Journal of Operator Theory

Volume 69, Issue 2, Spring 2013 pp. 339-358.

Norm closures of orbits of bounded operators

Authors: Piotr Niemiec
Author institution:Faculty of Mathematics and Computer Science: Institute of Mathematics, Jagiellonian University, ul. Lojasiewicza 6, Krakow, 30-348, Poland

Summary: To every bounded linear operator $A$ between Hilbert spaces $\CMcal{H}$ and $\CMcal{K}$ three cardinals $\iota_{\mathrm r}(A)$, $\iota_{\mathrm i}(A)$ and $\iota_{\mathrm f}(A)$ and a binary number $\iota_{\mathrm b}(A)$ are assigned in terms of which the descriptions of the norm closures of the orbits $\{G A L^{-1}\colon L \in \CMcal{G}_1,\ G \in \CMcal{G}_2\}$ are given for $\CMcal{G}_1$ and $\CMcal{G}_2$ (chosen independently) being the trivial group, the unitary group or the group of all invertible operators on $\CMcal{h}$ and $\CMcal{K}$, respectively.

Keywords: group action, closure of orbit, index of operator, Fredholm operator, semi-Fredholm operator, closed range operator, equivalence of operators

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