Remark on generalized $k$-quasihyponormal operators
Bull. Korean Math. Soc. 1998 Vol. 35, No. 4, 701-706
Eungil Ko
Ewha Womans University
Abstract : An operator $T \in {\cal{L}}(H)$ is generalized $k$-quasihyponormal if there exists a constant $M > 0$ such that \[ {T^{\ast}}^{k}[M^{2}(T-z)^{\ast}(T-z) - (T-z)(T-z)^{\ast}]T^{k} \geq 0 \] for some integer $k \geq 0$ and all $z \in {\bf C}$. In this paper, we show that if $T$ is a generalized $k$-quasihyponormal operator with the property $0 \notin \sigma(T)$, then $T$ is subscalar of order 2. As a corollary, we get that such a $T$ has a nontrivial invariant subspace if its spectrum has interior in {\bf C}.
Keywords : generalized $k$-quasihyponormal and subscalar operators, invariant subspaces
MSC numbers : 47B20, 47A15
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