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Equivalent characterization of centralizers on B(H)

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Abstract

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 Φ: B(H) → B(H) satisfies Φ(AB) = Φ(A)B = AΦ(B) for any A,B ∈ B(H) with AB = Z if and only if Φ(AB) = Φ(A)B = AΦ(B), ∀A,B ∈ B(H), that is, Φ is a centralizer. Similar results are obtained for Hilbert space nest algebras. In addition, we show that Φ(A 2) = AΦ(A) = Φ(A)A for any A ∈ B(H) with A 2 = 0 if and only if Φ(A) = AΦ(I) = Φ(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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Authors and Affiliations

  1. Department of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, P. R. China

    Wen Si Xu, Run Ling An & Jin Chuan Hou

Authors
  1. Wen Si Xu
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  2. Run Ling An
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  3. Jin Chuan Hou
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Corresponding author

Correspondence to Run Ling An.

Additional information

Supported by National Natural Foundation of China (Grant No. 11001194) and Provincial International Cooperation Project of Shanxi (Grant No. 2014081027–2)

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Xu, W.S., An, R.L. & Hou, J.C. Equivalent characterization of centralizers on B(H). Acta. Math. Sin.-English Ser. 32, 1113–1120 (2016). https://doi.org/10.1007/s10114-016-5610-4

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  • DOI: https://doi.org/10.1007/s10114-016-5610-4

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