Algebra Of Pseudodifferential Operators Research Articles

Spectral invariance of quasi-Banach algebras of matrices and pseudodifferential operators

Abstract We extend the stability and spectral invariance of convolution-dominated matrices to the case of quasi-Banach algebras p < 1 {p<1} . As an application, we construct a spectrally invariant quasi-Banach algebra of pseudodifferential operators with non-smooth symbols that generalize Sjöstrand’s results.

A groupoid approach to the Wodzicki residue

Originally, the noncommutative residue was studied in the 80's by Wodzicki in his thesis [33] and Guillemin [19]. In this article we give a definition of the Wodzicki residue, using the language of r-fibered distributions from [24], [30], in the context of filtered manifolds. We show that this groupoidal residue behaves like a trace on the algebra of pseudodifferential operators on filtered manifolds and coincides with the usual residue Wodzicki in the case where the manifold is trivially filtered. Moreover, in the context of Heisenberg calculus, we show that the groupoidal residue coincides with Ponge's definition [25] for contact and codimension 1 foliation Heisenberg manifolds.

Index theorems for Fredholm, semi-Fredholm, and almost periodic operators: all in one example

Based on operators borrowed from scattering theory, several concrete realizations of index theorems are proposed. The corresponding operators belong to some $C\*$-algebras of pseudo-differential operators with coefficients which either have limits at $\pm \infty$, or which are periodic or asymptotically periodic, or which are uniformly almost periodic. These various situations can be deduced from a single partial isometry which depends on several parameters. All computations are explicitly performed.

A [formula omitted] algebra of pseudodifferential operators on the half line

In this paper we employ a C⋆-algebra approach to the study of Fredholm properties of differential and pseudodifferential operators on the half line. The algebra investigated in this paper has compact commutators, so the Gelfand theory applies to the quotient algebra, and we obtain an explicit description of the corresponding maximal ideal space and necessary and sufficient conditions for Fredholmness of operators in the algebra.

AMBROSE–SINGER THEOREM ON DIFFEOLOGICAL BUNDLES AND COMPLETE INTEGRABILITY OF THE KP EQUATION

In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose–Singer theorem that enlarges the one stated in [J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math.128 (2004) 513–529], and conclude with a differential geometric treatment of KP hierarchy. The examples of Lie groups that are studied are principally those obtained by enlarging some graded Frölicher (Lie) algebras such as formal q-series of the quantum algebra of pseudo-differential operators. These deformations can be defined for classical pseudo-differential operators but they are used here on formal pseudo-differential operators in order to get a differential geometric framework to deal with the KP hierarchy that is known to be completely integrable with formal power series. Here, we get an integration of the Zakharov–Shabat connection form by means of smooth sections of a (differential geometric) bundle with structure group, some groups of q-deformed operators. The integration obtained by Mulase [Complete integrability of the Kadomtsev–Petviashvili equation Adv. Math.54 (1984) 57–66], and the key tools he developed, are totally recovered on the germs of the smooth maps of our construction. The tool coming from (classical) differential geometry used in this construction is the holonomy group, on which we have an Ambrose–Singer-like theorem: the Lie algebra is spanned by the curvature elements. This result is proved for any connection a diffeological principal bundle with structure group a regular Frölicher Lie group. The case of a (classical) Lie group modeled on a complete locally convex topological vector space is also recovered and the work developed in [J.-P. Magnot, Difféologie du fibré d'Holonomie en dimension infinie, Math. Rep. Canadian Roy. Math. Soc.28(4) (2006); J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math. 128 (2004) 513–529] is completed.

On the [formula omitted]-boundedness of pseudo-differential operators and their commutators with symbols in α-modulation spaces

The results of Sjöstrand [J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett. 1 (1994) 185–192] and Sugimoto [M. Sugimoto, L p -boundedness of pseudo-differential operators satisfying Besov estimates, I, J. Math. Soc. Japan 40 (1988) 105–122] on a mapping property of pseudo-differential operators are two different kinds of extensions of the pioneering work by Calderón and Vaillancourt [A.P. Calderón, R. Vaillancourt, On the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23 (1971) 374–378]. The objective of this paper is to show that these two results, which appeared to be independent ones, can be proved based on the same principle. For the purpose, we use the α-modulation spaces, a parameterized family of function spaces, which include Besov spaces and modulation spaces as special cases. As an application, we also discuss the L 2 -boundedness of the commutator [ σ ( X , D ) , a ] , where a ( x ) is a Lipschitz function and σ belongs to an α-modulation space.

The Schwinger cocycle on algebras with unbounded operators

In this article, we describe a class of algebras with unbounded operators on which the Schwinger cocycle extends. For this, we replace a space of bounded operators commonly used in the literature by some space of (maybe unbounded) tame operators, in particular by spaces of pseudo-differential operators, acting on the space of sections of a vector bundle E → M . We study some particular examples which we hope interesting or instructive. The case of classical and log-polyhomogeneous pseudo-differential operators is studied, because it carries other cocycles, defined with renormalized traces of pseudo-differential operators, that are some generalizations of the Khesin–Kravchenko–Radul cocycle. The present construction furnishes a simple proof of an expected result: The cohomology class of these cocycles are the same as cohomology class of the Schwinger cocycle. When M = S 1 , we show that the Schwinger cocycle is non-trivial on many algebras of pseudo-differential operators (these operators need not to be classical or bounded). These two results complete the work and extend the results of a previous work [J.-P. Magnot, Renormalized traces and cocycles on the algebra of S 1 -pseudo-differential operators, Lett. Math. Phys. 75 (2) (2006) 111–127]. When dim ( M ) > 1 , we furnish a new example of sign operator which could suggest that the framework of pseudo-differential operators is not adapted to all the cases. On this example, we have to work on some algebras of tame operators, in order to show that the Schwinger cocycle has a non-vanishing cohomology class.

Pseudodifference Operators on Weighted Spaces, and Applications to Discrete Schrödinger Operators

We study pseudodifference operators on ZN with symbols which are bounded on ZN×TN together with their derivatives with respect to the second variable. In the same way as partial differential operators on RN are included in an algebra of pseudodifferential operators, difference operators on ZN are included in an algebra of pseudodifference operators. Particular attention is paid to the Fredholm properties of pseudodifference operators on general exponentially weighted spaces lwp(ZN) and to Phragmen–Lindelof type theorems on the exponential decay at infinity of solutions to pseudodifference equations. The results are applied to describe the essential spectrum of discrete Schrodinger operators and the decay of their eigenfunctions at infinity.

Residues and homology for pseudodifferential operators on foliations

We study the Hochschild homology groups of the algebra of complete symbols on a foliated manifold $(M,F)$. The first step is to relate these groups to the Poisson homology of $(M,F)$ and of other related foliated manifolds. We then establish several general properties of the Poisson homology groups of foliated manifolds. As an example, we completely determine these Hochschild homology groups for the algebra of complete symbols on the irrational slope foliation of a torus (under some diophantine approximation assumptions). We also use our calculations to determine all residue traces on algebras of pseudodifferential operators along the leaves of a foliation.

Reduction of orders in boundary value problems without transmission property

Given an algebra of pseudo-differential operators on a manifold, an elliptic element is said to be a reduction of orders, if it induces isomorphisms of Sobolev spaces with a corresponding shift of smoothness. Reductions of orders on a manifold with boundary refer to boundary value problems. We employ specific smooth symbols of arbitrary real orders and with parameters, and we show that the associated operators induce isomorphisms between Sobolev spaces on a given manifold with boundary. Such operators for integer orders have the transmission property and belong to the calculus of Boutet de Monvel [1], cf. also [9]. In general, they fit to the algebra of boundary value problems without the transmission property in the sense of [17] and [24]. Order reducing elements of the present kind are useful for constructing parametrices of mixed elliptic problems. We show that order reducing symbols have the Volterra property and are parabolic of anisotropy 1; analogous relations are formulated for arbitrary anisotropies. We then investigate parameter-dependent operators, apply a kernel cut-off construction with respect to the parameter and show that corresponding holomorphic operator-valued Mellin symbols reduce orders in weighted Sobolev spaces on a cone with boundary. We finally construct order reducing operators on a compact manifold with conical singularities and boundary.

An index theorem for gauge-invariant families: The case of solvable groups

We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family \(\mathcal{G} \to B\) of Lie groups (these families are called ``gauge-invariant families'' in what follows). If the fibers of \(\mathcal{G} \to B\) are simply-connected and solvable, we compute the Chern character of the gauge-equivariant index, the result being given by an Atiyah–Singer type formula that incorporates also topological information on the bundle \(\mathcal{G} \to B\). The algebras of invariant pseudodifferential operators that we study, \(\psi _{{\text{inv}}}^\infty (Y)\) and \({\psi }_{{inv}}^\infty (Y)\), are generalizations of ``parameter dependent'' algebras of pseudodifferential operators (with parameter in R q), so our results provide also an index theorem for elliptic, parameter dependent pseudodifferential operators. We apply these results to study Fredholm boundary conditions on a simplex.

Invariance spectrale des algèbres d'opérateurs pseudodifférentiels

We construct and study several algebras of pseudodifferential operators that are closed under holomorphic functional calculus. This leads to a better understanding of the structure of inverses of elliptic pseudodifferential operators on certain non-compact manifolds. It also leads to decay properties for the solutions of these operators. To cite this article: R. Lauter et al., C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1095–1099.

ON THE HARMONIC SUPERSPACE LANGUAGE ADAPTED TO THE GELFAND–DICKEY ALGEBRA OF DIFFERENTIAL OPERATORS

Methods developed for the analysis of nonlinear integrable models are used in the harmonic superspace (HS) framework. These methods, when applied to the HS, can lead to obtain more information about the meaning of integrability in nonlinear physical problems. Among the results obtained, we give the basic ingredients towards building in the HS language the analogue of the Gelfand–Dickey (GD) algebra of pseudo-differential operators. Some useful conventional notations and algebraic structures are also introduced to make the use of the harmonic superspace techniques more accessible.

The structure of integrable Lax flows on nonlocal manifolds: Dynamic systems with sources

We consider a dynamic system on the extended phase space to the initial Lie algebra and study its generalized Hamiltonian and integrability in the cases when the initial Lie algebra coincides with the Grassmann algebra of pseudodifferential operators on the real line and on the centrally extended affine Lie algebra.

Complex Star Algebras

We describe a classification of star algebras on the cotangent bundle of a complex manifold, locally isomorphic to the algebra of pseudo-differential operators; this requires a slight extension of the usual definition of star algebras. We show that in dimension ≥ 3 these are essentially trivial and come from algebras of differential operators on X; in dimension 1 and 2 there are many more, which we describe.

Bihamiltonian reductions and ωn-algebras

We discuss the geometry of the Marsden-Ratiu reduction theorem for a bihamiltonian manifold. We consider the case of the manifolds associated with the Gel'fand-Dickey theory, i.e., loop algebras over sl(n+1). We provide an explicit identification, tailored on the MR reduction, of the Adler-Gel'fand-Dickey brackets with the Poisson brackets on the MR-reduced bihamiltonian manifold N. Such an identification relies on a suitable immersion of the space of sections of the cotangent bundle of N into the algebra of pseudo differential operators connected to geometrical features of the theory of (classical) W_n algebras.

Mellin and green pseudodifferential operators associated with non-compact edges

An algebra of pseudodifferential operators on an infinite wedge is discussed. A notion of ellipticity is given that is shown to be equivalent to the existence of a parametrix within the calculus. The symbols carry asymptotic information which will be needed for the description of the asymptotics of solutions to elliptic equations on manifolds with non-compact edges.

Γ -conformal algebras

Γ -conformal algebra is an axiomatic description of the operator product expansion of chiral fields with simple poles at finitely many points. We classify these algebras and their representations in terms of Lie algebras and their representations with an action of the group Γ. To every Γ-conformal algebra and a character of Γ we associate a Lie algebra generated by fields with the OPE with simple poles. Examples include twisted affine Kac–Moody algebras, the sin algebra (which is a “Γ-conformal” analogue of the general linear algebra) and its analogues, the algebra of pseudodifferential operators on the circle, etc.

Algebras of Pseudodifferential Operators in L2 Given by Smooth Measures on Hilbert Spaces

AbstractPseudodifferential operators with symbols on a Hilbert phase space are defined as symmetric operators in L2 given by a smooth measure. The main formulae of symbolic calculus are proved in this context.

P ∞ algebra of symmetries of the kadomtsev-petviashvili equation, free fermions, and 2-cocycles in the lie algebra of pseudo-differential operatorsalgebra of symmetries of the kadomtsev-petviashvili equation, free fermions, and 2-cocycles in the lie algebra of pseudo-differential operators

The symmetry algebraP ∞=W ∞ ⊕P ⊕I ∞ of integrable systems is defined. As an example, the classical Lie point symmetries of all higher Kadomtsev-Petviashvili equations are obtained. It is shown that of the point symmetries, the (positive) ones belong to theW ∞ symmetries, while the other (negative) ones belong toI ∞ symmetries. The corresponding action on the τ-function is obtained for the positive symmetries. The negative symmetries cannot be obtained from the free fermion algebra. A new embedding of the Virasoro algebra intogl(∞) describes the conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle.