The aim of this paper is to answer some questions concerning weak resolvents. Firstly, we investigate the domain of extension of weak resolvents Ω and find a formula linking Ω 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 (1) X ↦ U*XU, 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm173-2-2,
author = {Micha\l\ Jasiczak},
title = {Multidimensional weak resolvents and spatial equivalence of normal operators},
journal = {Studia Mathematica},
volume = {173},
year = {2006},
pages = {129-147},
zbl = {1098.47023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm173-2-2}
}
Michał Jasiczak. Multidimensional weak resolvents and spatial equivalence of normal operators. Studia Mathematica, Tome 173 (2006) pp. 129-147. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm173-2-2/