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∈LMn∞ be a matrix-valued rational function having a “matrix pole”, i.e., there exists α∈Dα∈D for which kerHΦ⊆(z−α)HCn2, where HΦHΦ denotes the Hankel operator with symbol Φ. If(i)TΦTΦ is hyponormal;(ii)ker[TΦ⁎,TΦ] is invariant for TΦTΦ, then TΦTΦ is normal. Hence in particular, if TΦTΦ is subnormal then TΦTΦ 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.