For <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$0< R < 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo><</mo> <mi>R</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, let A be the annulus <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\{z \in \mathbb {C}:R< |z|<1\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mi>R</mi> <mo><</mo> <mo stretchy="false">|</mo> <mi>z</mi> <mo stretchy="false">|</mo> <mo><</mo> <mn>1</mn> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\partial A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation> be the boundary of A. Here, we consider the operator of multiplication by <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation>, denoted by <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$M_{\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation>, on the space <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L^2(\partial A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>∂</mi> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We give a complete matricial description of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$M_{\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>M</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation> and show that it is the sum of two <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$2\times 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> operator matrices and the components of each of these matrices is a doubly infinite weighted Toeplitz matrix. We define the slant Toeplitz operator <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$U_{\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$L^2(\partial A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>∂</mi> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and compression of slant Toeplitz operator <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$V_{\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>V</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$H^2(\partial A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>∂</mi> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We show that if <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\varphi $</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> is a trigonometric polynomial on <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\partial A$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>A</mi> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$U_{\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>U</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$V_{\varphi }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>V</mi> <mi>φ</mi> </msub> </math></EquationSource> </InlineEquation> cannot be hyponormal unless <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\varphi \equiv 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>φ</mi> <mo>≡</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>.

Hyponormality of slant toeplitz operators on the hardy space of an annulus

期刊论文
发表日期：

2026年8月3日

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Munmun Hazarika,
Silpi Sikha Das

摘要

For $0< R < 1$ $0 < R < 1$ , let A be the annulus $\{z \in \mathbb {C}:R< |z|<1\}$ ${z \in C : R < | z | < 1}$ and $\partial A$ $\partial A$ be the boundary of A. Here, we consider the operator of multiplication by $\varphi $ $φ$ , denoted by $M_{\varphi }$ $M_{φ}$ , on the space $L^2(\partial A)$ $L^{2} (\partial A)$ . We give a complete matricial description of $M_{\varphi }$ $M_{φ}$ and show that it is the sum of two $2\times 2$ $2 \times 2$ operator matrices and the components of each of these matrices is a doubly infinite weighted Toeplitz matrix. We define the slant Toeplitz operator $U_{\varphi }$ $U_{φ}$ on $L^2(\partial A)$ $L^{2} (\partial A)$ and compression of slant Toeplitz operator $V_{\varphi }$ $V_{φ}$ on $H^2(\partial A)$ $H^{2} (\partial A)$ . We show that if $\varphi $ $φ$ is a trigonometric polynomial on $\partial A$ $\partial A$ , then $U_{\varphi }$ $U_{φ}$ and $V_{\varphi }$ $V_{φ}$ cannot be hyponormal unless $\varphi \equiv 0$ $φ \equiv 0$ .