Mortad, Mohammed Hichem Similarities involving unbounded normal operators. (English) Zbl 1206.47006 Tsukuba J. Math. 34, No. 1, 129-136 (2010). 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. Reviewer: Shanli Sun (Beijing) Cited in 1 Document 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.) Keywords:normal operator; unbounded operator; commutativity; spectral projections; Fuglede-Putnam theorem; numerical range; Hilbert space Citations:Zbl 0188.19701 PDFBibTeX XMLCite \textit{M. H. Mortad}, Tsukuba J. Math. 34, No. 1, 129--136 (2010; Zbl 1206.47006) Full Text: DOI Euclid