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

Title & Authors
WEYL'S THEOREM, TENSOR PRODUCT, FUGLEDE-PUTNAM THEOREM AND CONTINUITY SPECTRUM FOR k-QUASI CLASS An* OPERATO
Hoxha, Ilmi; Braha, Naim Latif;

Abstract
An operator $\small{T{\in}L(H)}$, is said to belong to k-quasi class $\small{A_n^*}$ operator if $\small{T^{*k}({\mid}T^{n+1}{\mid}^{\frac{2}{n+1}}-{\mid}T^*{\mid}^2)T^k{\geq}O}$ 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 $\small{A_n^*}$. Second, we consider the tensor product for k-quasi class $\small{A_n^*}$, giving a necessary and sufficient condition for $\small{T{\otimes}S}$ to be a k-quasi class $\small{A_n^*}$, when T and S are both non-zero operators. Then, the existence of a nontrivial hyperinvariant subspace of k-quasi class $\small{A_n^*}$ operator will be shown, and it will also be shown that if X is a Hilbert-Schmidt operator, A and $\small{(B^*)^{-1}}$ are k-quasi class $\small{A_n^*}$ operators such that AX = XB, then $\small{A^*X=XB^*}$. Finally, we will prove the spectrum continuity of this class of operators.
Keywords
k-quasi class $\small{A_n^*}$ operators;Weyl's theorem;a-Weyl's theorem;polaroid operators;tensor products;Fuglede-Putnam theorem;hyperinvariant;continuity spectrum;
Language
English
Cited by
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