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WEYL’S THEOREM, TENSOR PRODUCT, FUGLEDE-PUTNAM THEOREM AND CONTINUITY SPECTRUM FOR k-QUASI CLASS A ∗ n OPERATORS

Naim Latif Braha|Ilmi Hoxha
대한수학회 (The Korean Mathematical Society)|2014.09|대한수학회지 (Journal of the KMS)|제51권 제5호|pp. 1089-1104

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참고문헌

키워드

Fuglede-Putnam theoremtensor productsk-quasi class A ∗ n operatorsWeyl’s theorema-Weyl’s theorempolaroid operatorshyperinvariantcontinuity spectrum.
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초록

An operator T ∈ L(H), is said to belong to k-quasi class A∗n operator if T∗k(|Tn+1|2/n+1 - |T∗|2)Tk≥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 ⊗ 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.

Abstract

An operator T ∈ L(H), is said to belong to k-quasi class A∗n operator if T∗k(|Tn+1|2/n+1 - |T∗|2)Tk≥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 ⊗ 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.

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