NUMERICAL RANGES OF SOME FOGUEL OPERATORS

The Foguel operator is defined as FT = [S* T], where S is the right shift on a Hilbert 0 S space 7-l and T is an arbitrary bounded linear operator acting on 7-l. Obviously, the numerical range W (F0) of FT with T = 0 is the open unit disk, and it was suggested by Gau, Wang and Wu in their Linear Algebra and Applications (2021) paper that W (FaI) for non-zero a E C might be an elliptical disk. In this paper, we described W (FaI) explicitly and, as it happens, it is not.