Block Toeplitz operators with rational symbols

Toeplitz operators (or equivalently, Wiener-Hopf operators; more generally, block Toeplitz operators; and particularly, Toeplitz determinants) are of importance in connection with a variety of problems in physics, and in particular, in the field of quantum mechanics. For example, a study of solvable models in quantum mechanics uses the spectral theory of Toeplitz operators (cf. [Pr]); the one-dimensional Heisenberg Hamiltonian of ferromagnetism is written as a direct sum of the sums of Toeplitz operators and multiplicative potentials, so that a study on the spectral properties of Toeplitz operators is required in understanding this model (cf. [DMA]); a study of quantum spin chains uses Toeplitz determinants (cf. [KMN]); a study of the vicious walkers model uses the Toeplitz and Fredholm theory (cf. [HI]); and the theory of block Toeplitz determinants plays an important role in the study of high-temperature superconductivity (cf. [BE]). On the other hand, the theory of hyponormal operators is an extensive and highly developed area. In particular, a study of the spectral properties of hyponormal operators has made important contributions in the study of related mathematical physics problems. For example, if T is a hyponormal operator then the norm of ||Tn|| can be easily computed from the n power of ||T ||. Also hyponormal operators enjoy Weyl’s theorem, which is the statement that if T − λI is a non-invertible Fredholm operator of index zero then λ is an isolated eigenvalue of finite multiplicity, and vice versa. Besides, hyponormal operators possess many useful spectral properties. Consequently, it is quite informative to know the hyponormality of (block) Toeplitz operators. In this paper we are concerned with the hyponormality of block Toeplitz operators with rational symbols. A bounded linear operator A on an infinite dimensional complex Hilbert space H is said to be hyponormal if its self-commutator [A∗, A] = A∗A−AA∗ is positive (semidefinite). For φ in L∞(T) of the unit circle T = ∂ D, the (single) Toeplitz operator with symbol φ is the operator Tφ on the Hardy space H(T) defined by