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 anal...
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<SUB>M'n</SUB><SUP>~</SUP> be a matrix-valued rational function having a ''matrix pole'', i.e., there exists α@?D for which kerH<SUB>Φ</SUB>@?(z-α)H<SUB>C^n</SUB><SUP>2</SUP>, where H<SUB>Φ</SUB> denotes the Hankel operator with symbol Φ. If(i)T<SUB>Φ</SUB> is hyponormal; (ii)ker[T<SUB>Φ</SUB><SUP>@?</SUP>,T<SUB>Φ</SUB>] is invariant for T<SUB>Φ</SUB>, then T<SUB>Φ</SUB> is normal. Hence in particular, if T<SUB>Φ</SUB> is subnormal then T<SUB>Φ</SUB> 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.