ON HYPONORMALITY OF TOEPLITZ OPERATORS WITH POLYNOMIAL AND SYMMETRIC TYPE SYMBOLS

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 <TEX>${\Phi}_0$</TEX>, <TEX>${\Phi}_1$</TEX> and <TEX>${\Phi}_2$</TEX>. Here we explicitly evaluate the Schur's function <TEX>${\Phi}_3$</TEX>. Using this value we find necessary and sufficient conditions under which the Toeplitz operator <TEX>$T_{\varphi}$</TEX> is hyponormal, where <TEX>${\varphi}$</TEX> is a trigonometric polynomial given by <TEX>${\varphi}(z)$</TEX> = <TEX>${\sum}^N_{n=-N}a_nz_n(N{\geq}4)$</TEX> and satisfies the condition <TEX>$\bar{a}_N\(\array{a_{-1}\\a_{-2}\\a_{-4}\\{\vdots}\\a_{-N}}\)=a_{-N}\;\(\array{\bar{a}_1\\\bar{a}_2\\\bar{a}_4\\{\vdots}\\\bar{a}_N}\)$</TEX>. Finally we illustrate the easy applicability of the derived results with a few examples.