Hyponormality of Slant Weighted Toeplitz Operators on the Torus

Here we consider a sequence of positive numbers beta = {beta k}k is an element of Zn with beta 0 = 1, and assume that there exists 0 < r <= 1 such that for each i = 1, 2, ... , n and k = (k1,. .. , kn) is an element of Zn, we have r <= beta k beta k+epsilon i <= 1 if ki >= 0, and beta k+epsilon i r <= <= 1 if ki < 0. For such a weight sequence beta, we define the weighted sequence space L2(Tn, beta) to be the set of all f(z) = E beta k k is an element of Zn akzk for which Ek is an element of Zn |ak|2 beta 2k < infinity. Here T is the unit circle in the complex plane, and for n >= 1, Tn denotes the n-Torus which is the cartesian product of n copies of T. For phi is an element of L infinity(Tn, beta), we define the slant weighted Toeplitz operator A phi on L2(Tn, beta) and establish several properties of A phi. We also prove that A phi cannot be hyponormal unless phi equivalent to 0.