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 compact 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 $d \geqslant 2$, there exist spherical isometries with finite-dimensional joint cokernel, which are not essential spherical unitaries. We also discuss some strictly higher-dimensional obstructions in representing an essential spherical isometry as a compact perturbation of a spherical isometry.

DOI: 10.14232/actasm-018-335-6

AMS Subject Classification (1991): 47A13; 47B20

Keyword(s): spherical isometry, essential spherical isometry

Received September 21, 2018 and in final form February 11, 2019. (Registered under 85/2018.)