Munmun Hazarika and Ambeswar Phukon Tezpur University, Kokrajhar Govt. College

Abstract : In [6], it was shown that hyponormality for Toeplitz operators with polynomial symbols can be reduced to classical Schur's algorithm in function theory. In [6], Zhu has also given the explicit values of the Schur's functions $\Phi_0,\Phi_1$ and $\Phi_2$. Here we explicitly evaluate the Schur's function $\Phi_3$. Using this value we find necessary and sufficient conditions under which the Toeplitz operator $T_\varphi$ is hyponormal, where $\varphi$ is a trigonometric polynomial given by $\varphi(z)=\sum_{n=-N}^{N}a_nz_n\,(N\geq4)$ and satisfies the condition $\bar{a}_N\left( \begin{smallmatrix} a_{-1} \\ a_{-2} \\ a_{-4} \\ \vdots \\ a_{-N} \\ \end{smallmatrix} \right) =a_{-N}\left( \begin{smallmatrix} \bar{a}_1 \\ \bar{a}_2\\ \bar{a}_4 \\ \vdots \\ \bar{a}_N \\ \end{smallmatrix} \right) $. Finally we illustrate the easy applicability of the derived results with a few examples.