Abstract. A result due to Williams, Stampfli and Fillmore shows that an essential isometry $T$ on a Hilbert space $\mathcal H$ is a compact perturbation of an isometry if and only if ind$(T)\le 0$. A recent result of S. Chavan yields an analogous characterization of essential spherical isometries $T=(T_1,\dots ,T_n)\in \mathcal {B}(\mathcal {H})^n$ with $\dim (\bigcap _{i=1}^n\ker (T_i))\le \dim (\bigcap _{i=1}^n\ker (T_i^*))$. In the present note we show that in dimension $n>1$ the result of Chavan holds without any condition on the dimensions of the joint kernels of $T$ and $T^*$.
DOI: 10.14232/actasm-020-767-3
AMS Subject Classification
(1991): 47L05, 46B20
Keyword(s):
Birkhoff--James orthogonality,
left-symmetric point,
right-symmetric point,
supporting functional,
symmetric operator
received 17.5.2020, revised 28.5.2020, accepted 28.5.2020. (Registered under 517/2020.)
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