Journal of Advanced Studies in Topology eISSN: 2090-388X pISSN: 2090-8288 Vol. 3, No. 4, 2012, 1–7 c  2012 Modern Science Publishers www.m-sciences.com RESEARCH ARTICLE Multiplication Operators on Musielak-Orlicz Spaces of Bochner Type Kuldip Raj, Sunil K. Sharma ∗ and Anil Kumar School of Mathematics, Shri Mata Vaishno Devi University, Katra-182320, J&K, India. (Received: 29 December 2011, Accepted: 12 April 2012) The invertible, closed range, compact and Fredholm Multiplication operators are characterized in this paper. Keywords: Multiplication operator; Orlicz space; Musielak-Orlicz space; Musielak-Orlicz space of Bochner type; invertibility; compact operator; closed range and Fredholm operator. AMS Subject Classification: 47B38, 46A06. 1. Introduction and Preliminaries Let R, R + and N denote the set of reals, non-negative reals and the set of natural numbers respectively. Let ( G, ∑ ,µ ) be a σ-finite measure space and L 0 = L 0 ( G, ∑ ,µ ) be the space of all (equivalence classes of) complex-valued functions defined on G. By ϕ : R → [0, ∞] we denote an Orlicz function, i.e., ϕ is convex, even, continuous at zero and left hand side continuous in the extended sense (that is infinite limits are not excluded) on R + (see [1–5]). By M we denote a Musielak-Orlicz function, that is M : G × R → [0, ∞] and (1) M (t,.) is an Orlicz function for µ-a.e. t ∈ G, (2) M (.,u) ∈ L 0 for any u ∈ R. The function M generates on the space L 0 the convex modular  M (f )=  G M (t, |f (t)|)dµ. The space L M =  f ∈ L 0 :  M (λf ) < ∞ for some λ> 0  is called the Musielak-Orlicz space generated by M . Its subspace E M is defined as E M =  f ∈ L 0 :  M (λf ) < ∞ for any λ> 0  . The space L M endowed with the Luxemberg norm ||f || M = inf  λ> 0:  M ( f λ ) ≤ 1  * Corresponding author Email: sunilksharma42@yahoo.co.in