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Similarities involving unbounded normal operators. (English) Zbl 1206.47006

Unbounded operators \(N, H\) and \(K\) on a Hilbert space \(\mathcal H\) are said to have the property \(P\) if they are normal and \(N = HK = KH\).
In this paper, the author mainly shows the following result: Assume that \(N, H\) and \(K\) are unbounded operators having the property \(P\), and \(D(H)\subset D(K)\). If \(A\) is a bounded operator on \(\mathcal H\) for which \(0\notin W(A)\) and such that \(AH\subset KA\), then \(H =K\), where \(D(H)\) [resp., \(D(K)\)] denotes the domain of \(H\) [resp., \(K\)] and \(W(A)\) denotes the numerical range of \(A\).
This result is a generalization to unbounded operators of a result about some similarities involving bounded normal operators due to M.R.Embry [“Similarities involving normal operators on Hilbert space”, Pac.J.Math.35, 331–336 (1970; Zbl 0188.19701)]. Furthermore, it is shown by an example that, if \(A\) is also unbounded, the result fails to be true.

MSC:

47A12 Numerical range, numerical radius
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)

Citations:

Zbl 0188.19701
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