Hyponormal trigonometric Toeplitz operators

We investigate hyponormal Toeplitz operators Tφ with trigonometric polynomial symbols φ via the Caratheodory-Schur Interpolation Problem. We present several formulae for computing the rank of the selfcommutator [T ∗ φ ,Tφ ] in the cases where Tφ is a hyponormal operator. In addition we consider the hyponormal extension problem of Toeplitz operators. Mathematics subject classification (2010): Primary 47B20, 47B35, 47A20.