International Journal of Mathematical Archive-3(2), 2012, Page: 739-746 Available online through www.ijma.info ISSN 2229 – 5046 International Journal of Mathematical Archive- 3 (2), Feb. – 2012 739 WEIGHTED COMPOSITION OF k - QUASI - PARANORMAL OPERATORS D. Senthilkumar 1 , P. Maheswari Naik 2 and R. Santhi 3* 1 Post Graduate and Research Department of Mathematics, Government Arts College (Autonomous), Coimbatore - 641 018, Tamil Nadu, India E-mail: senthilsenkumhari@gmail.com 2 Post Graduate and Research Department of Mathematics, Government Arts College (Autonomous), Coimbatore - 641 018, Tamil Nadu, India E-mail: maheswarinaik21@gmail.com 3 Department of Mathematics, Sri Ramakrishna Engineering College, Vattamalaipalayam, Coimbatore - 641 022, India E-mail: santhinithyasena@yahoo.co.in (Received on: 05-01-12; Accepted on: 09-02-12) ________________________________________________________________________________________________ ABSTRACT An operator ( ) T BH ∈ is said to be k - quasi - paranormal operator if 2 1 2 k k k T x T x Tx + + ≤ for every x H ∈ , k is a natural number. In this paper, k - quasi - paranormal composition operators on 2 L space and Hardy space is characterized. Subject Classification: Primary 47B33; Secondary 47B37. Keywords: k - quasi - paranormal operators, Composition operators, Conditional expectation, Hardy space. ________________________________________________________________________________________________ 1. INTRODUCTION AND PRELIMINARIES Let H be an infinite dimensional complex Hilbert space and ( ) BH denote the algebra of all bounded linear operators acting on H . Every operator T can be decomposed into T = UT with a partial isometry U , where T = * TT . In this paper, T = UT denotes the polar decomposition satisfying the kernel condition ( ) NU = ( ) N T . An operator T is said to be positive (denoted 0 T ≥ ) if ( , ) 0 Tx x ≥ for all x H ∈ . The operator T is said to be a p - hyponormal operator if and only if ( ) ( ) * * p p TT TT ≥ for a positive number p . In [23], the class of log - hyponormal operators is defined as follows: T is called log - hyponormal if it is invertible and satisfies ( ) ( ) * * log log p p TT TT ≥ . Class of p - hyponormal operators and class of log hyponormal operators were defined as extension class of hyponormal operators, i.e., * * TT TT ≥ . It is well known that every p - hyponormal operator is a q - hyponormal operator for 0 p q ≥ > , by the Löwner - Heinz theorem " 0 A B ≥ ≥ ensures A B α α ≥ for any [0,1] α ∈ ", and every invertible p - hyponormal operator is a log - hyponormal operator since log ( ) ⋅ is an operator monotone function. An operator T is called paranormal if 2 2 Tx Tx x ≤ for all x H ∈ . It is also well known that there exists a hyponormal operator T such that 2 T is not hyponormal (see [14]). Furuta, Ito and Yamazaki [9] introduced class () Ak and absolute - k - paranormal operators for 0 k > as generalizations of class A and paranormal operators, respectively. An operator T belongs to class () Ak if ________________________________________________________________________________________________ * Corresponding author: R. Santhi 3* ,* E-mail: santhinithyasena@yahoo.co.in