Acta mathematica scientia,Series B ›› 2002, Vol. 22 ›› Issue (4): 517-525.

• Articles • Previous Articles     Next Articles

K-POTENT PRESERVING LINEAR MAPS

 HOU Sheng-Zhao, HOU Jin-Chuan   

  1. Department of Mathematics, Shanxi Teachers University, Linfen 041004, China
  • Online:2002-10-14 Published:2002-10-14
  • Supported by:

    The project is partially supported by NNSFC and PNSFS

Abstract:

Let B(X) be the Banach algebra of all bounded linear operators on a complex Banach space X. Let k  2 be an integer and  a weakly continuous linear surjective map from B(X) into itself. It is shown that  is k-potent preserving if and only if it is k-th-power preserving, and in turn, if and only if it is either an automorphism or an antiautomorphism on B(X) multiplied by a complex number  satisfying k−1 = 1. Let A be a von Neumann
algebra and B be a Banach algebra, it is also shown that a bounded surjective linear map from A onto B is k-potent preserving if and only if it is a Jordan homomorphism multipliedby an invertible element with (k − 1)-th power I

Key words: Banach space operator, k-potent operator, automorphism

CLC Number: 

  • 47B48
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