Commuting subnormal operators simultaneously quasisimilar to unilateral shifts
William W. Hastings · Illinois Journal of Mathematics · 1978
Let S (St, S,) be an n-tuple of pairwise commuting subnormal opera- tors on a separable, infinite dimensional Hilbert space H.An n-tuple T (Tt,..., T,) is a normal extension of S if Tt,..., T are normal commuting operators on a Hilbert space K H and T is a normal extension of St. T. Ito A vector Xo H is a joint cyclic vector for S if the smallest subspace of H containing Xo and invariant under S t, S is all of H.To construct examples, let t be a measure with compact support in C". (By measure is meant a positive, finite measure on the Borel subsets of C".) Let H2(/) be the L2(/)-closure of ., the polynomials in z (zt, z.).Define operators Wu on L2(/) by(Wiuf)(z) zif (z (multiplication by z,)and let V Wln2.).Then U--(Utu, U.u) is an n-tuple of pairwise commuting subnormal operators and the constant function 1 is a joint cyclic vector.Furthermore Wu (Wtu, Wu) is a minimal normal extension.THEOREM 0. Suppose S (St,..., Sn) is an n-tuple of pairwise commuting subnormal operators on H with a joint cyclic vector Xo of norm 1. Suppose that S has a commutin# normal extension T (Tt, T) on K H and suppose T is minimal.Then there exists a Borel probability measure bt with compact support in C and there exists a unitary operator V: K-, L2(/z)such that Vxo= 1, VH H2(/0 and TProof.Letbe the smallest *-subalgebra of Ae(K)which contains Tt, T and I.The theorem is just the spectral theorem applied to the algebra od plus the GelfandoNaimark theorem.