Extension with respect to a complement space (Q1902267)

For a pair of Hilbert spaces \(H\subseteq K\) (with possibly different inner products), denote by \(H\hookrightarrow K\) if the inclusion map from \(H\) into \(K\) is a contraction. In this paper, the author defines the class of subnormal operators (with respect to a complement space): an operator \(A\) on \(H\) is subnormal in this sense if there is a Hilbert space \(K\) with \(H\hookrightarrow K\) and a normal operator \(N\) on \(K\) such that \(A= N|H\). Then he proceeds to derive some elementary properties of such operators. For example, it is shown that \(A\) is in this class if and only if there are a quasiaffinity \(X\) (one-to-one and with dense range) and a subnormal operator (in the usual sense) \(S\) such that \(XA= SX\). [Reviewer's remark: The last result, Theorem 5, is not really relevant to the remaining ones. It is a special case of a more general result: an absolutely continuous contraction \(A\) satisfies \(XS= AX\) for some quasi affinity \(X\), where \(S\) is the simple unilateral shift, if and only if \(A\) is cyclic and is not of class \(C_0\) [cf. \textit{P. Y. Wu}, Trans. Am. Math. Soc. 291, 229-239 (1985; Zbl 0582.47026), Corollary 2.4].