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Separate and joint similarity to families of normal operators

Volume 149 / 2002

Piotr Niemiec Studia Mathematica 149 (2002), 39-62 MSC: 47B40, 47B15. DOI: 10.4064/sm149-1-3

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.

Authors

  • Piotr NiemiecInstitute of Mathematics
    Jagiellonian University
    Reymonta 4
    30-059 Kraków, Poland
    e-mail

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