Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
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GENERALIZED WEYL'S THEOREM FOR ALGEBRAICALLY $k$-QUASI-PARANORMAL OPERATORS

  • Senthilkumar, D. (Post Graduate and Research Department of Mathematics, Government Arts College (Autonomous)) ;
  • Naik, P. Maheswari (Department of Mathematics Hindusthan College of Arts and Science) ;
  • Sivakumar, N. (Department of Mathematics Hindusthan College of Arts and Science)
  • Published : 2012.11.15
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Abstract

An operator $T\;{\varepsilon}\;B(\mathcal{H})$ is said to be $k$-quasi-paranormal operator if $||T^{k+1}x||^2\;{\leq}\;||T^{k+2}x||\;||T^kx||$ for every $x\;{\epsilon}\;\mathcal{H}$, $k$ is a natural number. This class of operators contains the class of paranormal operators and the class of quasi - class A operators. In this paper, using the operator matrix representation of $k$-quasi-paranormal operators which is related to the paranormal operators, we show that every algebraically $k$-quasi-paranormal operator has Bishop's property ($\beta$), which is an extension of the result proved for paranormal operators in [32]. Also we prove that (i) generalized Weyl's theorem holds for $f(T)$ for every $f\;{\epsilon}\;H({\sigma}(T))$; (ii) generalized a - Browder's theorem holds for $f(S)$ for every $S\;{\prec}\;T$ and $f\;{\epsilon}\;H({\sigma}(S))$; (iii) the spectral mapping theorem holds for the B - Weyl spectrum of T.

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References

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