We study the composition of time-ffrequency localization operators (wavepacket operators) and develop a symbolic calculus of such operators on modulation spaces. The use of time-frequency methods (phase space methods) allows the use of rough symbols of ultra-rapid growth in place of smooth symbols in the standard classes. As the main application it is shown that, in general, a localization operators possesses the Fredholm property, and thus its range is closed in the target space.
This article provides a version of scale calculus geared towards a notion of (nonlinear) Fredholm maps between certain types of Frechet spaces, retaining as many as possible of the properties Fredholm maps between Banach spaces enjoy, and the existence of a constant rank theorem for such maps. It does so by extending the notion of linear Fredholm maps from [HWZ14] and [Weh12] to a setting where the Nash-Moser inverse function theorem can be applied and which also encompasses the necessary examples such as the reparametrisation action and (nonlinear) elliptic partial differential operators.
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.
The aim of the present work is to show how, using the differential calculus associated to Dirichlet forms, it is possible to construct Fredholm modules on post critically finite fractals by regular harmonic structures. The modules are d-summable, the summability exponent d coinciding with the spectral dimension of the generalized laplacian operator associated with the regular harmonic structures. The characteristic tools of the noncommutative infinitesimal calculus allow to define a d-energy functional which is shown to be a self-similar conformal invariant.
We consider special classes of linear bounded operators in Banach spaces and suggest certain operator variant of symbolic calculus. It permits to formulate an index theorem and to describe Fredholm properties of elliptic pseudo-differential operator on manifolds with non-smooth boundaries.
We discuss an extension of Toeplitz quantization based on polyanalytic functions. We derive isomorphism theorem for polyanalytic Toeplitz operators between weighted Sobolev-Fock spaces of polyanalytic functions, which are images of modulation spaces under polyanalytic Bargmann transforms. This generalizes well-known results from the analytic setting. Finally, we derive an asymptotic symbol calculus and present an asymptotic expansion of complex Weyl operators in terms of polyanalytic Toeplitz operators.
In this paper, we introduce and study a class of pseudo-differential operators on the lattice $\mathbb{Z}^n$. More preciously, we consider a weighted symbol class $M_{ρ, Λ}^m(\mathbb{ Z}^n\times \mathbb{T}^n), m\in \mathbb{R}$ associated to a suitable weight function $Λ$ on $\mathbb{ Z }^n$. We study elements of the symbolic calculus for pseudo-differential operators associated with $M_{ρ, Λ}^m(\mathbb{ Z}^n\times \mathbb{T}^n)$ by deriving formulae for the composition, adjoint, transpose. We define the notion of $M$-ellipticity for symbols belonging to $M_{ρ, Λ}^m(\mathbb{ Z}^n\times \mathbb{…
In this paper we develop the functional calculus for elliptic operators on compact Lie groups without the assumption that the operator is a classical pseudo-differential operator. Consequently, we provide a symbolic descriptions of complex powers of such operators. As an application, we give a constructive symbolic proof of the Gårding inequality for operators in $(ρ,δ)$-classes in the setting of compact Lie groups.
Let $P$ be a generalized laplacian on $R^{2n+1}$. It is known that $P$ is the generating functional of semigroups of measures $μ_{t}$ on the Heisenberg group $H^{n}$ and $ν_{t}$ on the Abelian group $R^{2n+1}$. Under some smoothness and growth conditions on the functional $P$ expressed in terms of its Abelian Fourier transform $\widehat{P}$ we show that the semigroup $μ_{t}$ is a kind of perturbation of the semigroup $ν_{t}$. More precisely, we give pointwise estimates for the difference of the densities of the measures $μ_{t}$ and $ν_{t}$. As a consequence we get a description of the asymptot…
Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and Melrose. In particular, we introduce certain symbols leading to a simple description of the Fredholm operators inside the calculus. When the leaves of the fibration `resolving' the foliation are compact, we also obtain an index formula for Fredholm perturbations of Dirac-type operators. Along the way, we obtain a formula for the adiabatic limit of the eta inva…
A hypoelliptic operator in the Heisenberg calculus on a compact contact manifold is a Fredholm operator. Its symbol determines an element in the K-theory of the noncommutative algebra of Heisenberg symbols. We construct a periodic cyclic cocycle which, when paired with the Connes-Chern character of the principal Heisenberg symbol, calculates the index. Our index formula is local, i.e. given as a local expression in terms of the principal symbol of the operator and a connection on TM and its curvature. We prove our index formula by reduction to Boutet de Monvel's index theorem for Toeplitz oper…
In this paper, we present first results of our investigation regarding symbolic pseudo-differential calculi on nilpotent Lie groups. On any graded Lie group, we define classes of symbols using difference operators. The operators are obtained from these symbols via the natural quantisation given by the representation theory. They form an algebra of operators which shares many properties with the usual Hormander calculus.
The extended Heisenberg algebra for a contact manifold has a symbolic calculus that accommodates both Heisenberg pseudodifferential operators as well as classical pseudodifferential operators. We derive here a formula for the index of Fredholm operators in this extended calculus. This formula incorporates in a single expression the Atiyah-Singer formula for elliptic operators, as well as Boutet de Monvel's Toeplitz index formula.
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 …
In this note we present a symbolic pseudo-differential calculus on the Heisenberg group. We particularise to this group our general construction [4,3,2] of pseudo-differential calculi on graded groups. The relation between the Weyl quantization and the representations of the Heisenberg group enables us to consider here scalar-valued symbols. We find that the conditions defining the symbol classes are similar but different to the ones in [1]. Applications are given to Schwartz hypoellipticity and to subelliptic estimates on the Heisenberg group.
In this paper we develop the global symbolic calculus of pseudo-differential operators generated by a boundary value problem for a given (not necessarily self-adjoint or elliptic) differential operator. For this, we also establish elements of a non-self-adjoint distribution theory and the corresponding biorthogonal Fourier analysis. We give applications of the developed analysis to obtain a-priori estimates for solutions of operators that are elliptic within the constructed calculus.
Solving analytic systems using inversion can be implemented in a variety of ways. One method is to use Lagrange inversion and variations. Here we present a different approach, based on dual vector fields. For a function analytic in a neighborhood of the origin in the complex plane, we associate a vector field and its dual, an operator version of Fourier transform. The construction extends naturally to functions of several variables. We illustrate with various examples and present an efficient algorithms readily implemented symbolically in Maple while suitable as well for numerical computations…
In this note we present a symbolic pseudo-differential calculus on graded nilpotent Lie groups and, as an application, a version of the sharp Garding inequality. As a corollary, we obtain lower bounds for positive Rockland operators with variable coefficients as well as their Schwartz-hypoellipticity.
In the setting of several commuting operators on a Hilbert space one defines the notions of invertibility and Fredholmness in terms of the associated Koszul complex. The index problem then consists of computing the Euler characteristic of such a special type of Fredholm complex. In this paper we investigate transformation rules for the index under the holomorphic functional calculus. We distinguish between two different types of index results: 1) A global index theorem which expresses the index in terms of the degree function of the "symbol" and the locally constant index function of the "coor…
We are consider domains in cotangent bundles with the property that the null foliation of their boundary is fibrating and the leaves satisfy a Bohr-Sommerfeld condition (for example, the unit disk bundle of a Zoll metric). Given such a domain, we construct an algebra of associated semiclassical pseudodifferential operators with singular symbols. The Schwartz kernels of the operators have frequency set contained in the union of the diagonal and the flow-out of the null foliation of the boundary of the domain. We develop a symbolic calculus, prove the existence of projectors (under a mild additi…
To a generalized tight continuous frame in a Hilbert space $\H$ indexed by a locally compact space $\Si$ endowed with a Radon measure, one associates a coorbit theory converting spaces of functions on $\Si$ in spaces of vectors comparable with $\H$. If the continuous frame is provided by the action of a suitable family of bounded operators on a fixed window, a symbolic calculus emerges, assigning operators in $\H$ to functions on $\Si$. We give some criteria of relative compactness for sets and for families of compact operators, involving tightness properties in terms of objects canonically as…
