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Bounded vectors and formally normal operators. (English) Zbl 0544.47020

Dilation theory, Toeplitz operators, and other topics, 7th int. Conf. Oper. theory, Timişoara & Herculane/Rom. 1982, Oper. Theory, Adv. Appl. 11, 363-370 (1983).
[For the entire collection see Zbl 0512.00019.]
For an unbounded operator S on a Hilbert space, a vector f in \(\cap^{\infty}_{n=1}D(S^ n)\) is said to be a bounded vector for S if there are positive numbers a and c such that \(\| S^ nf\| \leq ac^ n\) for \(n=1,2,... \). A densely defined operator N is said to be formally normal if \(D(N)\subseteq D(N^*)\) and \(\| NF\| =\| N^*f\|\) for \(f\in D(N)\). The authors prove that if N is formally normal and there is a dense linear manifold X which consists of bounded vectors for N such that N\(X\subseteq X\) and \(N^*X\subseteq X\), then the closure \(\bar N\) of N is normal. They also give a characterization of some unbounded subnormal operators which is similar to the Halmos-Bram one in the bounded case.
Reviewer: K.Takahashi

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
47B20 Subnormal operators, hyponormal operators, etc.

Citations:

Zbl 0512.00019