Fredholm Theory Research Articles

Wiener-Hopf Indices of Unitary-Valued Functions on the Imaginary Axis

This paper is concerned with the Wiener-Hopf indices of unitary-valued rational matrix functions on the imaginary axis. These indices play a role in the Fredholm theory for Wiener-Hopf integral operators. Our main result gives formulas for the Wiener-Hopf indices in terms of the matrices appearing in realizations of the factors in a Douglas-Shapiro-Shields factorization of the unitary-valued function. Two approaches to this problem are presented: one direct approach using operator theoretic methods, and a second approach using the Cayley transform which allows to use results for an analogous problem regarding unitary-valued functions on the unit circle and corresponding Toeplitz operators.

On the convergence of Fourier spectral methods involving noncompact operators

Abstract Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the existence of a sufficiently good approximation of this regulator. Importantly, the numerical method itself need not make use of this extra approximant. We apply the framework to Fourier finite-section and collocation-based numerical methods for solving differential equations with periodic boundary conditions and to solving Riemann–Hilbert problems on the unit circle. We also obtain improved results concerning the approximation of eigenvalues of differential operators with periodic coefficients.

Fredholm theory on Krein spaces and its application to pseudospectrum

In this note, we introduce a notion of the J -kernel of a bounded linear operator on a Krein space and study the J -Fredholm theory for Krein space operators. Using J -Fredholm theory, we discuss and study the J -essential pseudospectra of bounded linear operators on Krein spaces, and some of their properties are studied. Furthermore, we discuss the structure of the J -essential pseudospectra of infinite-dimensional Hamiltonian operators.

Stationarity and Fredholm Theory in Subextremal Kerr-de Sitter Spacetimes

In a recent paper, we proved that solutions to linear wave equations in a subextremal Kerr-de Sitter spacetime have asymptotic expansions in quasinormal modes up to a decay order given by the normally hyperbolic trapping, extending the results of Vasy (2013). One central ingredient in the argument was a new definition of quasinormal modes, where a non-standard choice of stationary Killing vector field had to be used in order for the Fredholm theory to be applicable. In this paper, we show that there is in fact a variety of allowed choices of stationary Killing vector fields. In particular, the horizon Killing vector fields work for the analysis, in which case one of the corresponding ergoregions is completely removed.

Fredholm Theory Relative to Any Algebra Homomorphisms

In this paper, we give another definition of Ruston elements and almost Ruston elements, which is equivalent to the definitions given by Mouton and Raubenheimer in the case that the homomorphism has a closed range and Riesz property. For two homomorphisms, we consider the preserver problems of Fredholm theory and Fredholm spectrum theory. In addition, we study the spectral mapping theorems of Fredholm (Weyl, Browder, Ruston, and almost Ruston) elements relative to a homomorphism. Last but not least, the dependence of Fredholm theory on three homomorphisms is considered, and meanwhile, the transitivity of Fredholm theory relative to three homomorphisms is illustrated. Furthermore, we consider the Fredholm theory relative to more homomorphisms.

Fredholm theory of the Toeplitz algebra on the space of all entire functions

AbstractWe prove that an aggregate Toeplitz operator on the Fréchet space of all entire functions is a Fredholm operator if and only if its symbol does not vanish. The result is motivated by and closely resembles the classical result of Gohberg and Douglas from the Hardy space theory of Toeplitz operators. There are however some subtle differences which we also discuss.

Contact instantons with Legendrian boundary condition: A priori estimates, asymptotic convergence and index formula

In this paper, we establish nonlinear ellipticity of the equation of contact instantons with Legendrian boundary condition on punctured Riemann surfaces by proving the a priori elliptic coercive estimates for the contact instantons with Legendrian boundary condition, and prove an asymptotic exponential [Formula: see text]-convergence result at a puncture under the uniform [Formula: see text] bound. We prove that the asymptotic charge of contact instantons at the punctures under the Legendrian boundary condition vanishes. This eliminates the phenomenon of the appearance of spiraling cusp instanton along a Reeb core, which removes the only remaining obstacle towards the compactification and the Fredholm theory of the moduli space of contact instantons in the open string case, which plagues the closed string case. Leaving the study of [Formula: see text]-estimates and details of Gromov-Floer-Hofer style compactification of contact instantons to [27], we also derive an index formula which computes the virtual dimension of the moduli space. These results are the analytic basis for the sequels [27]–[29] and [36] containing applications to contact topology and contact Hamiltonian dynamics.

Some new results about order Fredholm theory in Banach lattices

Some new results about order Fredholm theory in Banach lattices

A spectral dichotomy for commuting m-isometries with negative core operator

Let T=(T1,T2) be a pair of commuting operators such that the Taylor spectrum σ(T) of T is contained in the closed unit bidisc D‾2 and the left spectrum of either of T1 or T2 is contained in the unit circle ∂D. If the core operator of T is negative, then we show that σ(T) is either contained in ∂D2 or equal to D‾2. This fact applies in particular to a commuting pair of m-isometries with negative core operator. Our method of proof relies on a strictly 2-variable fact about the topological boundary of the Taylor spectrum. We also present several applications of this dichotomy to the multivariate Fredholm theory.

Ruelle–Taylor resonances of Anosov actions

Combining microlocal methods and a cohomological theory developed by J. Taylor, we define for Anosov \mathbb{R}^{\kappa} -actions a notion of joint Ruelle resonance spectrum. We prove that these Ruelle–Taylor resonances fit into a Fredholm theory, are intrinsic and form a discrete subset of \mathbb{C}^{\kappa} , with \lambda=0 being always a leading resonance. The joint resonant states at 0 give rise to some new measures of SRB type and the mixing properties of these measures are related to the existence of purely imaginary resonances. The spectral theory developed in this article applies in particular to the case of Weyl chamber flows and provides a new way to study such flows.

On new approach to semi-Fredholm theory in unital C<sup>*</sup>-algebras

Axiomatic Fredholm theory in unital C*-algebras was established in [D. Kečkić, Z. Lazović, Acta Sci. Math., 83(3-4):629--655, 2017]. Following the pure algebraic approach by Kečkić and Lazović, in the author's paper [S. Ivković, Banach J. Math. Anal., 17:51, 2023] we extended further this theory to axiomatic semi-Fredholm and semi-Weyl theory in unital C*-algebras. However, recently, in [S. Ivković, arXiv:2306.01133] we developed another approach to axiomatic Fredholm theory in unital C*-algebras which is based on the theory of Hilbert modules and is equivalent to the algebraic approach by Kečkić and Lazović. In this paper, we extend further that new Hilbert-module approach from Fredholm theory to semi-Fredholm and semi-Weyl theory in unital C*-algebras. Hence, we provide new proofs to the results in [S. Ivković, Banach J. Math. Anal., 17:51, 2023].

Essential Approximate Pseudospectra of Multivalued Linear Relations

One of the fundamental ideas investigated in A. Ammar, A. Jeribi and K. Mahfoudhi in [?] is that of providing conditions under which the essential approximate pseudospectrum of closed, densely defined linear operators have a relationship with Fredholm theory and perturbation theory. In this paper the approximate pseudospectrum and the essential approximate pseudospectrum of closed, densely defined multivalued linear relations are introduced and studied, and work done in the aforementioned papers are extended to general multivalued linear relations

Geometric singular perturbation analysis to the coupled Schrödinger equations

In this paper, we study the solitary wave solutions of the perturbed coupled Schrödinger equations iut+uxx+b1f∗u|u|2+d|v|2u−τu(|u|2)x=0,ivt+vxx+b2g∗v|v|2+d|u|2v−τv(|v|2)x=0. By combining the geometric singular perturbation theory, invariant manifold theory and Fredholm theory, we construct an invariant manifold for the associated differential equations and use this invariant manifold to obtain a homoclinic orbit. Furthermore, the solitary wave solutions of the perturbed coupled Schrödinger equations are obtained.

Eigenvalues and eigenfunctions of nonuniform beams with a new method based on perturbation

By using a new method based on perturbation method and Fredholm theory, the eigenvalues and eigenfunctions of nonuniform Euler-Bernoulli beam vibration are solved analytically. The eigenvalues and eigenfunctions under boundary conditions are given for the tension beam problem which considers the variation of tension in vertical marine riser from top to bottom caused by gravity effect. The asymptotic analytical results are compared with classical results to verify the effectiveness of the new method based on perturbation method. The influence of the change of the bottom tension on the perturbation method is analyzed. The results show that: for each order of natural frequency of riser, the natural frequency will gradually increase with the increase of bottom tension, and the obtained natural frequency will become more and more accurate; when the bottom tension is constant, the new method based on perturbation method in this paper can give good accuracy for higher order natural frequency rather than lower order.

Modified Green-Hyperbolic Operators

Green-hyperbolic operators - partial differential operators on globally hyperbolic spacetimes that (together with their formal duals) possess advanced and retarded Green operators - play an important role in many areas of mathematical physics. Here, we study modifications of Green-hyperbolic operators by the addition of a possibly nonlocal operator acting within a compact subset $K$ of spacetime, and seek corresponding '$K$-nonlocal' generalised Green operators. Assuming the modification depends holomorphically on a parameter, conditions are given under which $K$-nonlocal Green operators exist for all parameter values, with the possible exception of a discrete set. The exceptional points occur precisely where the modified operator admits nontrivial smooth homogeneous solutions that have past- or future-compact support. Fredholm theory is used to relate the dimensions of these spaces to those corresponding to the formal dual operator, switching the roles of future and past. The $K$-nonlocal Green operators are shown to depend holomorphically on the parameter in the topology of bounded convergence on maps between suitable Sobolev spaces, or between suitable spaces of smooth functions. An application to the LU factorisation of systems of equations is described.

Boundary value problems for a second‐order elliptic partial differential equation system in Euclidean space

Let be a bounded regular domain, let be the standard Dirac operator in , and let be the Clifford algebra constructed over the quadratic space . For fixed, denotes the space of ‐vectors in . In the framework of Clifford analysis, we consider two boundary value problems for a second‐order elliptic system of partial differential equations of the form in , where is a smooth ‐vector valued function. The boundary conditions of the problems contain the inner and outer products of the ‐vector solution with both the Dirac operator and the normal vector to , ensuring the well‐posedness for the problems. Investigation of the spectral properties of the sandwich operator is considered by using the Fredholm theory. Finally, it is shown that satisfactory problem‐solving properties, in general, fail when we replace the standard Dirac operator by those, obtained via unusual orthogonal bases of .

Semi-Fredholm theory in $$C^{*}$$-algebras

Kečkić and Lazović introduced an axiomatic approach to Fredholm theory by considering Fredholm type elements in a unital $$C^{*}$$ -algebra as a generalization of $$C^{*}$$ -Fredholm operators on the standard Hilbert $$C^{*}$$ -module introduced by Mishchenko and Fomenko and of Fredholm operators on a properly infinite von Neumann algebra introduced by Breuer. In this paper, we establish semi-Fredholm theory in unital $$C^{*}$$ -algebras as a continuation of the approach by Kečkić and Lazović. We introduce the notion of a semi-Fredholm type element and a semi-Weyl type element in a unital $$C^{*}$$ -algebra. We prove that the difference between the set of semi-Fredholm elements and the set of semi-Weyl elements is open in the norm topology, that the set of semi-Weyl elements is invariant under perturbations by finite type elements, and several other results generalizing their classical counterparts. Also, we illustrate applications of our results to the special case of properly infinite von Neumann algebras and we obtain a generalization of the punctured neighbourhood theorem in this setting.

Hodge theory on ALG∗ manifolds

Abstract We develop a Fredholm theory for the Hodge Laplacian in weighted spaces on ALG∗ manifolds in dimension four. We then give several applications of this theory. First, we show the existence of harmonic functions with prescribed asymptotics at infinity. A corollary of this is a non-existence result for ALG∗ manifolds with non-negative Ricci curvature having group Γ = { e } \Gamma=\{e\} at infinity. Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG∗ manifold. A corollary of this is vanishing of the first Betti number for any ALG∗ manifold with non-negative Ricci curvature. Another application of our analysis is to determine the optimal order of ALG∗ gravitational instantons.

On the meromorphic continuation of Eisenstein series

Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a “soft” proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. The proof relies on the principle of meromorphic continuation. It is close in spirit to Selberg’s later proofs.

Instantons on Sasakian 7-manifolds

AbstractWe study a natural contact instanton equation on gauge fields over 7-dimensional Sasakian manifolds, which is closely related to both the transverse Hermitian Yang–Mills (HYM) condition and the G2-instanton equation. We obtain, by Fredholm theory, a finite-dimensional local model for the moduli space of irreducible solutions. Following the approach by Baraglia and Hekmati in five dimensions [1], we derive cohomological conditions for smoothness and express its dimension in terms of the index of a transverse elliptic operator. Finally, we show that the moduli space of self-dual contact instantons is Kähler, in the Sasakian case. As an instance of concrete interest, we specialize to transversely holomorphic Sasakian bundles over contact Calabi–Yau 7-manifolds, as studied by Calvo-Andrade, Rodríguez and Sá Earp [8], and we show that in this context the notions of contact instanton, integrable G2-instanton and HYM connection coincide.