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...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 0<dim,H0<dimg H for any Especially, a set of infinite dimensions are found out such that where m, n are integers with n>m. 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.展开更多
文摘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 0<dim,H0<dimg H for any Especially, a set of infinite dimensions are found out such that where m, n are integers with n>m. 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.
