Abstract.The symbols of hyponormal Toeplitz operators are completely described and those are also studied, being related with the extremal problems of Hardy spaces. Moreover, we discuss Halmos's question about a subnormal Toeplitz operator when the self-commutator is finite rank.

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Abstract. We show a polynomially boundend operator T is similar to a unitary operator if there is a singular unitary operator W and an injection X such that XT = W X. If, in addition, T is of class C̺, then T itself is unitary.According to Sz.-Nagy and Foiaş [5], a (bounded linear) operator T on a separable Hilbert space H is said to be of class C ̺ with ̺ > 0 if there exists a unitary operator U on a Hilbert space K (⊃ H) such that T n = ̺P H U n |H for n = 1, 2, . . . , where P H is the orthogonal projection of K onto H. For ̺ = 2 it is known (see [5, Chapter I, Proposition 11.2]) that T is of class C 2 if and only if its numerical radius w(T ) (:= sup{|(T x, x)| : x ≤ 1}) is not greater than one. In this paper we show that if T is of class C ̺ and there exist a singular unitary operator W and an injection X such that XT = W X, then T is unitary. Here a unitary operator is singular by definition if its spectral measure is singular with respect to the (linear) Lebesgue measure on the unit circle T. Such a situation occurs in connection with a compact operator A, as observed by Watanabe [6], which satisfies |(Ax, x)| ≤ (|A|x, x) for all x. Our result gives an affirmative answer to a conjecture that such an operator A is normal. Clearly, if T is of class C ̺ , then T is polynomially bounded, i.e., there exists a constant M such that p(T ) ≤ M max{|p(z)| : |z| = 1} for every polynomial p. In our main result (Theorem 1) an assertion for the case of a polynomially bounded operator T is also included. Though this part can be derived from a result of Mlak [2] (see also [3]), our proof is quite different from Mlak's.Let A(T) be the disk algebra, that is, A(T) is the norm closure of polynomials in the algebra C(T) of all continuous functions on T with norm 1991 Mathematics Subject Classification: Primary 47A20.

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