In this article thc concept of local conjugation of a C^1 mapping between two Banach manifolds is introduced. Thcn a rank theorem for nonlinear scmi-Fredholm operators between two Banach manifolds and a finite rank th...In this article thc concept of local conjugation of a C^1 mapping between two Banach manifolds is introduced. Thcn a rank theorem for nonlinear scmi-Fredholm operators between two Banach manifolds and a finite rank theorem are established in global analysis.展开更多
In this article, we introduce the concept of demicompactness with respect to a closed densely defined linear operator, as a generalization of the class of demicompact operator introduced by Petryshyn in [24] and we es...In this article, we introduce the concept of demicompactness with respect to a closed densely defined linear operator, as a generalization of the class of demicompact operator introduced by Petryshyn in [24] and we establish some new results in Fredholm theory. Moreover, we apply the obtained results to discuss the incidence of some perturbation results on the behavior of relative essential spectra of unbounded linear operators acting on Banach spaces. We conclude by characterizations of the relative Schechter＇s and approximate essential spectrum.展开更多
When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be a...When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋ （B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋ （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.展开更多
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
基金This research was supported by the National Natural Science Foundation of China (10271053)the Doctoral Programme Foundation of the Ministry of Education of China
文摘In this article thc concept of local conjugation of a C^1 mapping between two Banach manifolds is introduced. Thcn a rank theorem for nonlinear scmi-Fredholm operators between two Banach manifolds and a finite rank theorem are established in global analysis.
文摘In this article, we introduce the concept of demicompactness with respect to a closed densely defined linear operator, as a generalization of the class of demicompact operator introduced by Petryshyn in [24] and we establish some new results in Fredholm theory. Moreover, we apply the obtained results to discuss the incidence of some perturbation results on the behavior of relative essential spectra of unbounded linear operators acting on Banach spaces. We conclude by characterizations of the relative Schechter＇s and approximate essential spectrum.
文摘When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋ （B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋ （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.
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.
