One may trace the idea that spectral flow should be given as the integral of a one form back to the 1974 Vancouver ICM address of I.M. Singer. Our main theorem gives analytic formulae for the spectral flow along a norm differentiable path of self-adjoint bounded Breuer-Fredholm operators in a semi-finite von Neumann algebra. These formulae have a geometric interpretation which derives from the proof. Namely we define a family of Banach submanifolds of all bounded self-adjoint Breuer-Fredholm operators and on each submanifold define global one forms whose integral on a norm differentiable path contained in the submanifold calculates the spectral flow along this path. We emphasise that our methods do not give a single globally defined one form on the self adjoint Breuer- Fredholms whose integral along all paths is spectral flow rather, as the choice of the plural `forms' in the title suggests, we need a family of such one forms in order to confirm Singer's idea. The original context for this result concerned paths of unbounded self-adjoint Fredholm operators. We therefore prove analogous formulae for spectral flow in the unbounded case as well. The proof is a synthesis of key contributions by previous authors, whom we acknowledge in detail in the introduction, combined with an additional important recent advance in the differential calculus of functions of non-commuting operators.
Muraleetharan and Thirulogasanthan in (J. Math. phys. 59, No. 10, 103506, 27p. (2018)) introduced the concept of Calkin Sspectrum of a bounded quaternionic linear operators. The study of this spectrum is establisched using the Fredholm operators theory. Motivated by this, we study the general framework of the Fredholm element with respect to a quaternionic Banach algebra homomorphism. First, we investigate the Fredholm S-spectrum of the sum of two elements in quaternionic Banach algebra by means of the Fredholm S-spectrum of the two elements. Next, we prove a perturbation result on this spectr…
We describe essential (in particular Fredholm and semi-Fredholm) spectra of operators on Banach lattices of the form $T=wU$, where $w$ is a central operator and $U$ is a disjointness preserving operator such that its spectrum $σ(U)$ is a subset of the unit circle.
Given a (not necessarily continuous) homomorphism between Banach algebras $\T\colon\A\to\B$, an element $a\in\A$ will be said to be B-Fredholm (respectively generalized B-Fredholm) relative to $\T$, if $\T(a)\in \B$ is Drazin invertible (respectively Koliha-Drazin invertible). In this article, the aforementioned elements will be characterized and their main properties will be studied. In addition, perturbation properties will be also considered.
Schur coupling (SC) and equivalence after extension (EAE) are important relations for bounded operators on Banach spaces. It has been known for 30 years that the former implies the latter, but only recently Ter Horst, Messerschmidt, Ran and Roelands disproved the converse by constructing a pair of Fredholm operators which are EAE, but not SC. Motivated by this result, we investigate when EAE and SC coincide for Fred\-holm operators. Fredholm operators which are EAE have the same Fredholm index. Surprisingly, we find that for each integer $k$ and every pair of Banach spaces $(\mathcal{X},\mathc…
Let $L$ be a self-adjoint invertible operator in a Hilbert space such that $L^{-1}$ is $p$-summable. Under a certain discrete dimension spectrum assumption on $L$, we study the relation between the (regularized) Fredholm determinant, $\det_{p}(I+z\cdot L^{-1})$, on the one hand and the zeta regularized determinant, $\det_ζ(L+z)$, on the other. One of the main results is the formula \begin{equation*} \frac{\det_ζ (L + z)}{\det_ζ (L)} = \exp \left( \sum_{j=1}^{p-1} \frac{z^j}{j!} \cdot \frac{d^j}{dz^j} \log \det\nolimits_ζ (L+z) |_{z=0} \right) \cdot \det\nolimits_{p}(I + z \cdot L^{-1} ). \end{…
We prove necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients on weighted Banach function spaces. These conditions are formulated in terms of indices of submultiplicative functions associated with local properties of the space, of the curve, and of the weight. As an example, we consider weighted Nakano spaces $L^{p(\cdot)}_w$ (weighted Lebesgue spaces with variable exponent). Moreover, our necessary conditions become also sufficient for weighted Nakano spaces over nice curves whenever $w$ is a Khvedelidze weight, and the variable exponent …
We introduce a new concept of perturbation of closed linear subspaces and operators in Banach spaces called uniform lambda-adjustment which is weaker than perturbations by small gap, operator norm, q-norm, and K2-approximation. In arbitrary Banach spaces some of the classical Fredholm stability theorems remain true under uniform lambda-adjustment, while other fail. However, uniformly lambda-adjusted subspaces and linear operators retain their (semi--)Fredholm properties in a Banach space which dual is Fréchet-Urysohn in weak* topology. We also introduce another concept of perturbation called u…
We discuss several natural metrics on spaces of unbounded self--adjoint operators and their relations, among them the Riesz and the graph metric. We show that the topologies of the spaces of Fredholm operators resp. invertible operators depend heavily on the metric. Nevertheless we prove that in all cases the spectral flow is up to a normalization the only integer invariant of non-closed paths which is path additive and stable under homotopies with endpoints varying in the space of invertible self-adjoint operators. Furthermore we show that for certain Riesz continuous paths of self--adjoint F…
An analytic definition of a $\mathbb{Z}_2$-valued spectral flow for paths of real skew-adjoint Fredholm operators is given. It counts the parity of the number of changes in the orientation of the eigenfunctions at eigenvalue crossings through $0$ along the path. The $\mathbb{Z}_2$-valued spectral flow is shown to satisfy a concatenation property and homotopy invariance, and it provides an isomorphism on the fundamental group of the real skew-adjoint Fredholm operators. Moreover, it is connected to a $\mathbb{Z}_2$-index pairing for suitable paths. Applications concern the zero energy bound sta…
In this article we give a comprehensive treatment of a `Clifford module flow' along paths in the skew-adjoint Fredholm operators on a real Hilbert space that takes values in KO${}_{*}(\mathbb{R})$ via the Clifford index of Atiyah-Bott-Shapiro. We develop its properties for both bounded and unbounded skew-adjoint operators including an axiomatic characterization. Our constructions and approach are motivated by the principle that \[ \text{spectral flow} = \text{Fredholm index}. \] That is, we show how the KO--valued spectral flow relates to a KO-valued index by proving a Robbin-Salamon type resu…
In this paper, we define an analytical index for continuous families of Fredholm operators parameterized by a topological space $\mathbb{X}$ into a Banach space $X.$ We also consider the Weyl spectrum for continuous families of bounded linear operators and we study its continuity.
We show that the perturbation class for the upper semi-Fredholm operators between two Banach spaces X and Y coincides with the strictly singular operators when X is subprojective and that the perturbation class for the lower semi-Fredholm operators coincides with the strictly cosingular operators when Y is superprojective. Similar results were previously obtained under stronger conditions for X and Y.
In this paper we consider A-Fredholm and semi-A-Fredholm operators on Hilbert C*-modules over a W*-algebra A defined in [3],[10]. Using the assumption that A is a W*-algebra (and not an arbitrary C*-algebra), we obtain several results such as generalization of Schechter-Lebow characterization of semi-Fredholm operators and generalization of "punctured neighbourhood" theorem, as well as some other results that generalize their classical counterparts. We consider both adjointable and non adjointable semi-Fredholm operators over W*-algebras. Moreover, we also work with general bounded, adjointabl…
Starting from the definition of A-Fredholm and semi-A-Fredholm operator on the standard module over a unital C*- algebra A, introduced in [8] and [4], we construct various generalizations of these operators and obtain several results as an analogue or a generalization of the results in [1], [2], [3],[7]. Moreover, we study also non-adjointable semi-A-Fredholm operators as a natural continuation of the work in [6] on non-adjointable A-Fredholm operators and obtain an analogue or a generalization in this setting of the results in [4], [5].
We consider Toeplitz and Hankel operators with piecewise continuous generating functions on $l^p$-spaces and the Banach algebra generated by them. The goal of this paper is to provide a transparent symbol calculus for the Fredholm property and a handy formula for the Fredholm index for operators in this algebra.
Let $A$ be a linear bounded operator in a Hilbert space $H$, $N(A)$ and $R(A)$ its null-space and range, and $A^*$ its adjoint. The operator $A$ is called Fredholm iff $dim N(A)= dim N(A^*):=n<\infty$ and $R(A)$ and $R(A^*)$ are closed subspaces of $H$. A simple and short proof is given of the following known result: $A$ is Fredholm iff $A=B+F$, where $B$ is an isomorphism and $F$ is a finite-rank operator. The proof consists in reduction to a finite-dimensional linear algebraic system which is equivalent to the equation $Au=f$ in the case of Fredholm operators.
We investigate stability of Fredholm properties on interpolation scales of quasi-Banach spaces. This analysis is motivated by problems arising in PDE's and several applications are presented.
It is known that two Banach space operators that are Schur coupled are also equivalent after extension, or equivalently, matricially coupled. The converse implication, that operators which are equivalent after extension or matricially coupled are also Schur coupled, was only known for Fredholm Hilbert space operators and Fredholm Banach space operators with index 0. We prove that this implication also holds for Hilbert space operators with closed range, generalizing the result for Fredholm operators, and Banach space operators that can be approximated in operator norm by invertible operators. …
Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvents of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil $λ\rightarrow T-λS$ are provided and an explicit expression of the generalized resolvent is given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators a…
In this work it is introduced the notion of regular Fredholm pair, i.e. a Fredholm pair whose operators are regular. The main properties of these objects are studied, and what is more, they are entirely classified. Furthermore, the index of a Fredholm pair turns out to be an extremely useful tool in the description of the aforementioned objects. Finally, regular Fredholm pairs are characterized in terms of regular Fredholm symmetrical pairs, exact chains of multiplication operators, and invertible Banach space operators.
In this article poles, isolated spectral points, group, Drazin and Koliha-Drazin invertible elements in the context of quotient Banach algebras or in ranges of (not necessarily continuous) homomorphism between complex unital Banach algebras will be characterized using Fredholm and Riesz Banach algebra elements. Calkin algebras on Banach and Hilbert spaces will be also considered.
Based on the recently introduced uniform $λ-$adjustment for closed subspaces of Banach spaces we extend the concept of the strictly singular and finitely strictly singular operators to the sequences of closed subspaces and operators in Banach spaces and prove theorems about lower semi--Fredholm stability. We also state some new open questions related to strict singularity and the geometry of Banach spaces.
Let $X$ a Banach space and $T$ a bounded linear operator on $X.$ We denote by $S(T)$ the set of all $λ\in \cit$ such that $T$ does not have the single-valued extension property at $λ$. In this note we prove equality up to $S(T)$ between the left Drazin spectrum and the left B-Fredholm spectrum and between the semi-essential approximate point spectrum and the left Drazin spectrum. As applications we investigate generalized Weyl's theorem for operator matrices and multipliers operators.
We study adjointable, bounded operators on the direct sum of two copies of the standard Hilbert C*-module over a unital C*-algebra A that are given by upper triangular 2 by 2 operator matrices. Using the definition of A-Fredholm and semi-A-Fredholm operators given in [3], [4], we obtain conditions relating semi-A-Fredholmness of these operators and that of their diagonal entries, thus generalizing the results in [1], [2]. Moreover, we generalize the notion of the spectra of operators by replacing scalars by the center of the C*-algebra A denoted by Z(A).Considering these new spectra in Z(A) of…
The Evans function is a well known tool for locating spectra of differential operators in one spatial dimension. In this paper we construct a multidimensional analogue as the modified Fredholm determinant of a ratio of Dirichlet-to-Robin operators on the boundary. This gives a tool for studying the eigenvalue counting functions of second-order elliptic operators that need not be self-adjoint. To do this we use local representation theory for meromorphic operator-valued pencils, and relate the algebraic multiplicities of eigenvalues of elliptic operators to those of the Robin-to-Robin and Robin…
In noncommutative geometry one is interested in invariants such as the Fredholm index or spectral flow and their calculation using cyclic cocycles. A variety of formulae have been established under side conditions called summability constraints. These can be formulated in two ways, either for spectral triples or for bounded Fredholm modules. We study the relationship between these by proving various properties of the map on unbounded self adjoint operators $D$ given by $f(D)=D(1+D^2)^{-1/2}$. In particular we prove commutator estimates which are needed for the bounded case. In fact our methods…
We give a definition of the spectral flow for continuous paths in the space of bounded and essentially hyperbolic operators. We provide a homotopical characterization of the spectral flow in terms of a group homomorphism of the fundamental group of the projectors of the Calkin algebra with the infinite cyclic group Z. This characterization helps us to exhibit examples of infinite-dimensional Banach spaces where the spectral flow is not injective nor surjective. We prove that a path with spectral flow equal to an integer m exists if and only if there exists a projector P connected by an arc to …
Let $\mathcal H$ be a complex infinite-dimensional separable Hilbert space, and let $\mathcal K(\mathcal H)$ be the $C^*$-algebra of compact linear operators in $\mathcal H$. Let $(E,\|\cdot\|_E)$ be a symmetric sequence space. If $\{μ(n,x)\}$ are the singular values of $x\in\mathcal K(\mathcal H)$, let $\mathcal C_E=\{x\in\mathcal K(\mathcal H): \{μ(n,x)\}\in E\}$ with $\|x\|_{\mathcal C_E}=\|\{μ(n,x)\}\|_E$, $x\in\mathcal C_E$, be the Banach ideal of compact operators generated by $E$. Let $\mathcal C_E^h=\{x\in\mathcal C_E : x=x^*\}$ be the real Banach subspace of self-adjoint operators in …
We study the gap (= "projection norm" = "graph distance") topology of the space of (not necessarily bounded) self--adjoint Fredholm operators in a separable Hilbert space by the Cayley transform and direct methods. In particular, we show that the space is connected contrary to the bounded case. Moreover, we present a rigorous definition of spectral flow of a path of such operators (actually alternative but mutually equivalent definitions) and prove the homotopy invariance. As an example, we discuss operator curves on manifolds with boundary.
In this paper, we report on new results related to the existence of an adjoint for operators on separable Banach spaces and discuss a few interesting applications. (Some results are new even for Hilbert spaces.) Our first two applications provide an extension of the Poincaré inequality and the Stone-von Neumann version of the spectral theorem for a large class of $C_0$-generators of contraction semigroups on separable Banach spaces. Our third application provides a natural extension of the Schatten-class of operators to all separable Banach spaces. As a part of this program, we introduce a new…
For 1 ≤ p < ∞, it is known that the set K^*_p contains of all Lambert multipliers acting between L^p-spaces is a Banach space. In this study, we introduce a new induced norm by conditional expectation operators to show that K^*_p is a commutative Banach algebra with respect to this norm. Furthermore, in main result, the Fredholm *-multiplication operators on L^p-spaces are characterized, and some more results are obtained.
We study the simplest singular points of Fredholm maps of index zero between Banach spaces, i.e. when the kernel of the Frechet derivative of the map has dimension one. Even in this relatively simple case we have a rich variety of singularities which are completely classified under the natural geometric assumption of transversality of the map. In fact we have, locally, a suitable stratification of the singular points that allows us to identify three kinds of singularities: a) the ordinary or k-singularities (the infinite-dimensional analogues of the well-known Morin singularities) and two new …
Quantitative Fredholm backstepping and rapid stabilization
Peking University（北京大学）；Sorbonne Université, CNRS, Inria équipe CAGE（索邦大学，法国国家科学研究中心，Inria CAGE团队）；Université de Lorraine, CNRS, Inria（洛林大学，法国国家科学研究中心，Inria）；CERMICS, École des Ponts ParisTech（巴黎路桥学院 CERMICS）；LS2N, Ecole Centrale de Nantes & CNRS UMR 6004（南特中央理工学院 LS2N 与法国国家科学研究中心 UMR 6004）
An Explicit Representation of the Dominant Eigenstructure for Positive Operators on Banach Lattices
Kogakuin University（工学大学）；National Institute of Population and Social Security Research（国立社会保障·人口问题研究所）；National Yang Ming Chiao Tung University（国立阳明交通大学）
利用部分 Bell 多项式显式构造满足 Doeblin 条件的正算子的 Riesz 投影和对应特征函数，通过标量函数 D(λ) 的导数提供归一化常数，并得到特征函数的显式泛函表示。
If $p\in [1,+\infty]$ and $T$ is a linear operator with $p$-nuclear adjoint from a Banach space $ X$ to a Banach space $Y$ then if one of the spaces $X^*$ or $Y^{***}$ has the approximation property, then $T$ belongs to the ideal $N^p$ of operators which can be factored through diagonal oparators $l_{p'}\to l_1.$ On the other hand, there is a Banach space $W$ such that $W^{**}$ has a basis and such that for each $p\in [1,+\infty], p\neq 2,$ there exists an operator $T: W^{**}\to W$ with $p$-nuclear adjoint that is not in the ideal $N^p,$ as an operator from $W^{**}$ to $ W.$
If $ s\in (0,1]$ and $ T$ is a linear operator with $ s$-nuclear adjoint from a Banach space $ X$ to a Banach space $ Y$ and if one of the spaces $ X^*$ or $ Y^{***}$ has the approximation property of order $s,$ $AP_s,$ then the operator $ T$ is nuclear. The result is in a sense exact. For example, it is shown that for each $r\in (2/3, 1]$ there exist a Banach space $Z_0$ and a non-nuclear operator $ T: Z_0^{**}\to Z_0$ so that $Z_0^{**}$ has a Schauder basis, $ Z_0^{***}$ has the $AP_s$ for every $s\in (0,r)$ and $T^*$ is $r$-nuclear.
Generalizing earlier results about the set of idempotents in a Banach algebra, or of self-adjoint idempotents in a $C^*$-algebra, we announce constructions of nice connecting paths in the connected components of the set of elements in a Banach algebra, or of self-adjoint elements in a $C^*$-algebra, that satisfy a given polynomial equation, without multiple roots. In particular, we will prove that in the Banach algebra case every such non-central element lies on a complex line, all of whose points satisfy the given equation. We also formulate open questions.
We present the current results in the study of weighted composition operators on weighted Banach spaces of an unbounded, locally finite metric space. Specifically, we determine characterizations of bounded and compact weighted composition operators, including the operator and essential norms. In addition, we characterize the weighted composition operators that are injective, are bounded below, have closed range, and have bounded inverse. We characterize the isometries and surjective isometries among the weighted composition operators, as well as those that satisfy the Fredholm condition. Lastl…
We discuss the essential self-adjointness of wave operators, as well as the limiting absorption principle, in generalizations of asymptotically Minkowski settings. This is obtained via using a Fredholm framework for inverting the spectral family first, and then refining its conclusions to show its dense range in L^2 when acting on an appropriate subdomain.
Given $X$ a complex Banach space, $L$ a complex nilpotent finite dimensional Lie algebra, and $ρ\colon L\to L(X)$, a representation of $L$ in $X$ such that $ρ(l)\in K(X)$ for all $l\in L$, the Taylor, the Slodkowski, the Fredholm, the split and the Fredholm split joint spectra of the representation $ρ$ are computed.
We prove novel results on interpolation of Fredholm operators including an abstract factorization theorem. The main result of this paper provides sufficient conditions on the parameters $θ\in (0,1)$ and $q\in \lbrack 1,\infty ]$ under which an operator $A$ is a Fredholm operator from the real interpolation space $(X_{0},X_{1})_{θ,q}$ to $(Y_{0},Y_{1})_{θ,q} $ for a given operator $A\colon (X_{0},X_{1})\rightarrow (Y_{0},Y_{1})$ between compatible pairs of Banach spaces such that its restrictions to the endpoint spaces are Fredholm operators. These conditions are expressed in terms of the corre…
Let X be a Banach Space over K=R or C, and let f:=F+C be a weakly coercive operator from X onto X, where F is a C^1-operator, and C a C^1 compact operator. Sufficient conditions are provided to assert that the perturbed operator f is a C^1-diffeomorphism. Three corollaries are given. The first one, when F is a linear homeomorphism. The second one, when F is a k-contractive perturbation of the identity. The third one, when X is a Hilbert space and F a particular linear operator. The proof of our results is based on properties of Fredholm operators, as well as on local and global inverse mapping…
