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On class A operators. (English) Zbl 1196.47021
Let $\mathcal{H}$ and $\mathcal{L(H)}$ be a complex separable Hilbert space and an algebra of all bounded linear operators on $\mathcal{H}$, respectively. An operator $T\in \mathcal{L(H)}$ is said to belong class A if $|T|^{2}\leq |T^{2}|$ holds. The authors obtain some properties of class A operators, in particular, that every class A operator is subscalar of order $12$. As a corollary of that result, the authors obtain partial solutions of nontrivial invariant subspace problem: Let $T\in \mathcal{L(H)}$ be a class A operator. (1) If $\sigma (T)$ has nonempty interior in $\mathbb{C}$, then $T$ has a nontrivial invariant subspace, and (2) there exists a positive integer $K$ such that for all positive integers $k\geq K$, $T^{2k}$ has a nontrivial invariant subspace. For $T\in \mathcal{L(H)}$ and $x\in \mathcal{H}$, $\mathcal{O}(x,T)=\{T^{n}x\in \mathcal{H}: n=0,1,\dots\}$ is called the orbit of $x$ under $T$. If $\mathcal{O}(x,T)$ is dense in $\mathcal{H}$, then $x$ is called a hypercyclic vector for $T$. An operator $T$ is hypertransitive if every nonzero vector in $\mathcal{H}$ is hypercyclic for $T$. The nonhypertransitive problem is known as the question whether every operator is nonhypertransitive or not. Lastly, the authors obtain the following result: If $T$ is a class A operator, then it is nonhypertransitive.
47B20Subnormal operators, hyponormal operators, etc.
47A11Local spectral properties
47A15Invariant subspaces of linear operators
47A16Cyclic vectors, hypercyclic and chaotic operators
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