Multilinear mappings and estimates of multiplicity

A method for estimating multiplicity of operators in terms of certain multilinear mappings is introduced. Several applications are given which include the following result: IfT is a completely non-unitary contraction on a complex Hilbert space, and for some vectorx₀, then there exists a constant >0, such that for every positive integern, the multiplicity of the operatorTⁿ is not less thann. This extends a result of B. Sz. Nagy and Foia [10].