Hindawi Publishing Corporation Chinese Journal of Mathematics Volume 2013, Article ID 834637, 4 pages http://dx.doi.org/10.1155/2013/834637 Research Article Characterizations of Strong Strictly Singular Operators C. Ganesa Moorthy 1 and C. T. Ramasamy 2 1 Department of Mathematics, Alagappa University, Karaikudi 630 005, India 2 Department of Mathematics, H. H. Te Rajah’s College, Pudukkottai, Tamil Nadu 622001, India Correspondence should be addressed to C. T. Ramasamy; ctrams83@gmail.com Received 26 August 2013; Accepted 12 October 2013 Academic Editors: W. Klingenberg, X. Tang, and C. Yin Copyright © 2013 C. Ganesa Moorthy and C. T. Ramasamy. 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. A new class of operators called strong strictly singular operators on normed spaces is introduced. Tis 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 Te 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 specifcally stated. An operator  is called strictly singular if it does not have a bounded inverse on any infnite 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]). Te collection of all strictly singular (precompact) operators from  into  forms a closed subspace of the normed space B(,), the collection of all bounded linear operators from  into  (cf: [1]). Te collection of strictly singular (precompact) operators on  (from  into ) forms a closed ideal of the normed algebra B()(= B(,)) (cf: [1]). A linear transformation :→ has a bounded inverse if and only if ‖‖ ≤ ‖‖ for all ∈, for some >0. 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 fxed >0 (see [2]). Tis gives a motivation to defne 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  . Te 1-1 operator  :  → induced by  is defned 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  0 is an infnite dimensional subspace of  such that the restriction of  on  0 has a bounded inverse. Ten, there is a positive constant  such that ‖‖ ≤ ‖‖ for every ∈ 0 . Since 