Journal of Operator Theory

Volume 52, Issue 1, Summer 2004 pp. 113-132.

Linear spans of unitary and similarity orbits of a Hilbert space operator

Authors: K.R. Davidson (1) and L.W. Marcoux (2)
Author institution: (1) Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
(2) Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Summary: In this note, we show that if a bounded linear operator $T$ acting on an infinite dimensional, separable, complex Hilbert space $\mathcal{H}$ is not of the form scalar plus compact, then every bounded linear operator on $\mathcal{H}$ can be written as a linear combination of $144 or fewer operators unitarily equivalent to $T$, as a linear combination of $6$ or fewer operators similar to $T$, and as a sum of $8$ or fewer operators similar to $T$. When $T$ is not polynomially compact, the set of all sums of 2 operators similar to $T$ is dense in $\mathcal{B}(\mathcal{H})$, while if $T$ is polynomially compact, but not of the form scalar plus compact, then the set of sums of $3$ operators similar to $T$ is dense in $\mathcal{B}(\mathcal{H})$.

Keywords: Unitary orbit, similarity orbit, numerical range, Lie ideals, polynomially compact

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