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})$.

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