Subnormal and quasinormal Toeplitz operators with matrix-valued rational symbols

In this paper we deal with the subnormality and the quasinormality of Toeplitz operators with matrix-valued rational symbols. In particular, in view of Halmos’s Problem 5, we focus on the question: Which subnormal Toeplitz operators are normal or analytic ? We first prove: Let Φ ∈ L∞Mn be a matrix-valued rational function having a “matrix pole,” i.e., there exists α ∈ D for which kerHΦ ⊆ (z−α)H Cn , where HΦ denotes the Hankel operator with symbol Φ. If (i) TΦ is hyponormal; (ii) ker [T ∗ Φ, TΦ] is invariant for TΦ, then TΦ is normal. Hence in particular, if TΦ is subnormal then TΦ is normal. Next, we show that every pure quasinormal Toeplitz operator with a matrix-valued rational symbol is unitarily equivalent to an analytic Toeplitz operator.