<p>In this note we provide a sufficient condition on when the composition operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$C_{\Phi }:A^2_{a}(\mathbb {D}^2)\rightarrow A^2_{\beta }(\mathbb {D}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi mathvariant="normal">Φ</mi> </msub> <mo>:</mo> <msubsup> <mi>A</mi> <mi>a</mi> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">D</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">→</mo> <msubsup> <mi>A</mi> <mi>β</mi> <mn>2</mn> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">D</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is bounded, whenever <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a\ge -1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≥</mo> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\beta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>β</mi> </math></EquationSource> </InlineEquation> is positive, with the assumption that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> is induced by non-smooth Rational Inner Functions.</p>

Composition Operators and Rational Inner Functions II: Boundedness Between Two Different Bergman Spaces

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2026年4月30日

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Athanasios Beslikas

摘要

In this note we provide a sufficient condition on when the composition operator $C_{\Phi }:A^2_{a}(\mathbb {D}^2)\rightarrow A^2_{\beta }(\mathbb {D}^2)$ $C_{Φ} : A_{a}^{2} (D^{2}) \to A_{β}^{2} (D^{2})$ is bounded, whenever $a\ge -1$ $a \geq - 1$ and $\beta $ $β$ is positive, with the assumption that $\Phi $ $Φ$ is induced by non-smooth Rational Inner Functions.