Multicyclicity of unbounded normal operators and polynomial approximation in \(\mathbb{C}\)
In [Duke Math.\ J.\ 22, 75--94 (1955; Zbl 0064.11603)], \textit{J.\,Bram} showed that every star-cyclic bounded normal operator \(N\) on a separable Hilbert space \(H\) has a cyclic vector, and that every multiplication operator by the independent variable \(z\) on \(L^2(m)\), for some finite compact measure \(m\), has a cyclic vector in \(L^\infty(m)\). The author of the present paper extends this result to the case of a general unbounded normal operator, implying by this that the classical multiplicity and the multicyclicity of the operator coincide. Furthermore, he obtains a couple of consequences. The proof is a modification and development of Bram's method devloped in the above cited paper, but the extension seems to be far from being obvious and straightforward.
polynomial approximation in \(L^{2}(\mathbb{C}, m)\)
scientific article; zbMATH DE number 5604498