Sets of bounded linear operators , ⊂ ℬ(H) (ℋ is a Hilbert space) are similar if there exists an invertible (in ℬ(H)) operator G such that . A bounded operator is scalar if it is similar to a normal operator. is jointly scalar if there exists a set ⊂ ℬ(H) of normal operators such that and are similar. is separately scalar if all its elements are scalar. Some necessary and sufficient conditions for joint scalarity of a separately scalar abelian set of Hilbert space operators are presented (Theorems 3.7, 4.4 and 4.6). Continuous algebra homomorphisms between the algebra of all complex-valued continuous functions on a compact Hausdorff space and the algebra of all bounded operators in a Hilbert space are studied.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-1-3,
author = {Piotr Niemiec},
title = {Separate and joint similarity to families of normal operators},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {39-62},
zbl = {1016.47027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-1-3}
}
Piotr Niemiec. Separate and joint similarity to families of normal operators. Studia Mathematica, Tome 151 (2002) pp. 39-62. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-1-3/