On hyponormality of Toeplitz operators with polynomial and symmetric type symbols

摘要

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 $T_arphi$ is hyponormal, where $arphi$ is a trigonometric polynomial given by $arphi(z)=sum_{n=-N}^{N}a_nz_n,(Ngeq4)$ and satisfies the condition $ar{a}_Nleft( egin{smallmatrix} a_{-1} a_{-2} a_{-4} dots a_{-N} end{smallmatrix} ight) =a_{-N}left( egin{smallmatrix} ar{a}_1 ar{a}_2 ar{a}_4 dots ar{a}_N end{smallmatrix} ight) $. Finally we illustrate the easy applicability of the derived results with a few examples.