For an n-tuple, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\varvec{\lambda }:= (\lambda _1, \lambda _2, \ldots , \lambda _n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">λ</mi> </mrow> <mo>:</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>λ</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathbb {C}}^n,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msup> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> we introduce the <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\textbf{k}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">k</mi> </math></EquationSource> </InlineEquation>th-order <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varvec{\lambda }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">λ</mi> </mrow> </math></EquationSource> </InlineEquation>-slant Hankel operators on the Lebesgue space <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L^2({\mathbb {T}}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\mathbb {T}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">T</mi> </math></EquationSource> </InlineEquation> is the unit circle and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">${\textbf{k}}:= (k_1,k_2, \ldots ,k_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold">k</mi> <mo>:</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>k</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>k</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is an n-tuple with each <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$k_i \ge 1,$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>k</mi> <mi>i</mi> </msub> <mo>≥</mo> <mn>1</mn> <mo>,</mo> </mrow> </math></EquationSource> </InlineEquation> an integer (<InlineEquation ID="IEq11"> <EquationSource Format="TEX">$1 \le i \le n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>≤</mo> <mi>i</mi> <mo>≤</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>). These operators are described in terms of solutions of a system of operator equations. Further we study these operators with reference to the Calkin algebra.

Generalized Slant Hankel Operators of Multivariate Order

期刊论文
发表日期：

2026年4月6日

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Gopal Datt,
Bhawna Bansal Gupta

摘要

For an n-tuple, $\varvec{\lambda }:= (\lambda _1, \lambda _2, \ldots , \lambda _n)$ $λ : = (λ_{1}, λ_{2}, \dots, λ_{n})$ in ${\mathbb {C}}^n,$ $C^{n},$ we introduce the ${\textbf{k}}$ $k$ th-order $\varvec{\lambda }$ $λ$ -slant Hankel operators on the Lebesgue space $L^2({\mathbb {T}}^n)$ $L^{2} (T^{n})$ , where ${\mathbb {T}}$ $T$ is the unit circle and ${\textbf{k}}:= (k_1,k_2, \ldots ,k_n)$ $k : = (k_{1}, k_{2}, \dots, k_{n})$ is an n-tuple with each $k_i \ge 1,$ $k_{i} \geq 1,$ an integer ( $1 \le i \le n$ $1 \leq i \leq n$ ). These operators are described in terms of solutions of a system of operator equations. Further we study these operators with reference to the Calkin algebra.