Abstract
A theorem of Fillmore, Stampfli and Williams asserts that a bounded linear Hilbert space operator is an essential isometry if and only if it is a com- pact perturbation of either an isometry or a coisometry with finite-dimensional kernel. In this note, we discuss the spherical analog of this result. It turns out that the spherical analog of this result does not hold verbatim, and this failure may be attributed to the fact that in dimension d ≥ 2, there exist spherical isometries with finite-dimensional joint cokernel, which are not essential spher- ical unitaries. We also discuss some strictly higher-dimensional obstructions in representing an essential spherical isometry as a compact perturbation of a spherical isometry.
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Communicated by L. Kérchy
Acknowledgment.
The author conveys his sincere thanks to Rajeev Gupta and Shubhankar Podder for some useful inputs that improved the earlier presentation.
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Chavan, S. Essential spherical isometries. ActaSci.Math. 85, 589–594 (2019). https://doi.org/10.14232/actasm-018-335-6
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DOI: https://doi.org/10.14232/actasm-018-335-6