FREDHOLM OPERATORS ON THE SPACE OF BOUNDED SEQUENCES

Necessary and sufficient conditions are studied that a bounded operator Tx =(xTx, x(1)(*)x, x(2)(*)x, ... ) on the space l(infinity), where x(n)(*) is an element of l(infinity)(*), is lower or upper semi-Fredholm; in particular, topological properties of the set {x(1)x, x(2)x, ...} are investigated. Various estimates of the defect d(T) = codimR(T), where R(T) is the range of T, are given. The case of x(n)* = d(n)x(tn)(*), where d(n) is an element of R and x(tn)(*) >= 0 are extreme points of the unit ball B-l infinity*, that is t(n) is an element of beta N, is considered. In terms of the sequence {t(n)}, the conditions of the closedness of the range R(T) are given and the value d(T) is calculated. For example, the condition {n : 0 < vertical bar d(n)vertical bar < delta} = circle divide for some delta is sufficient and if for large n points to are isolated elements of the sequence ttnl, then it is also necessary for the closedness of R(T) (t(n0), is isolated if there is a neighborhood U of t(n0) satisfying t(n) is not an element of U for all n not equal n(0)). If : { n : vertical bar d(n)vertical bar < delta} = circle divide, then d(T) is equal to the defect delta{t(n)} of {t(n)}. It is shown that if d(T) = infinity and R(T) is closed, then there exists a sequence {A(n)} of pairwise disjoint subsets of N satisfying chi(An) is not an element of R(T).