Two conditions for subnormality of unbounded operators (Q5890197)

An (unbounded) closed densely defined operator \(T\) on a Hilbert space is shown to be subnormal if there exists a core of \(cm\)-vectors or if the linear span \({\mathcal E}\) of quasi-analytic vectors is a core together with the property \(\sum^n_{k=0} \langle T^k f,T^{n-k}f\rangle\geq 0\) for any \(f\in{\mathcal E}\) and any \(n> 0\). Here, A vector \(f\in{\mathcal D}^\infty(T)\) is called a \(cm\)-vector for \(T\), if there exists a constant \(a_f> 0\) such that \(\sum^n_{k=0} (-1)^k a^k_f{n\choose k}\|T^{k+m} f\|^2\geq 0\) for any \(n,m\geq 0\), and it is called quasi-analytic for \(T\), if \(\sum^\infty_{n=1}\|T^n f\|^{-{1\over n}}=\infty\). The questions of definition, existence, and uniqueness of minimal normal extensions are discussed.