<p>Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the algebra of all bounded linear operators on an infinite-dimensional Hilbert space <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. A map <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation>, from <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ \mathcal {B}(\mathcal {H}) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> into a closed subset of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$ \mathbb {C}\ $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">C</mi> <mspace width="4pt" /> </mrow> </math></EquationSource> </InlineEquation> is said to be a <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\partial $</EquationSource> <EquationSource Format="MATHML"><math> <mi>∂</mi> </math></EquationSource> </InlineEquation>-spectrum if <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\partial \sigma (T) \subseteq \Delta (T) \subseteq \sigma (T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$T \in \mathcal {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>∈</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\sigma (T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> denotes the spectrum of <i>T</i> and <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\partial \sigma (T)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>∂</mi> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> its boundary. Fix a <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\partial $</EquationSource> <EquationSource Format="MATHML"><math> <mi>∂</mi> </math></EquationSource> </InlineEquation>-spectrum map <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\Delta $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation>. In this paper, we characterize all maps <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\phi :\mathcal {B}(\mathcal {H}) \rightarrow \mathcal {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϕ</mi> <mo>:</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> whose ranges contain all operators of rank at most two and that satisfy either <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\Delta (TS^) = \Delta \big (\phi (T)\phi (S)^\big )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <msup> <mi>S</mi> <mo>∗</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mi>ϕ</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mo>∗</mo> </msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$T,S \in \mathcal {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>,</mo> <mi>S</mi> <mo>∈</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\Delta (TS^T) = \Delta \big (\phi (T)\phi (S)^\phi (T)\big )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Δ</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <msup> <mi>S</mi> <mo>∗</mo> </msup> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mi>ϕ</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mo>∗</mo> </msup> <mi>ϕ</mi> <mrow> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$T,S \in \mathcal {B}(\mathcal {H}).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>,</mo> <mi>S</mi> <mo>∈</mo> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation></p>

Maps preserving the $\partial $-spectrum of skew products of operators

期刊论文
发表日期：

2026年6月15日

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Hassane Benbouziane,
Aukacha Daoudi,
Mustapha Ech-Chérif El Kettani,
Ismail El Khchin

摘要

Let $\mathcal {B}(\mathcal {H})$ $B (H)$ be the algebra of all bounded linear operators on an infinite-dimensional Hilbert space $\mathcal {H}$ $H$ . A map $\Delta $ $Δ$ , from $ \mathcal {B}(\mathcal {H}) $ $B (H)$ into a closed subset of $ \mathbb {C}\ $ $C$ is said to be a $\partial $ $\partial$ -spectrum if $\partial \sigma (T) \subseteq \Delta (T) \subseteq \sigma (T)$ $\partial σ (T) \subseteq Δ (T) \subseteq σ (T)$ for all $T \in \mathcal {B}(\mathcal {H})$ $T \in B (H)$ , where $\sigma (T)$ $σ (T)$ denotes the spectrum of T and $\partial \sigma (T)$ $\partial σ (T)$ its boundary. Fix a $\partial $ $\partial$ -spectrum map $\Delta $ $Δ$ . In this paper, we characterize all maps $\phi :\mathcal {B}(\mathcal {H}) \rightarrow \mathcal {B}(\mathcal {H})$ $ϕ : B (H) \to B (H)$ whose ranges contain all operators of rank at most two and that satisfy either $\Delta (TS^*) = \Delta \big (\phi (T)\phi (S)^*\big )$ $Δ (T S^{*}) = Δ (ϕ (T) ϕ {(S)}^{*})$ for all $T,S \in \mathcal {B}(\mathcal {H})$ $T, S \in B (H)$ or $\Delta (TS^*T) = \Delta \big (\phi (T)\phi (S)^*\phi (T)\big )$ $Δ (T S^{*} T) = Δ (ϕ (T) ϕ {(S)}^{*} ϕ (T))$ for all $T,S \in \mathcal {B}(\mathcal {H}).$ $T, S \in B (H) .$

Maps preserving the \(\partial \)-spectrum of skew products of operators

摘要