For paranormal operator T on a separable complex Hilbert space H, we show that (1) Weyl's theorem holds for T , i.e., σ(T ) \ w(T ) = π00(T ) and (2) every Riesz idempotent E with respect to a non-zero isolated point λ of σ(T ) is self-adjoint(i.e., it is an orthogonal projection) and satisfies that ranE = ker(T − λ) = ker(T − λ) * .

Abstract. Let T be p-hyponormal or log-hyponormal on a Hilbert space H. Then we have XT ¼ T Ã X whenever XT Ã ¼ TX for some X 2 BðHÞ. This is an extension of Patel's result. Also for p-hyponormal or log-hyponormal T Ã , dominant S and any X 2 BðHÞ such that XT ¼ SX, we have XT Ã ¼ S Ã T.

Abstract. In this paper, we show that Weyl's theorem holds for class A operators under a certain condition. We also show that a class A operator whose Weyl spectrum equals to the one-point set {0} is always compact and normal. (2000): 47A53, 47B20. Key words and phrases: Class A operators, w -hyponormal operators, continuity of spectra, Weyl's theorem. Mathematics subject classification R E F E R E N C E S

Abstract. We shall show that the Riesz idempotent E λ of every * -paranormal operator T on a complex Hilbert space H with respect to each isolated point λ of its spectrum σ(T ) is self-adjoint and satisfies E λ H = ker(T − λ) = ker(T − λ) * . Moreover, Weyl's theorem holds for * -paranormal operators and more general for operators T satisfying the norm condition T x n ≤ T n x x n−1 for all x ∈ H. Finally, for this more general class of operators we find a sufficient condition such that E λ H = ker(T − λ) = ker(T − λ) * holds.

Let T be a (p, k)-quasihyponormal operator on a complex Hilbert space, i.e., T * k ((T * T ) p − (T T * ) p )T k 0 for a positive number 0 < p 1 and a positive integer k. Let λ 0 be an isolated point of σ (T ) and E the Riesz idempotent for λ 0 . In this paper, we prove that (1) if λ 0 / = 0, then E is self-adjoint and EH = ker(T − λ 0 ) = ker((T − λ 0 ) * ), (2) if λ 0 = 0, then EH = ker(T k ).