Separate and joint similarity to families of normal operators

Piotr Niemiec¹

¹ Institute of Mathematics Jagiellonian University Reymonta 4 30-059 Kraków, Poland

Studia Mathematica, Tome 149 (2002) no. 1, pp. 39-62

Résumé

Sets of bounded linear operators ${\cal S},{\cal T} \subset \cal B(H)$ ($\cal H$ is a Hilbert space) are similar if there exists an invertible (in $\cal B(H)$) operator $G$ such that $G^{-1}\cdot {\cal S}\cdot G=\cal T$. A bounded operator is scalar if it is similar to a normal operator. $\cal S$ is jointly scalar if there exists a set ${\cal N}\subset {\cal B(H)}$ of normal operators such that $\cal S$ and $\cal N$ are similar. $\cal S$ 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.

Détail

DOI : 10.4064/sm149-1-3

Keywords: sets bounded linear operators cal cal subset cal cal hilbert space similar there exists invertible cal operator cdot cal cdot cal bounded operator scalar similar normal operator cal jointly scalar there exists set cal subset cal normal operators cal cal similar cal separately scalar its elements scalar necessary sufficient conditions joint scalarity separately scalar abelian set hilbert space operators presented theorems continuous algebra homomorphisms between algebra complex valued continuous functions compact hausdorff space algebra bounded operators hilbert space studied