Toeplitz Operators generated by Continuous Vanishing Oscilation functions

[gulugur]

Hi,

I am studying Toeplitz Operators on Hardy spaces and I need an
example of an H\infty function of the upper half plane whose boundary
value function is in CVO(R) (space of continuous functions f on R for
which lim_x-->\infty sup{|f(x+h)-f(x)|:|h|<1}=0), discontinuous at
infinity and the commutator of the Toeplitz operator generated by the
function with its adjoint is compact on H^2 Hardy space of the upper
half-plane. Does anyone know an example of such a function?
Any comments would be greatly appreciated. Thank you very much in
advance.

Best regards

Ugur Gul.