SOME PROPERTIES OF CLASS A(k) OPERATORS AND THEIR HYPONORMAL TRANSFORMS

Abstract

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)

WEYL'S THEOREM FOR CLASSA(k) OPERATORS

“…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].…”

“…Hence the proof is complete. Limit Condition [18]. For each λ ∈ σ a (T) and a corresponding sequence {y n } of unit vectors,T satisfies the condition lim n→∞ |T| 2 y n = |λ| 2 where T is a class A(k) operator, k > 1 andT is its hyponormal operator transform.…”

“…That is,T − λ is normaloid and hence r(T − λ) = sup {|λ| : λ ∈ σ (T − λ)} = sup {|λ| : λ ∈ σ (T) − λ} This shows that T − λ is normaloid and hence T is transaloid. Also T has the SVEP [18,Theorem 11]. Therefore Weyl's theorem holds for f (T), for every f ∈ H(σ (T)).…”

“…In [18] [Theorems 6 and 7, Corollaries 3 and 5], we have proved that if a class A(k) operator T with k > 1 satisfies the Limit Condition, We say that T ∈ B(H) is isoloid if every isolated point of σ (T) is in the point spectrum of T [3]. Also if every restriction T| M to its reducing subspace M is isoloid, then we say that T satisfies the condition (α ) [3].…”