ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2012, Vol. 33, No. 2, pp. 165–169. c  Pleiades Publishing, Ltd., 2012. Multiplication Operators on Weighted Hardy Spaces Sunil Kumar Sharma 1* and B. S. Komal 2** (Submitted by F.G. Avkhadiev) 1 Indus International University, Una (H.P.), India 2 M.I.E.T. Kot-Bhalwal, Jammu, India Received August 19, 2011 Abstract—The main purpose of this paper is to study multiplication operators on weighted Hardy spaces. DOI: 10.1134/S1995080212020126 Keywords and phrases: Adjoint of an operator, Compact operator, Multiplication operator, Weighted Hardy space. 1. INTRODUCTION Let {β n } be a sequence of positive real numbers with β(0) = 1. For 1 ≤ p< ∞, let H p (β) be the space of formal series {f : f (z)= ∑ ∞ n=0 f n z n }, where {f n } ∞ n=0 is a sequence of complex numbers such that ∑ ∞ n=0 |f n | p β p n < ∞. Then H p (β) is a Banach space under the norm ||f || p β = ∑ ∞ n=0 |∧(n)| p β p (n) < ∞. For p =2, the space H 2 (β) is a Hilbert space under the inner product defined as 〈f,g〉 = ∑ ∞ n=0 a n ¯ b n β 2 n , where f (z)= ∑ ∞ n=0 a n z n and g(z)= ∑ ∞ n=0 b n z n are elements of H 2 (β). Let Ω be a region in C and let H (Ω) denote the Banach space of holomorphic functions defined on Ω. If θ :Ω → C is a mapping such that θ · f ∈ H (Ω) whenever f ∈ Ω, then a multiplication transformation M θ : H (Ω) → H (Ω) is defined by M θ f = θ · f . In case M θ is continuous,we call it a multiplication operator induced by θ. The importance of multiplication operators arose from the spectral theorem which states that every normal operator on Hilbert space H is unitarily equavilent to a multiplication operator. A study of multiplication operators on L p (λ) has been carried out by Singh and Kumar [6], Komal and Pathania [4], where it has been noticed that most of the properties of composition operators are stated in terms of properties of multiplication operators via the relation C ∗ φ C φ = M f 0 , where M f 0 is the multiplication operator induced by f 0 = dλφ −1 dλ ,the Radon-Nikodym derivative of the measure λφ −1 with respect to the measure. The main purpose of this paper is to study multiplication operators on weighted Hardy spaces. The symbol B(H) denote the Banach algebra of all bounded linear operator on H into itself and N o denote the set {0, 1, 2, 3,... }. * E-mail: sunilshrm167@gmail.com ** E-mail: bskomal2@yahoo.co.in 165