Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Studia Mathematica / All issues

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.