DIMENSIONS OF SUBSPACES OF A HILBERT-SPACE AND INDEX OF THE SEMI-FREDHOLM OPERATOR

Let 2(H) denote the set of all dosed subspaces of the Hilbert space H. The generalized dimension, dim(g)H(0) for any H-0 is an element of 2( H), is introduced. Then an order is defined in [2(H)], the set of gene ralized dimensions of 2(H). It makes [2(H)] totally ordered such that 0 less than or equal to dim(g)H(0) less than or equal to(g)H for any H -0 is an element of 2(H). Especially, a set of infinite dimensions are found out such that infinity<infinity*<infinity where m, n are integ ers with n>m. Based on these facts. the generalized index, ind(g)A is defined for any A epsilon SF(H) (the set of all semi-Fredholm operator s) and Ind(g)A = infinity (-infinity*) is proved for any pure semi-Fr edholm operator A epsilon SF_(H)(SF_(H)). The generalized index and di mension defined here are topological and geometric, similar to the ind ex of a Fredholm operator and the finite dimension. Some calculus of a nalysis an be performed on them (usually, infinity, and infinity(n), n=0, 1, 2,..., are identified with infinity. A known result deduced fr om this fact is not very proper, as will be shown later). For example, considering isometric operators in I(H) it is proved that V'(1), V-2 are arcwise connected in B-l(x)(H). (the set of all operators with lef t inverses) if and only if Ind(g)T(l) =Ind(g)V(2). It follows that A, B epsilon SF+(H)(SF-(H)) are arcwise connected in SF+(H)(SF-(H)) if an d only if Ind(g)A=lnd(g)B. The stability of Ind(g) under compactor sma ll perturbations and the continuity of the mapping Ind(g):SF(H)--> Z b oolean OR{- infinity, infinity*} also hold. Thus the study of SF(B) i s strictly based on geometric and analytic sense.