Equivalent Characterization of Centralizers on B（H）

Let H be a Hilbert space with dimH ≥2 and Z ∈ B（H） be an arbitrary but fixed operator. In this paper we show that an additive map （I）： B（H）→ B（H） satisfies Ф（AB） = Ф（A）B = AФ（B） for any A, B ∈ B（H） with AB = Z if and only if Ф（AB） = Ф（A）B = AФ（B）, A, B ∈ B（H）, that is, （I） is a centralizer. Similar...

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Veröffentlicht in:	数学学报：英文版 2016 (9), p.1113-1120
1. Verfasser:	Wen Si XU Run Ling AN Jin Chuan HOU
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Schlagworte:	dim 希尔伯特空间扶正器无限维添加剂等价刻画线性代数运营商
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description	Let H be a Hilbert space with dimH ≥2 and Z ∈ B（H） be an arbitrary but fixed operator. In this paper we show that an additive map （I）： B（H）→ B（H） satisfies Ф（AB） = Ф（A）B = AФ（B） for any A, B ∈ B（H） with AB = Z if and only if Ф（AB） = Ф（A）B = AФ（B）, A, B ∈ B（H）, that is, （I） is a centralizer. Similar results are obtained for Hilbert space nest algebras. In addition, we show that Ф（A2） = AФ（A） = Ф（A）A for any A ∈ B（H） with A2 = 0 if and only if Ф（A） = AФ（I） = Ф（I）A, A ∈ B（H）, and generalize main results in Linear Algebra and its Application, 450, 243-249 （2014） to infinite dimensional case. New equivalent characterization of centralizers on B（H） is obtained.
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title	Equivalent Characterization of Centralizers on B（H）
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