Spectra and semigroup smoothing for non-elliptic quadratic operators

We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space intrinsically associated to their Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal space. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the case of global ellipticity. We also describe the large time behavior of contraction semigroups generated by these operators.     

Initial value problem for pseudo-differential operators

The algebra of generalized Weyl symbols is used in the proof of the continuity of the semigroupexptĤ in the Schwartz space of test functions. Fundamental results on algebras of differentiable Weyl symbols are presented. New examples of σ-temperate Riemannian metrics are constructed. Such metrics form a basis for construction of algebras of differentiable Weyl symbols. Conditions for the existence of semigroups of operators, conditions for pseudo-differential operators to be sectorial, and conditions for the continuity of such semigroups in spaces of test functions and distributions are established. Initial value problems for second-order differential operators are considered. Bibliography: 16 titles. Translated fromProblemy Matematicheskogo Analiza, No. 18, 1998, pp. 3–42.     

Weyl composition of symbols in large dimension

This paper is concerned with the Weyl composition of symbols in large dimension. We specify a class of symbols in order to estimate the Weyl symbol of the product of two Weyl h-pseudodifferential operators, with constants independent of the dimension. The proof includes regularized and hybrid compositions, together with a decomposition formula. We also analyze, in this context, the remainder term of the semiclassical expansion of the Weyl composition. The class of symbols contains symbols of Schrödinger semigroups in large dimension, typically for nearest neighbors or mean field interaction potentials. The Weyl composition is applied with Kac operators.     

Toeplitz operators on the Fock space: Radial component effects

The paper is devoted to the study of specific properties of Toeplitz operators with (unbounded, in general) radial symbolsa=a(r). Boundedness and compactness conditions, as well as examples, are given. It turns out that there exist non-zero symbols which generate zero Toeplitz operators. We characterize such symbols, as well as the class of symbols for whichT _a =0 impliesa(r)=0 a.e. For each compact setM there exists a Toeplitz operatorT _a such that spT _a =ess-spT _a =M. We show that the set of symbols which generate bounded Toeplitz operators no longer forms an algebra under pointwise multiplication.Besides the algebra of Toeplitz operators we consider the algebra of Weyl pseudodifferential operators obtained from Toeplitz ones by means of the Bargmann transform. Rewriting our Toeplitz and Weyl pseudodifferential operators in terms of the Wick symbols we come to their spectral decompositions.This work was partially supported by CONACYT Project 27934-E, México.The first author acknowledges the RFFI Grant 98-01-01023, Russia.     

Weighted norm inequalities for convolution operators and links with the Weiss Conjecture

We study convolution operators on weighted Lebesgue spaces and obtain weight characterisations for boundedness of these operators with certain kernels. Our main result is Theorem 3 which enables us to obtain results for certain kernel functions supported on bounded intervals; in particular we get a direct proof of the known characterisations for Steklov operators in Section 3 by using the weighted Hardy inequality. Our methods also enable us to obtain new results for other kernel functions in Section 4. In Section 5 we demonstrate that these convolution operators are related to operators arising from the Weiss Conjecture (for scalar-valued observation functionals) in linear systems theory, so that results on convolution operators provide elementary examples of nearly bounded semigroups not satisfying the Weiss Conjecture. Also we apply results on the Weiss Conjecture for contraction semigroups to obtain boundedness results for certain convolution operators.     

Elliptic operators and maximal regularity on periodic little-H?lder spaces

We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is satisfied in the setting of periodic little-H?lder spaces, provided the coefficients of the differential operator satisfy minimal regularity assumptions. We address parameter-dependent elliptic equations, deriving invertibility and resolvent bounds which lead to results on generation of analytic semigroups. We also demonstrate that the techniques and results of the paper hold for elliptic differential operators with operator-valued coefficients, in the setting of vector-valued functions.     

On a class of exponential-type operators and their limit semigroups

The paper is mainly focused upon the study of a class of second order degenerate elliptic operators on unbounded intervals.We show that these operators generate strongly continuous semigroups in suitable weighted spaces of continuous functions.Furthermore, we represent the semigroups as limits of iterates of the so-called exponential-type operators.In a particular case, starting from the stochastic differential equations associated with these operators, we also find an integral representation of the semigroup and determine its asymptotic behaviour.     

Pseudodifferential operators on ultra-modulation spaces

The boundedness of pseudodifferential operators on modulation spaces defined by the means of almost exponential weights is studied. The results are applied to symbol class with almost exponential bounds including polynomial and ultra-polynomial symbols. The Weyl correspondence is used and it is noted that the results can be transferred to the operators with appropriate anti-Wick symbols. It is proved that a class of elliptic pseudodifferential operators can be almost diagonalized by the elements of Wilson bases, and estimates for their eigenvalues are given. Furthermore, it is shown that the same can be done by using Gabor frames.     

Representations of regularized determinants of exponentials of differential operators by functional integrals

Representations of regularized determinants of elements of one-parameter operator semigroups whose generators are second-order elliptic differential operators by Lagrangian functional integrals are obtained. Such semigroups describe solutions of inverse Kolmogorov equations for diffusion processes. For self-adjoint elliptic operators, these semigroups are often called Schrödinger semigroups, because they are obtained by means of analytic continuation from Schrödinger groups. It is also shown that the regularized determinant of the exponential of the generator (this exponential is an element of a one-parameter semigroup) coincides with the exponential of the regularized trace of the generator.     

A trace formula for canonical differential expressions

We prove a trace formula for pairs of self-adjoint operators associated to canonical differential expressions. An important role is played by the associated Weyl function.     

Pseudo-differential operators and Markov semigroups on compact Lie groups

We extend the Ruzhansky-Turunen theory of pseudo-differential operators on compact Lie groups into a tool that can be used to investigate group-valued Markov processes in the spirit of the work in Euclidean spaces of N. Jacob and collaborators. Feller semigroups, their generators and resolvents are exhibited as pseudo-differential operators and the symbols of the operators forming the semigroup are expressed in terms of the Fourier transform of the transition kernel. The symbols are explicitly computed for some examples including the Feller processes associated to stochastic flows arising from solutions of stochastic differential equations on the group driven by Lévy processes. We study a family of Lévy-type linear operators on general Lie groups that are pseudo-differential operators when the group is compact and find conditions for them to give rise to symmetric Dirichlet forms.     

Natural equivariant transversally elliptic Dirac operators

We introduce a new class of natural, explicitly defined, transversally elliptic differential operators over manifolds with compact group actions. Under certain assumptions, the symbols of these operators generate all the possible values of the equivariant index. We also show that the components of the representation-valued equivariant index coincide with those of an elliptic operator constructed from the original data.     

An algebra of meromorphic corner symbols

Operators on manifolds with corners that have base configurations with geometric singularities can be analysed in the frame of a conormal symbolic structure which is in spirit similar to the one for conical singularities of Kondrat'ev's work. Solvability of elliptic equations and asymptotics of solutions are determined by meromorphic conormal symbols. We study the case when the base has edge singularities which is a natural assumption in a number of applications. There are new phenomena, caused by a specific kind of higher degeneracy of the underlying symbols. We introduce an algebra of meromorphic edge operators that depend on complex parameters and investigate meromorphic inverses in the parameter-dependent elliptic case. Among the examples are resolvents of elliptic differential operators on manifolds with edges.     

SVEP and Generalized Weyl’s Theorem

We use localised single-valued extension property to prove generalized Weyl/Browder and a-generalized Weyl/Browder type theorems for Banach space operators.     

Expansion of the boundaries of the solutions of the linear Volterra integral equations with convolution kernel

We solve a certain differential equation and system of integral equations. As applications, we characterize holomorphic symbols of commuting Toeplitz operators on the pluriharmonic Bergman space. In addition, pluriharmonic symbols of normal Toeplitz operators are characterized. Also, zero semi-commutators for certain classes of Toeplitz operators are characterized.This research is partially supported by KOSEF(98-0701-03-01-5).     

一类半轴上的锥Sobolev空间

Meromorphic Mellin symbols arise in questions of characterizing elliptic regularity and the asymptotics of solutions to differential equations on spaces with conical singularities, and other singularities as well, and the construction of pseudo-differential parametrices to elliptic operators.     

Conditions for Admissibility of Observation Operators and Boundedness of Hankel Operators

We derive new necessary and sufficient conditions for admissibility of observation operators for certain C ₀-semigroups. We also prove a new sufficient criterion for admissibility for observation operators with infinite-dimensional output space on contraction semigroups. If the contraction semigroup is completely non-unitary and its co-generator has finite defect indices, then this criterion is also necessary. In the case of the right shift semigroup on L ²(0,), these conditions translate into conditions for the boundedness of Hankel operators.     

Boundary value problems with global projection conditions

Parametrices of elliptic boundary value problems for differential operators belong to an algebra of pseudodifferential operators with the transmission property at the boundary. However, generically, smooth symbols on a manifold with boundary do not have this property, and several interesting applications require a corresponding more general calculus. We introduce here a new algebra of boundary value problems that contains Shapiro-Lopatinskij elliptic as well as global projection conditions; the latter ones are necessary, if an analogue of the Atiyah-Bott obstruction does not vanish. We show that every elliptic operator admits (up to a stabilisation) elliptic conditions of that kind. Corresponding boundary value problems are then Fredholm in adequate scales of spaces. Moreover, we construct parametrices in the calculus.     

Form-bounded perturbations of generators of sub-Markovian semigroups

We develop perturbation theory of generators of sub-markovian semigroups by relatively form-bounded perturbations. The L ^p-smoothing properties of semigroups and the uniqueness problem are considered. Applications to operators of mathematical physics are given.     

Spectral Estimates for Resolvent Differences of Self-Adjoint Elliptic Operators

The concept of quasi boundary triples and Weyl functions from extension theory of symmetric operators in Hilbert spaces is developed further and spectral estimates for resolvent differences of two self-adjoint extensions in terms of general operator ideals are proved. The abstract results are applied to self-adjoint realizations of second order elliptic differential operators on bounded and exterior domains, and partial differential operators with δ-potentials supported on hypersurfaces are studied.