Hyponormality of Toeplitz operators with polynomial symbols

Abstract

A bounded linear operator A on a Hilbert space H with inner product (·, ·) is said to be hyponormal if its selfcommutator [A∗, A] = A∗A − AA∗ induces a positive semidefinite quadratic form on H via ξ 7→ ([A∗, A]ξ, ξ), for ξ ∈ H. Let H(T) denote the Hardy space of the unit circle T = ∂D in the complex plane. Recall that given φ ∈ L∞(T), the Toeplitz operator with symbol φ is the operator Tφ on H(T) defined by Tφf = P (φ · f), where f ∈ H(T) and P denotes the projection that maps L(T) onto H(T). The hyponormality of Toeplitz operators has been studied by C. Cowen [1],[2], P. Fan [4], C. Gu [8], T. Ito and T. Wong [9], T. Nakazi and K. Takahashi [11], D. Yu [13], K. Zhu [14], R. Curto, D. Farenick, the second and the third named authors [3],[5],[6],[10] and others. An elegant theorem of C. Cowen [2] characterizes the hyponormality of a Toeplitz operator Tφ on H(T) by properties of the symbol φ ∈ L∞(T). K. Zhu [14] reformulated Cowen’s criterion and then showed that the hyponormality of Tφ with polynomial symbols φ can be decided by a method based on the classical interpolation theorem of I. Schur [12]. Also Farenick and the third named author [5] characterized the hyponormality of Tφ in terms of the Fourier coefficients of the trigonometric polynomial φ in the cases that the outer coefficients of φ have the same modulus. But the case of arbitrary trigonometric polynomials φ, though solved in principle by Cowen’s theorem or Zhu’s theorem, is in practice very complicated. On the other hand, Nakazi and Takahashi [11, Corollary 5] showed that if φ(z) = ∑N n=−m anz n is a trigonometric polynomial with m ≤ N and if for every zero ζ of zφ such that |ζ| > 1, the number 1/ζ is a zero of zφ in the open unit disk D of multiplicity greater than or equal to the multiplicity of ζ, then Tφ is hyponormal. But the converse is not true in general. To see this consider the following trigonometric polynomial: φ(z) = z−2(z−2)(z−1)(z− 15 )(z− 13 ). Then φ(z) = 2 15z −2 − 19 15z + 55 15 − 53 15z + z. Using an argument of P. Fan [4, Theorem 1] – for every trigonometric polynomial φ of the form φ(z) = ∑2 n=−2 anz ,