The spectral multiplicity of the solutions of polynomial operator equations

A. V. Lipin

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XVI, Tome 157 (1987), pp. 157-164

Résumé

We consider linear operators $T$ on a Hilbert space which satisfy the polynomial operator equation $p(T)=A$, where $p$ is a polynomial with complex coefficients and $A$ is a normal operator which is assumed to be either reductive or unitary. We calculate spectral characteristics of $T$: the multiplicity of the spectrum, the “disc”, the lattice of invariant subspaces, and others. The main example of the $T$'s considered is given by the weighted substitution operator $Tf=\varphi (f\circ\omega)$ on $L^2(X,\nu)$, where $\omega$ is a periodic automorphism of a measure space $(X,\nu)$ and $\varphi\in L^\infty(X,\nu)$.