<p>This paper investigates the algebraic, spectral, and compactness properties of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varvec{k}^{\text {th}} $</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="bold-italic">k</mi> </mrow> <mtext>th</mtext> </msup> </math></EquationSource> </InlineEquation>-order slant Hankel operators acting on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ \mathscr {L}^2(\mathbb {T}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="script">L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, along with their compressions to the Hardy space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ \mathscr {H}^2(\mathbb {T}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi mathvariant="script">H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We present a comprehensive structural analysis of these operators by examining their infinite matrix representations and deriving necessary and sufficient conditions for both normality and hyponormality. The paper further characterizes the commutativity of pairs of slant Hankel operators, establishing that such operators commute if and only if their symbol functions are scalar multiples of each other. Our compactness results reveal that these operators are non-compact except in trivial cases. In addition, we conduct a detailed investigation of their spectral properties, offering exact descriptions of the point spectrum, essential spectrum, and spectral radius. The study introduces new normality criteria grounded in the Hermitian structure of associated infinite matrices, and it resolves or disproves multiple conjectures concerning the role of symbol functions in determining operator behavior.</p>

ON THE STRUCTURE AND SPECTRUM OF $K^\text {TH}$-ORDER SLANT HANKEL OPERATORS

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发表日期：

2026年4月28日

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Oinam Nilbir Singh,
M. P. Singh,
Thokchom Sonamani Singh

摘要

This paper investigates the algebraic, spectral, and compactness properties of the $\varvec{k}^{\text {th}} $ $k^{th}$ -order slant Hankel operators acting on $ \mathscr {L}^2(\mathbb {T}) $ $L^{2} (T)$ , along with their compressions to the Hardy space $ \mathscr {H}^2(\mathbb {T}) $ $H^{2} (T)$ . We present a comprehensive structural analysis of these operators by examining their infinite matrix representations and deriving necessary and sufficient conditions for both normality and hyponormality. The paper further characterizes the commutativity of pairs of slant Hankel operators, establishing that such operators commute if and only if their symbol functions are scalar multiples of each other. Our compactness results reveal that these operators are non-compact except in trivial cases. In addition, we conduct a detailed investigation of their spectral properties, offering exact descriptions of the point spectrum, essential spectrum, and spectral radius. The study introduces new normality criteria grounded in the Hermitian structure of associated infinite matrices, and it resolves or disproves multiple conjectures concerning the role of symbol functions in determining operator behavior.

ON THE STRUCTURE AND SPECTRUM OF \(K^\text {TH}\)-ORDER SLANT HANKEL OPERATORS

摘要