Multidimensional weak resolvents and spatial equivalence of normal operators

Tom 173 / 2006

Micha/l Jasiczak Studia Mathematica 173 (2006), 129-147 MSC: 47B15, 47C05, 47L99. DOI: 10.4064/sm173-2-2


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.


  • Micha/l JasiczakFaculty of Mathematics and Computer Science
    Adam Mickiewicz University
    Umultowska 87
    61-614 Poznań, Poland

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