%0 Journal Article
%T Characterizations of Strong Strictly Singular Operators
%A C. Ganesa Moorthy
%A C. T. Ramasamy
%J Chinese Journal of Mathematics
%D 2013
%R 10.1155/2013/834637
%X A new class of operators called strong strictly singular operators on normed spaces is introduced. This class includes the class of precompact operators, and is contained in the class of strictly singular operators. Some properties and characterizations for these operators are derived. 1. Introduction The spaces and will denote normed spaces, and will denote a bounded linear mapping from a normed space into a normed space in this paper. Completeness is assumed only when it is specifically stated. An operator is called strictly singular if it does not have a bounded inverse on any infinite dimensional subspace contained in . If is totally bounded in , where is the open unit ball in , then is called a precompact operator. If , closure of , is compact in , then is called a compact operator. Every precompact operator is strictly singular (cf: [1]). The collection of all strictly singular (precompact) operators from into forms a closed subspace of the normed space , the collection of all bounded linear operators from into (cf: [1]). The collection of strictly singular (precompact) operators on (from into ) forms a closed ideal of the normed algebra (cf: [1]). A linear transformation has a bounded inverse if and only if for all , for some . It is easy to see as a consequence of the open mapping theorem that a continuous linear transformation from a Banach space into a Banach space has closed range, if and only if for given , there is an element such that and , for some fixed (see [2]). This gives a motivation to define a new class of operators called strong strictly singular operators. Write , where is the null space of , as the quotient space endowed with the quotient norm. Let denote the quotient map from onto . The 1-1 operator induced by is defined by . Note that is 1-1 and linear with range same as the range of . If is 1-1 then . Also, and . So, we have the following conclusions.(a) is precompact on if and only if is precompact on .(b) is compact on if and only if is compact on . 2. Definition Definition 1. An operator is said to be strong strictly singular if for any subspace of such that dim and the null space there is no positive number with the property: for a given , there is an element such that and . Lemma 2. Every strong strictly singular operator is strictly singular. Proof. Let be strong strictly singular. Suppose is an infinite dimensional subspace of such that the restriction of on has a bounded inverse. Then, there is a positive constant such that for every . Since is 1-1 on , we have . Let us consider that . Then, . Also, for given , there
%U http://www.hindawi.com/journals/cjm/2013/834637/