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

• Published in: Journal of the Korean Mathematical Society Vol. 51; no. 5; pp. 1089 - 1104
• Main Authors: Hoxha, Ilmi, Braha, Naim Latif
• Format: Journal Article
• Language: Korean
• Published: 2014
• Subjects: k-quasi class $A_n^$ operators; tensor products; a-Weyl's theorem; hyperinvariant; Weyl's theorem; Fuglede-Putnam theorem; polaroid operators; continuity spectrum
• ISSN: 0304-9914

An operator $T{\in}L(H)$, is said to belong to k-quasi class $A_n^*$ operator if $$T^{*k}({\mid}T^{n+1}{\mid}^{\frac{2}{n+1}}-{\mid}T^*{\mid}^2)T^k{\geq}O$$ 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 $A_n^*$. Second, we consider the tensor product for k-quasi class $A_n^*$, giving a necessary and sufficient condition for $T{\otimes}S$ to be a k-quasi class $A_n^*$, when T and S are both non-zero operators. Then, the existence of a nontrivial hyperinvariant subspace of k-quasi class $A_n^*$ operator will be shown, and it will also be shown that if X is a Hilbert-Schmidt operator, A and $(B^*)^{-1}$ are k-quasi class $A_n^*$ operators such that AX = XB, then $A^*X=XB^*$. Finally, we will prove the spectrum continuity of this class of operators.