Abstract. In this paper we shall first show that if T is a class A(k) operator then its operator transformT is hyponormal. Secondly we prove some spectral properties of T viaT. Finally we show that T has property (β).2000 Mathematics Subject Classification. 47A10, 47A63.Let H be a complex Hilbert space and L(H) the algebra of all bounded linear operators on H. An operator T ∈ L(H) has a unique polar decomposition T = U|T| where |T| = (T * T) 1 2 and U is the partial isometry satisfyingAn operator T ∈ L(H) is said to be hyponormal if T * T ≥ TT * where T * is the adjoint of T. As a generalisation of hyponormal operators, p-hyponormal and loghyponormal operators are defined in [2] and [9] respectively. An operator T is said to be p-hyponormal if and only if (T * T) p ≥ (TT * ) p for a positive number p and loghyponormal if and only if T is invertible and log(T * T) ≥ log(TT * ). An operator T is said to be of class A if and only if |T 2 | ≥ |T| 2 . See [9]. As a generalisation of class A, class A(k) and class A (s, t)

Abstract. In this article we show that the following properties hold for n-power normal operators T : (i) T has the Bishop's property(β).(ii) T is isoloid.

In this paper, we give the characterizations of n-power Quasi-isometry and n-power normal composition operators. Further, we also discuss the characterization of the n-power Quasi-isometry composite multiplication operator.

Abstract. In this paper we shall show that Weyl's theorem holds for class A(k) operators T where k ≥ 1, via its hyponormal transformT. Next we shall prove some applications of Weyl's theorem on class A(k) operators. . As a generalisation of class A(k) operators, Fujii et al. [12] introduced class A(s, t) operators. For positive numbers s and t, T belongs to class A(s, t) if (|TIt has been shown that a class A(k, 1) operator is a class A(k) operator [22]. Since many properties of hyponormal operators are known, by giving a hyponormal transform from a class A(k) operator T to a hyponormal operatorT, we can study the properties of T viaT [18].The following inclusion relation holds among these operators.