In this paper, we study the properties of the q-numerical ranges <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$W_q(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for an operator <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$T\in {\mathcal L}(\mathcal H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In particular, if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$T^{*}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>T</mi> <mrow> <mrow /> <mo>∗</mo> </mrow> </msup> </math></EquationSource> </InlineEquation> is similar to T or <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$T^{-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>T</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> via an invertible operator <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$X\in \mathcal {L}(\mathcal H)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>∈</mo> <mi mathvariant="script">L</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, respectively, where <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$0\not \in \overline{W_q(X^{-1})}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>∉</mo> <mover> <mrow> <msub> <mi>W</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi>X</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>¯</mo> </mover> </mrow> </math></EquationSource> </InlineEquation>, we investigate the spectrum of T and properties of T. Moreover, we prove that if T is compact and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$0\in W_q(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>∈</mo> <msub> <mi>W</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$W_q(T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>W</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is closed. Furthermore, we study about the continuity of <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\overline{W_q(\cdot )}$</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mrow> <msub> <mi>W</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation> and the Hausdorff distance. Finally, we consider properties of q-numerical ranges for special classes of operators.

On q-Numerical Ranges of Operators

期刊论文
发表日期：

2026年6月2日

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Rewayat Khan,
Eungil Ko,
Ji Eun Lee

摘要

In this paper, we study the properties of the q-numerical ranges $W_q(T)$ $W_{q} (T)$ for an operator $T\in {\mathcal L}(\mathcal H)$ $T \in L (H)$ . In particular, if $T^{*}$ $T^{*}$ is similar to T or $T^{-1}$ $T^{- 1}$ via an invertible operator $X\in \mathcal {L}(\mathcal H)$ $X \in L (H)$ , respectively, where $0\not \in \overline{W_q(X^{-1})}$ $0 \notin \bar{W_{q} (X^{- 1})}$ , we investigate the spectrum of T and properties of T. Moreover, we prove that if T is compact and $0\in W_q(T)$ $0 \in W_{q} (T)$ , then $W_q(T)$ $W_{q} (T)$ is closed. Furthermore, we study about the continuity of $\overline{W_q(\cdot )}$ $\bar{W_{q} (\cdot)}$ and the Hausdorff distance. Finally, we consider properties of q-numerical ranges for special classes of operators.