Let H be a complex Hilbert space. Denote by <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\cal{B}}(H)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\cal{B}}_{s}(H)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="script">B</mi> </mrow> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> the algebra of all bounded linear operators on H and the Jordan algebra of all self-adjoint operators in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\cal{B}}(H)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>, respectively. In this paper, the authors first give some elementary properties about higher dimensional numerical range of generalized (Jordan) product of operators in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\cal{B}}(H)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>. Based on these results, all surjective maps on <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\cal{B}}(H)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> (respectively, <InlineEquation ID="IEq6"> <EquationSource Format="TEX">${\cal{B}}_{s}(H)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="script">B</mi> </mrow> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>) that preserve higher dimensional numerical range of generalized (Jordan) product of operators are completely characterized. Particularly, they give complete characterizations of all surjective maps preserving higher dimensional numerical range of operator products, Jordan semi-triple products and Jordan products on <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\cal{B}}(H)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mrow> <mi mathvariant="script">B</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">${\cal{B}}_{s}(H)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mrow> <mrow> <mi mathvariant="script">B</mi> </mrow> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>H</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation>, respectively.

Higher Dimensional Numerical Range of Generalized (Jordan) Product of Operators

期刊论文
发表日期：

2026年6月24日

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Shaoxing Sun,
Xiaofei Qi,
Ting Zhang

摘要

Let H be a complex Hilbert space. Denote by ${\cal{B}}(H)$ $B (H)$ and ${\cal{B}}_{s}(H)$ $B_{s} (H)$ the algebra of all bounded linear operators on H and the Jordan algebra of all self-adjoint operators in ${\cal{B}}(H)$ $B (H)$ , respectively. In this paper, the authors first give some elementary properties about higher dimensional numerical range of generalized (Jordan) product of operators in ${\cal{B}}(H)$ $B (H)$ . Based on these results, all surjective maps on ${\cal{B}}(H)$ $B (H)$ (respectively, ${\cal{B}}_{s}(H)$ $B_{s} (H)$ ) that preserve higher dimensional numerical range of generalized (Jordan) product of operators are completely characterized. Particularly, they give complete characterizations of all surjective maps preserving higher dimensional numerical range of operator products, Jordan semi-triple products and Jordan products on ${\cal{B}}(H)$ $B (H)$ and ${\cal{B}}_{s}(H)$ $B_{s} (H)$ , respectively.