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

Abstract

Let H be a complex Hilbert space. Denote by ${\cal{B}}(H)$ and ${\cal{B}}_{s}(H)$ the algebra of all bounded linear operators on H and the Jordan algebra of all self-adjoint operators in ${\cal{B}}(H)$, respectively. In this paper, the authors first give some elementary properties about higher dimensional numerical range of generalized (Jordan) product of operators in ${\cal{B}}(H)$. Based on these results, all surjective maps on ${\cal{B}}(H)$ (respectively, ${\cal{B}}_{s}(H)$) 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 ${\cal{B}}(H)$ and ${\cal{B}}_{s}(H)$, respectively.

Keywords

k-Dimensional numerical range / Generalized product / Generalized Jordan product / 47B49

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