Spectral multiplicity of the solutions of polynomial operator equations

Abstract

In the note one considers operators T, acting in a Hilbert space and satisfying an equation of the form ρ(T)=A, where ρ is a polynomial, while A is a given normal operator, assumed to be either reductive or unitary. Under these conditions one computes some spectral characteristics of the operator T (spectral multiplicity, disc, lattice of invariant subspaces, etc.). Fundamental examples are the weighted substitution operators (T∶L²(X,ν)→L²(X,ν), Tf=ϕ·(f·ω), where ω is a periodic automorphism of (X,ν), ϕ∈L_∞ (X, ν).