Finite operators

Williams, J. P.

doi:10.1090/S0002-9939-1970-0264445-6

Finite operators
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by J. P. Williams

Proc. Amer. Math. Soc. 26 (1970), 129-136

DOI: https://doi.org/10.1090/S0002-9939-1970-0264445-6

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Abstract:

A bounded linear operator $A$ on a Hilbert space $H$ is called finite if $||AX - XA - 1|| \geqq 1$ for each $X \in B(H)$. The class of finite operators is uniformly closed, contains every normal operator, every operator with a compact direct summand, and the entire ${C^ \ast }$-algebra generated by each of its members. These results imply that the set of operators with a finite dimensional reducing subspace is not uniformly dense. It is also shown that the set of self-commutators is uniformly closed.

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