AN INTEGRAL REPRESENTATION FOR SEMIGROUPS OF UNBOUNDED NORMAL OPERATORS

Abstract. An integral representation for semigroups {Us}s∈S of unbounded normal operators in a Hilbert space H is presented which admits a signifi-cantly larger class of semigroups S than usual. In particular, S need not have a topology and so the traditional assumption that the functions s 7 → 〈Usx, y〉, for suitable elements x, y ∈ H, are continuous is no longer a requirement. The classical spectral theorem for a single (unbounded) normal or selfadjoint oper-ator is a consequence of the main result; the point is that the techniques used do not rely on the fact that a normal operator has a spectral decomposition via its resolution of the identity