Multidimensional weak resolvents and spatial equivalence of normal operators

Micha/l Jasiczak¹

¹ Faculty of Mathematics and Computer Science Adam Mickiewicz University Umultowska 87 61-614 Poznań, Poland

Studia Mathematica, Tome 173 (2006) no. 2, pp. 129-147

Résumé

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.

Détail

DOI : 10.4064/sm173-2-2

Keywords: paper answer questions concerning weak resolvents firstly investigate domain extension weak resolvents omega formula linking omega taylor spectrum equality weak resolvents operator tuples results isomorphism algebras generated these operators although isomorphism form mapsto * tag* where isometry normal operators always possible large subspace which unitary similarity holds observation prove infinite inflation spatial isomorphism between algebras generated inflations respectively does have form these facts generalized other necessarily commuting operators mostly self adjoint