Let f be a C^1 map between two Banach spaces E and F. It has been proved that the concept of generalized regular points of f, which is a generalization of the notion of regular points of f, has some crucial applicatio...Let f be a C^1 map between two Banach spaces E and F. It has been proved that the concept of generalized regular points of f, which is a generalization of the notion of regular points of f, has some crucial applications in nonlinearity and global analysis. We characterize the generalized regular points of f using the three integer-valued （or infinite） indices M（x0）, Mc（x0） and Mr（x0） at x0 ∈ E generated by f and by analyzing generalized inverses of bounded linear operators on Banach spaces, that is, iff ＇（x0） has a generalized inverse in the Banach space B（E, F） of all bounded linear operators on E into F and at least one of the indices M（x0）, Mc（x0） and Mr（x0） is finite, then xo is a generalized regular point off if and only if the multi-index （M（x）, Me（x）, Mr（x）） is continuous at X0.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
A version of the ＂Fredholm index = spectral flow＂ theorem is proved for general families of elliptic operators （A（t）}t∈R on closed （compact and without boundary） manifolds. Here we do not require that A（t）,...A version of the ＂Fredholm index = spectral flow＂ theorem is proved for general families of elliptic operators （A（t）}t∈R on closed （compact and without boundary） manifolds. Here we do not require that A（t）, t ∈R or its leading part is self-adjoint.展开更多
A closed subspace M of the Hardy space H^(2)(D^(2))over the bidisk is called submodule if it is invariant under multiplication by coordinate functions z and w.Whether every finitely generated submodule is Hilbert-Schm...A closed subspace M of the Hardy space H^(2)(D^(2))over the bidisk is called submodule if it is invariant under multiplication by coordinate functions z and w.Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem.This paper proves that every finitely generated submodule M containing(z)-Φ(w)is Hilbert-Schmidt,where 0(z),p(w)are two finite Blaschke products.展开更多
基金The National Natural Science Foundation of China(No10271053)the Foundation of Nanjing University of Finance andEconomics (NoB0556)
文摘Let f be a C^1 map between two Banach spaces E and F. It has been proved that the concept of generalized regular points of f, which is a generalization of the notion of regular points of f, has some crucial applications in nonlinearity and global analysis. We characterize the generalized regular points of f using the three integer-valued （or infinite） indices M（x0）, Mc（x0） and Mr（x0） at x0 ∈ E generated by f and by analyzing generalized inverses of bounded linear operators on Banach spaces, that is, iff ＇（x0） has a generalized inverse in the Banach space B（E, F） of all bounded linear operators on E into F and at least one of the indices M（x0）, Mc（x0） and Mr（x0） is finite, then xo is a generalized regular point off if and only if the multi-index （M（x）, Me（x）, Mr（x）） is continuous at X0.
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.
文摘A version of the ＂Fredholm index = spectral flow＂ theorem is proved for general families of elliptic operators （A（t）}t∈R on closed （compact and without boundary） manifolds. Here we do not require that A（t）, t ∈R or its leading part is self-adjoint.
基金supported by the National Nature Science Foundation of China (Grant Nos.12031002,11971086)supported by the Dalian High-Level Talent Innovation Project (Grant No.2020RD09).
文摘A closed subspace M of the Hardy space H^(2)(D^(2))over the bidisk is called submodule if it is invariant under multiplication by coordinate functions z and w.Whether every finitely generated submodule is Hilbert-Schmidt is an unsolved problem.This paper proves that every finitely generated submodule M containing(z)-Φ(w)is Hilbert-Schmidt,where 0(z),p(w)are two finite Blaschke products.
