Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> be a complex Hilbert space and <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> the algebra of all bounded linear operators on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. We denote by <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathcal {R}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> the lattice of operator ranges R(A) for all <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$A\in \mathcal {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</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>. We first characterize the lattice isomorphisms of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {R}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. As applications, we classify those maps which preserve operator range inclusion relation in both directions. Moreover, we determine the structure of all bijections <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathcal {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> satisfying <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$R(\Phi (A))\subseteq R(\Phi (B)+\Phi (C))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> if and only if <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$R(A)\subseteq R(B+C)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>⊆</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mo>+</mo> <mi>C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$A,B,C\in \mathcal {B}(\mathcal {H})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>,</mo> <mi>C</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>.

Maps preserving certain operator range inclusion relation on $\mathcal {B}(\mathcal {H})$

期刊论文
发表日期：

2026年4月7日

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Yihan Ding,
Guoxing Ji,
Weijuan Shi

摘要

Let $\mathcal {H}$ $H$ be a complex Hilbert space and $\mathcal {B}(\mathcal {H})$ $B (H)$ the algebra of all bounded linear operators on $\mathcal {H}$ $H$ . We denote by $\mathcal {R}(\mathcal {H})$ $R (H)$ the lattice of operator ranges R(A) for all $A\in \mathcal {B}(\mathcal {H})$ $A \in B (H)$ . We first characterize the lattice isomorphisms of $\mathcal {R}(\mathcal {H})$ $R (H)$ . As applications, we classify those maps which preserve operator range inclusion relation in both directions. Moreover, we determine the structure of all bijections $\Phi $ $Φ$ on $\mathcal {B}(\mathcal {H})$ $B (H)$ satisfying $R(\Phi (A))\subseteq R(\Phi (B)+\Phi (C))$ $R (Φ (A)) \subseteq R (Φ (B) + Φ (C))$ if and only if $R(A)\subseteq R(B+C)$ $R (A) \subseteq R (B + C)$ for all $A,B,C\in \mathcal {B}(\mathcal {H})$ $A, B, C \in B (H)$ .

Maps preserving certain operator range inclusion relation on \(\mathcal {B}(\mathcal {H})\)

摘要