<p>We investigate the Schatten class membership of the difference of composition operators on the standard weighted Hilbert-Bergman space over the unit disk. Our main result is to characterize Schatten <i>p</i>-class differences of composition operators for the range <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and to provide a sufficient condition for the range <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Our characterization for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> extends the known characterization for Hilbert-Schmidt differences. Our approach employs a discretization technique involving lattices and localized averaging over pseudohyperbolic disks. However, such an approach do not seem to work well for the range <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> and whether the sufficient condition for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p<2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo><</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> is also necessary remains open.</p>

Schatten Class Difference of Composition Operators on the Bergman Space

期刊论文
发表日期：

2026年5月20日

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Boo Rim Choe,
Koeun Choi,
Hyungwoon Koo,
Inyoung Park

摘要

We investigate the Schatten class membership of the difference of composition operators on the standard weighted Hilbert-Bergman space over the unit disk. Our main result is to characterize Schatten p-class differences of composition operators for the range $p\ge 2$ $p \geq 2$ and to provide a sufficient condition for the range $p<2$ $p < 2$ . Our characterization for $p\ge 2$ $p \geq 2$ extends the known characterization for Hilbert-Schmidt differences. Our approach employs a discretization technique involving lattices and localized averaging over pseudohyperbolic disks. However, such an approach do not seem to work well for the range $p<2$ $p < 2$ and whether the sufficient condition for $p<2$ $p < 2$ is also necessary remains open.