Toeplitz operators on the Fock space: Radial component effects. (Q1865925)

This interesting paper is devoted to the study of Toeplitz operators defined on the Fock space by means of radial symbols \(a=a(r)\) such that \(\int_0^\infty | a(r)| r^ne^{-r^2}\,dr<\infty\) for each \(n=0,1,2,\dots\). It is proved that for such symbols the corresponding Toeplitz operator is unitarily equivalent to a diagonal operator on \(\ell^2\). It turns out that there exist non-zero symbols that generate the zero Toeplitz operator and unbounded symbols that generate compact Toeplitz operators, and these classes are completely characterized. It is also shown that the set of symbols generating bounded Toeplitz operators is not an algebra. Spectral properties of Toeplitz operators and Weyl pseudodifferential operators are also given.