Weyl's and Browder's theorem for an elementary operator

Authors

  • F. Lombarkia Department of Mathematics, Faculty of Science, University of Batna, 05000, Batna, Algeria Author
  • A. Bachir Department of Mathematics, Faculty of Science, King Khalid University, Abha, P.O.Box 9004, Saudi Arabia Author

Keywords:

Elementary Operators, p-hyponormal, log-hyponormal, Weyl's Theorem, single valued extension property

Subjects:

47B47, 47A30, 47B20

Abstract

Let H be a separable infinite dimensional complex Hilbert space and letB(H) denote the algebra of bounded operators on H into itself. Thegeneralized derivation δA,B is defined by δA,B(X)=AXXB. For pairsC=(A1,A2) and D=(B1,B2) of operators, we define the elementaryoperator ΦC,D by ΦC,D(X)=A1XB1A2XB2. If A2=B2=I, weget the elementary operator ΔA1,B1(X)=A1XB1X. LetdA,B=δA,B or ΔA,B. We prove that if A,B arelog-hyponormal, then f(dA,B) satisfies (generalized) Weyl's Theorem for eachanalytic function f on a neighborhood of σ(dA,B), we also prove thatf(ΦC,D) satisfies Browder's Theorem for each analytic function f on aneighborhood of σ(ΦC,D).

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Published

2007-07-15