Multidimensional weak resolvents and spatial equivalence of normal operators
Volume 173 / 2006
Abstract
The aim of this paper is to answer some questions concerning weak resolvents. Firstly, we investigate the domain of extension of weak resolvents $\Omega$ and find a formula linking $\Omega$ with the Taylor spectrum. We also show that equality of weak resolvents of operator tuples $A$ and $B$ results in isomorphism of the algebras generated by these operators. Although this isomorphism need not be of the form $$ X \mapsto U^{*}XU, \tag*{(1)}$$ where $U$ is an isometry, for normal operators it is always possible to find a “large” subspace on which unitary similarity holds. This observation is used to prove that the infinite inflation of the spatial isomorphism between algebras generated by inflations of $A$ and $B$, respectively, does have the form (1). These facts are generalized to other not necessarily commuting operators. We deal mostly with the self-adjoint case.
