Separate and joint similarity to families of normal operators
Volume 149 / 2002
Abstract
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.
