Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denote the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C^*$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mo>∗</mo> </msup> </math></EquationSource> </InlineEquation>-algebra of bounded operators on a complex Hilbert space <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. For a nonempty set <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> and a function <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$F:\Omega \rightarrow \mathbb {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>F</mi> <mo>:</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">→</mo> <mi mathvariant="double-struck">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we introduce the joint numerical range <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {W}(F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">W</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and the joint algebraic numerical range <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathcal {V}(F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">V</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> as sets of complex functions on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>. If <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation> is a compact Hausdorff space and F is continuous, then <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {W}(F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">W</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathcal {V}(F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">V</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> are totally bounded sets in <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathscr {C}(\Omega )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">C</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Ω</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, the space of continuous complex functions on <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\Omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Ω</mi> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mathcal {V}(F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">V</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> equals <InlineEquation ID="IEq15"> <EquationSource Format="TEX">${\overline{\textrm{co}}}\mathcal {W}(F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mtext>co</mtext> <mo>¯</mo> </mover> <mi mathvariant="script">W</mi> <mrow> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, the closed convex hull of <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\mathcal {W}(F)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">W</mi> <mo stretchy="false">(</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Naturally, every n-tuple <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$(T_1,\dotsc ,T_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>T</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\mathbb {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is interpreted as a function of <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\{1,\dotsc ,n\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>n</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\mathbb {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$i\mapsto T_i$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>i</mi> <mo>↦</mo> <msub> <mi>T</mi> <mi>i</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. Therefore, this work generalizes the results related to operator tuples to a broad class of infinite systems.

Joint numerical ranges of continuous operator-valued functions

期刊论文
发表日期：

2025年12月15日

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Mortaza Abtahi,
Amin Azadi

摘要

Let $\mathbb {B}(\mathcal {H})$ $B (H)$ denote the $C^*$ $C^{*}$ -algebra of bounded operators on a complex Hilbert space $\mathcal {H}$ $H$ . For a nonempty set $\Omega $ $Ω$ and a function $F:\Omega \rightarrow \mathbb {B}(\mathcal {H})$ $F : Ω \to B (H)$ , we introduce the joint numerical range $\mathcal {W}(F)$ $W (F)$ and the joint algebraic numerical range $\mathcal {V}(F)$ $V (F)$ as sets of complex functions on $\Omega $ $Ω$ . If $\Omega $ $Ω$ is a compact Hausdorff space and F is continuous, then $\mathcal {W}(F)$ $W (F)$ and $\mathcal {V}(F)$ $V (F)$ are totally bounded sets in $\mathscr {C}(\Omega )$ $C (Ω)$ , the space of continuous complex functions on $\Omega $ $Ω$ , and $\mathcal {V}(F)$ $V (F)$ equals ${\overline{\textrm{co}}}\mathcal {W}(F)$ $\bar{co} W (F)$ , the closed convex hull of $\mathcal {W}(F)$ $W (F)$ . Naturally, every n-tuple $(T_1,\dotsc ,T_n)$ $(T_{1}, \dots, T_{n})$ in $\mathbb {B}(\mathcal {H})$ $B (H)$ is interpreted as a function of $\{1,\dotsc ,n\}$ ${1, \dots, n}$ to $\mathbb {B}(\mathcal {H})$ $B (H)$ , where $i\mapsto T_i$ $i \mapsto T_{i}$ . Therefore, this work generalizes the results related to operator tuples to a broad class of infinite systems.