Dimensions of subspaces of a Hilbert space and index of the semi-Fredholm operator

Abstract

Let 2" denote the set of all closed subspaces of the Hilbert space H. The generalized dimension, dim gH0 for any , is introduced. Then an order is defined in [2H], the set of generalized dimensions of 2H. It makes [2H] totally ordered such that 0dim,H0dimg H for any Especially, a set of infinite dimensions are found out such that where m, n are integers with nm. Based on these facts, the generalized index, indg is defined for any A SF(H) (the set of all semi-Fredholm operators) and Tnd gA = is proved for any pure semi-Fredholm operator A SF+(H)(SF.(H)). The generalized index and dimension defined here are topologjcal and geometric, similar to the index of a Fredholm operator and the finite dimension. Some calculus of analysis can be performed on them (usually, and 1, 2,..., are identified with A known result deduced from this fact is not very proper, as will be shown later). For example, considering isometric operators in I (H) it is proved that V1,V2 are arcwise connected in B1x (H) (the set of all operators with left inverses) if and only if Ind, V1 = IndgV2. It follows that A, BeSF+(H)(SF_(H)) are arcwise connected in SF+(H)(SF_(H)) if and only if IndgAl=IndgB. The stability of Ind9 under compact or small perturbations and the continuity of the mapping Indg:SF(H) also hold. Thus the study of SF(H) is strictly based on geometric and analytic sense.