Stability of index for semi-Fredholm chains

We extend the recent stability results of Ambrozie for Fredholm essential complexes to the semi-Fredholm case. Let X,Y be Banach spaces. By an operator we always mean a bounded linear operator. The set of all operators from X to Y will be denoted by L(X,Y ). Denote by N(T ) and R(T ) the kernel and range of an operator T ∈ L(X, Y ). Recall that an operator T : X → Y is called semi-Fredholm if it has closed range and at least one of the defect numbers α(T ) = dim N(T ), β(T ) = codim R(T ) is finite. If both of them are finite then T is called Fredholm. The index of a semi-Fredholm operator is defined by ind (T ) = α(T )− β(T ). We list the most important classical stability results for semi-Fredholm operators: Let T : X → Y be a semi-Fredholm operator. Then (1) There exists ε > 0 such that ind T ′ = ind T for every (semi-Fredholm) operator T ′ ∈ L(X,Y ) with ‖T ′ − T‖ < ε. (2) There exists ε > 0 such that α(T ′) ≤ α(T ) and β(T ′) ≤ β(T ) for every (semiFredholm) operator T ′ ∈ L(X, Y ) with ‖T ′ − T‖ < ε. (3) ind (T ′) = ind (T ) for every (semi-Fredholm) operator T ′ ∈ L(X, Y ) such that T − T ′ is compact. (the condition that T ′ is semi-Fredholm is satisfied automatically for operators close enough to T ; this will not be the case in more general situations). These results were generalized for Banach space complexes. By a complex it is meant an object of the following type: K : 0−→X0 δ0 −→X1 δ1 −→ · · · δn−2 −→Xn−1δ −→Xn−→0 where Xi are Banach spaces and δi operators such that δi+iδi = 0 for every i. The complex K is semi-Fredholm if the operators δi have closed ranges and the index of K, ind (K) = n ∑ i=0 (−1)αi(K), where αi(K) = dim(N(δi)/R(δi−1)) is well-defined. It was shown in [1], [14] that the index and the defect numbers αi of semi-Fredholm complexes exhibit properties (1) and (2). Property (3) proved to be surprisingly difficult. Some partial results were obtained in [11] and for Fredholm complexes (or better to say for Fredholm essential complexes) it was proved recently by Ambrozie [2], [3]. The aim of this paper is to extend the above mentioned results to semi-Fredholm chains (for the definition see below). * The research was supported by the grant No. 201/96/0411 of GA ČR.