Research Article Composition Operators on Cesàro Function Spaces Kuldip Raj, Suruchi Pandoh, and Seema Jamwal School of Mathematics, Shri Mata Vaishno Devi University, Katra 182320, India Correspondence should be addressed to Kuldip Raj; kuldeepraj68@redifmail.com Received 17 May 2013; Accepted 20 November 2013; Published 30 January 2014 Academic Editor: Satit Saejung Copyright © 2014 Kuldip Raj et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te compact, invertible, Fredholm, and closed range composition operators are characterized. We also make an efort to compute the essential norm of composition operators on the Ces` aro function spaces. 1. Introduction and Preliminaries Let (, , ) be a -fnite measure space and let  0 = 0 () denote the set of all equivalence classes of complex valued measurable functions defned on , where  = [0,1] or  = [0, ∞). Ten, for 1≤<∞, the Ces` aro function space is denoted by Ces  () and is defned as Ces  () = { ∈  0 () : ∫  ( 1  ∫  0      ()      ())   () < ∞} . (1) Te Ces` aro function space Ces  () is a Banach space under the norm          = (∫  ( 1  ∫  0      ()      ())   ()) 1/ ; (2) see [1]. Te Ces` aro functions spaces Ces  [0, ∞) for 1≤≤∞ were considered by Shiue [2], Hassard and Hussein [3], and Sy et al. [4]. Te space Ces ∞ [0, 1] appeared already in 1948 and it is known as the Korenblyum, Krein, and Levin space  (see [5, 6]). Recently, in [7], it is proved that, in contrast to Ces` aro sequence spaces, the Ces` aro function spaces Ces  () on both  = [0, 1] and  = [0, ∞) for 1<<∞ are not refexive and they do not have the fxed point property. In [8], Astashkin and Maligranda investigated Rademacher sums in Ces  [0, 1] for 1≤≤∞. Te description is diferent for 1≤<∞ and =∞. Let :→ be a nonsingular measurable trans- formation; that is,  −1 () = ( −1 ()) = 0, for each ∈, whenever () = 0. Tis condition means that the measure  −1 is absolutely continuous with respect to . Let  0 =  −1 / be the Radon-Nikodym derivative. In addition, we assume that  0 is almost everywhere fnite valued or equivalently that (,  −1 (s), ) is -fnite. An atom of the measure  is an element ∈ with () > 0 such that, for each ∈, if ⊂, then either () = 0 or () = (). Let  be an atom. Since  is -fnite, it follows that () < ∞. Also every -measurable function  on  is constant almost everywhere on . It is a well-known fact that every sigma fnite measure space (, , ) can be decomposed into two disjoint sets  1 and  2 such that  is atomic over  1 and  2 is a countable collection of disjoint atoms (see [9]). Any nonsingular measurable transformation  induces a linear operator   from Ces  () into the linear space of equivalence classes of -measurable functions on  defned by   =∘, ∈ Ces  (). Hence, the nonsingularity of  guarantees that the operator   is well defned. If   takes Ces  () into itself, then we call that   is a composition operator on Ces  (). By (Ces  ()), we denote the set of all bounded linear operators from Ces  () into itself. So far as we know, the earliest appearance of a composi- tion transformation was in 1871 in a paper of Schrljeder [10], where it is asked to fnd a function  and a number  such that ( ∘ ) () =  () , (3) Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 501057, 6 pages http://dx.doi.org/10.1155/2014/501057