Fredholm Pairs Research Articles

Parity as Z2‐valued spectral flow

This note is about the topology of the path space of linear Fredholm operators on a real Hilbert space. Fitzpatrick and Pejsachowicz introduced the parity of such a path, based on the Leray-Schauder degree of a path of parametrices. Here an alternative analytic approach is presented which reduces the parity to the ${\mathbb Z}_2$-valued spectral flow of an associated path of chiral skew-adjoints. Furthermore the related notion of ${\mathbb Z}_2$-index of a Fredholm pair of chiral complex structures is introduced and connected to the parity of a suitable path. Several non-trivial examples are provided. One of them concerns topological insulators, another an application to the bifurcation of a non-linear partial differential equation.

The Maslov Index in Symplectic Banach Spaces

We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying (weak) symplectic structures. Assuming vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we define the curve's Maslov index by symplectic reduction to the classical finite-dimensional case. We prove the transitivity of repeated symplectic reductions and obtain the invariance of the Maslov index under symplectic reduction, while recovering all the standard properties of the Maslov index. As an application, we consider curves of elliptic operators which have varying principal symbol, varying maximal domain and are not necessarily of Dirac type. For this class of operator curves, we derive a desuspension spectral flow formula for varying well-posed boundary conditions on manifolds with boundary and obtain the splitting of the spectral flow on partitioned manifolds.

Kadison’s Pythagorean Theorem and Essential Codimension

Kadison’s Pythagorean theorem (Proc Natl Acad Sci USA, 99(7):4178–4184, 2002; Proc Natl Acad Sci USA 99(8):5217–5222, 2002) provides a characterization of the diagonals of projections with a subtle integrality condition. Arveson (Proc Natl Acad Sci USA 104(4):1152–1158, 2007), Kaftal, Ng, Zhang (J Funct Anal 257(8):2497–2529, 2009), and Argerami (Integral Equ Oper Theory 82(1):33–49, 2015) all provide different proofs of that integrality condition. In this paper we interpret the integrality condition in terms of the essential codimension of a pair of projections introduced by Brown et al. (Proceedings of a conference on operator theory, Lecture notes in mathematics, Springer, Berlin, 1973), or, equivalently of the index of a Fredholm pair of projections introduced by Avron, Seiler and Simon (J Funct Anal 120(1):220–237, 1994). The same techniques explain the integer occurring in the characterization of diagonals of selfadjoint operators with finite spectrum by Bownik and Jasper (Trans Am Math Soc 367(7):5099–5140, 2015).

Euler-Poincaré pairing, Dirac index and elliptic pairing for Harish-Chandra modules

Let G be a connected real reductive group with maximal compact subgroup K of equal rank, and let M be the category of Harish-Chandra modules for G. We relate three differentely defined pairings between two finite length modules X and Y in M : the Euler-Poincare pairing, the natural pairing between the Dirac indices of X and Y , and the elliptic pairing of [2]. (The Dirac index IDir(X) is a virtual finite dimensional representation of K, the spin double cover of K.) Analogy with the case of Hecke algebras studied in [7] and [6] and a formal (but not rigorous) computation lead us to conjecture that the first two pairings coincide. In the second part of the paper, we show that they are both computed as the indices of Fredholm pairs (defined here in an algebraic sense) of operators acting on the same spaces. We construct Euler-Poincare functions fX for any finite length Harish-Chandra module X. These functions are very cuspidal in the sense of Labesse, and their orbital integrals on elliptic elements coincide with the character of X. From this we deduce that the Dirac index pairing coincide with the elliptic pairing. These results are the archimedean analog of results of Schneider-Stuhler [21] for p-adic groups.

Operators which are the difference of two projections

We study the set D of differencesD={A=P−Q:P,Q∈P}, where P denotes the set of orthogonal projections in H. We describe models and factorizations for elements in D, which are related to the geometry of P. The study of D throws new light on the geodesic structure of P (we show that two projections in generic position are joined by a unique minimal geodesic). The topology of D is examined, particularly its connected components are studied. Also we study the subsets Dc⊂DF, where Dc are the compact elements in D, and DF are the differences A=P−Q such that the pair (P,Q) is a Fredholm pair ((P,Q) is a Fredholm pair if QP|R(P):R(P)→R(Q) is a Fredholm operator).

Pairs of Projections: Geodesics, Fredholm and Compact Pairs

Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Cientificas y Tecnicas. Oficina de Coordinacion Administrativa Saavedra 15. Instituto Argentino de Matematica; Argentina

The Maslov index in weak symplectic functional analysis

We recall the Chernoff-Marsden definition of weak symplectic structure and give a rigorous treatment of the functional analysis and geometry of weak symplectic Banach spaces. We define the Maslov index of a continuous path of Fredholm pairs of Lagrangian subspaces in continuously varying Banach spaces. We derive basic properties of this Maslov index and emphasize the new features appearing.

On Regular Fredholm Relation Pairs

In this paper,we extend the Fredholm pairs of single valued operators to the category of multivalued linear operators,and discuss some elementary properties of Fredholm multivalued linear operator pairs.Under some appropriate conditions,we get some results for Fredholm multivalued linear operator pairs which are similar to that for the Fredholm operator pairs.

The Fredholm index of a pair of commuting operators, II

We first show that an inequality on Hilbert modules, obtained by Douglas and Yan in 1993, is always an equality. This allows us to establish the semi-continuity of the generalized Samuel multiplicities for a pair of commuting operators. Then we discuss the general structure of a Fredholm pair, aiming at developing a model theory. For application we prove that the Samuel additivity formula on Hilbert spaces of holomorphic functions is equivalent to a generalized Gleason problem. As a consequence it follows the additivity of Samuel multiplicity, in its full generality, on the symmetric Fock space. During the course we discover that a variant e ′ ( ⋅ ) of the classic algebraic Samuel multiplicity might be more suitable for Hilbert modules and can lead to better results.

Subelliptic SpinℂDirac operators, III. The Atiyah–Weinstein conjecture

In this paper we extend the results obtained in [9], [10] to manifolds with Spin C -structures defined, near the boundary, by an almost complex structure. We show that on such a manifold with a strictly pseudoconvex boundary, there are modified ¯ ∂-Neumann boundary conditions defined by projection operators, R eo , which give subelliptic Fredholm problems for the Spin C -Dirac operator, ð eo . We introduce a generalization of Fredholm pairs to the “tame” category. In this context, we show that the index of the graph closure of (ð eo , R eo ) equals

The Dichotomy Theorem for evolution bi-families

We prove that the operator G, the closure of the first-order differential operator − d / d t + D ( t ) on L 2 ( R , X ) , is Fredholm if and only if the not well-posed equation u ′ ( t ) = D ( t ) u ( t ) , t ∈ R , has exponential dichotomies on R + and R − and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D ( t ) = A + B ( t ) , A is the generator of a bi-semigroup, B ( ⋅ ) is a bounded piecewise strongly continuous operator-valued function. Also, we prove some perturbations results and consider various examples of not well-posed problems.

Subelliptic boundary conditions for Spin ℂ –Dirac operators, gluing, relative indices, and tame Fredholm pairs

Let X be a Spin manifold with boundary, such that the Spin structure is defined near the boundary by an almost complex structure, which is either strictly pseudoconvex or pseudoconcave (and hence contact). Using generalized Szego projectors, we define modified partial differential-Neumann boundary conditions, Reo, for spinors, which lead to subelliptic Fredholm boundary value problems for the Spin-Dirac operator, eth(eo). To study the index of these boundary value problems we introduce a generalization of Fredholm pairs to the "tame" category. In this context, we show that the index of the graph closure of (eth(eo), Reo) equals the tame relative index, on the boundary, between Reo and the Calderon projector. Let X0 and X1 be strictly pseudoconvex, Spin manifolds, as above. Let phi : bX1 --> bX0, be a contact diffeomorphism, S0, S1 denote generalized Szego projectors on bX0, bX1, respectively, and R0(eo), R1(eo), the subelliptic boundary conditions they define. If X1 is the manifold X1 with its orientation reversed, then the glued manifold X = X0 coproduct operator(phi) X1 has a canonical Spin structure and Dirac operator, ethX(eo). Applying these results we obtain a formula for the relative index, R-Ind(S0, phi*S1), [formula: see text]. As a special case, this formula verifies a conjecture of Atiyah and Weinstein [(1997) RIMS Kokyuroku 1014:1-14] for the index of the quantization of a contact transformation between cosphere bundles.

The geometry of Kato Grassmannians

Abstract We discuss Fredholm pairs of subspaces and associated Grassmannians in a Hilbert space. Relations between several existing definitions of Fredholm pairs are established as well as some basic geometric properties of the Kato Grassmannian. It is also shown that the so-called restricted Grassmannian can be endowed with a natural Fredholm structure making it into a Fredholm Hilbert manifold.

Fredholm theory for pairs of closed subspaces of a Banach space

We study the Fredholm theory for pairs of closed subspaces of a Banach space developed by Kato. We define the strictly singular and the strictly cosingular pairs of subspaces, and we show that some of the results of operator theory can be extended to this context. However, there are some notable differences. On the one hand, the perturbation classes problem has a positive answer in this context: the upper and lower semi-Fredholm pairs are stable under strictly singular and strictly cosingular perturbations, respectively, and this stability characterizes the strictly singular and the strictly cosingular pairs. Note that it has been proved recently that the perturbation classes problem for continuous semi-Fredholm operators has a negative answer. On the other hand, unlike in the case of operators, the Fredholm pairs are not stable under perturbation by strictly singular or strictly cosingular pairs. We also show the stability under composition of the compact, the strictly singular and the strictly cosingular pairs of subspaces.

Ordinary differential operators in Hilbert spaces and Fredholm pairs

Let $A(t)$ be a path of bounded operators on a real Hilbert space, hyperbolic at $\pm \infty$ . We study the Fredholm theory of the operator $F_A=d/dt-A(t)$ . We relate the Fredholm property of $F_A$ to the stable and unstable linear spaces of the associated system $X^{\prime}=A(t)X$ . Several examples are included to point out the differences with respect to the finite dimensional case, in particular concerning the role of the spectral flow. We define a general class of paths A for which many properties typical of the finite dimensional framework still hold. Our motivation is to develop the linear theory which is necessary for the set-up of Morse homology on Hilbert manifolds.

On the Morse index of Lagrangian systems

We give a new proof of the identity between the Morse index of a periodic solution of a Lagrangian system, positive definite in the velocities, and its Maslov index. Furthermore, we give an interpretation of the Maslov index when the Lagrangian system is not positive definite in the velocities. Our approach consists in relating the Morse index of the solution to a relative Morse index obtained passing to the Hamiltonian formulation. The main concepts used are the notion of commensurable subspaces, relative dimension, Fredholm pairs and relative index of strongly indefinite bilinear forms.

Maslov index in the infinite dimension and a splitting formula for a spectral flow

First, we prove a local spectral flow formula (Theorem 3.7) for a differentiable curve of selfadjoint Fredholm operators. This formula enables us to prove in a simple way a general spectral flow formula. Secondly, we prove a splitting formula (Theorem 4.12) for the spectral flow of a curve of selfadjoint elliptic operators on a closed manifold, which we decompose into two parts with commom boundary. Then the formula says that the spectral flow is a sum of two spectral flows on each part of the separated manifold with naturally introduced elliptic boundary conditions. In the course of proving this formula, we investigate a property of the Maslov index for paths of Fredholm pairs of Lagrangian subspaces

On Bojarski’s index formula for nonsmooth interfaces

Let D D be a Dirac type operator on a compact manifold M {\mathcal {M}} and let Σ \Sigma be a Lipschitz submanifold of codimension one partitioning M {\mathcal {M}} into two Lipschitz domains Ω ± \Omega _{\pm } . Also, let H ± p ( Σ , D ) {\mathcal {H}}^{p}_{\pm }(\Sigma ,D) be the traces on Σ \Sigma of the ( L p L^{p} -style) Hardy spaces associated with D D in Ω ± \Omega _{\pm } . Then ( H − p ( Σ , D ) , H + p ( Σ , D ) ) ({\mathcal {H}}^{p}_{-}(\Sigma ,D),{\mathcal {H}}^{p}_{+}(\Sigma ,D)) is a Fredholm pair of subspaces for L p ( Σ ) L^{p}(\Sigma ) (in Kato’s sense) whose index is the same as the index of the Dirac operator D D considered on the whole manifold M {\mathcal {M}} .

On Fredholm index in Banach spaces

We prove the stability of the index of a Fredholm complex of Banach spaces under those compact perturbations which are uniform limits of finite-rank operators (Theorem 3.3). This result is a consequence of some similar statements (Theorems 3.1 and 3.2) concerning more general objects, namely the Fredholm pairs (Definition 1.1).

The Euler characteristic is stable under compact perturbations

We prove in the general case the stability under compact perturbations of the index (i.e. the Euler characteristic) of a Fredholm complex of Banach spaces. In particular, we obtain the corresponding stability property for Fredholm multioperators. These results are the consequence of a similar statement, concerning more general objects called Fredholm pairs.