WEYL'S THEOREM, TENSOR PRODUCT, FUGLEDE-PUTNAM THEOREM AND CONTINUITY SPECTRUM FOR k-QUASI CLASS An*OPERATO

Abstract

An operator <TEX>$T{\in}L(H)$</TEX>, is said to belong to k-quasi class <TEX>$A_n^*$</TEX> operator if <TEX>$$T^{*k}({\mid}T^{n+1}{\mid}^{\frac{2}{n+1}}-{\mid}T^*{\mid}^2)T^k{\geq}O$$</TEX> for some positive integer n and some positive integer k. First, we will see some properties of this class of operators and prove Weyl's theorem for algebraically k-quasi class <TEX>$A_n^*$</TEX>. Second, we consider the tensor product for k-quasi class <TEX>$A_n^*$</TEX>, giving a necessary and sufficient condition for <TEX>$T{\otimes}S$</TEX> to be a k-quasi class <TEX>$A_n^*$</TEX>, when T and S are both non-zero operators. Then, the existence of a nontrivial hyperinvariant subspace of k-quasi class <TEX>$A_n^*$</TEX> operator will be shown, and it will also be shown that if X is a Hilbert-Schmidt operator, A and <TEX>$(B^*)^{-1}$</TEX> are k-quasi class <TEX>$A_n^*$</TEX> operators such that AX = XB, then <TEX>$A^*X=XB^*$</TEX>. Finally, we will prove the spectrum continuity of this class of operators.