily invariant spaces on the Euclidean unit ball $\mathbb B_d$ $B_{d}$ . We establish a dichotomy between the spaces $\mathcal {H}_\gamma $ $H_{γ}$ with reproducing kernel $(1 - \langle z,w \rangle )^{-\gamma }$ ${(1 - ⟨ z, w ⟩)}^{- γ}$ for $\gamma > 0$ $γ > 0$ , and all other spaces. Whereas the former admit many unitary weighted composition operators, the latter only admit trivial ones. This extends results of Martín, Mas and Vukotić from the disc to the ball. Some of our results continue to hold when $d = \infty $ $d = \infty$ .