Multicyclicity of unbounded normal operators and polynomial approximation in C

A remarkable and much cited result of Bram [J. Bram, Subnormal operators, Duke Math. J. 22 (1955) 75–94] shows that a star-cyclic bounded normal operator in a separable Hilbert space has a cyclic vector. If, in addition, the operator is multiplication by the variable in a space L2(m) (not only unitarily equivalent to it), then it has a cyclic vector in L∞(m). We extend Bram's result to the case of a general unbounded normal operator, implying by this that the (classical) multiplicity and the multicyclicity of the operator (cf. [N.K. Nikolski, Operators, Functions and Systems: An Easy Reading, vol. 2, Math. Surveys Monogr., vol. 93, Amer. Math. Soc., Providence, 2002]) coincide. It follows that if m is a sigma-finite Borel measure on C (possibly with noncompact support), then there is a nonnegative finite Borel measure τ equivalent to m and such that L2(C,τ) is the norm-closure of the polynomials in z.