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.)