On Eigenvalues of Pseudo-Differential Operators

Conditions are given to ensure the existence of finitely manyeigenvalues on the left-hand side of the bottom of the essentialspectrum. Estimates for the number of such eigenvalues are alsodiscussed.     

K-Theory of Suspended Pseudo-Differential Operators

We compute the K-theory groups of Melrose's algebra of 1-suspended pseudo-differential operators. The boundary map in the six-term long exact sequence turns out to be related to both the eta invariant of Melrose and to the index of elliptic operators. The proof is based on a new identity between the formal trace and the Wodzicki residue trace on the suspended algebra.Partially supported by the European Commission RTN HPRN-CT-1999-00118 Geometric Analysis.     

A Parameter-Dependent Symbolic Calculus for Pseudo-Differential Operators with Negative-Definite Symbols

Hoh's calculus for pseudo-differential operators with negative-definitesymbols is extended to the case of parameter-dependent symbols.Then this parameter-dependent calculus is applied to the studyof subordinate sub-Markovian semigroups.     

Exponential Function of Pseudo-Differential Operators in Gevrey Spaces

The article is devoted to the construction of the exponential function of the matrix pseudo-differential operator, which does not satisfy conditions of any known theorem (see, e.g. Sec. 8 Ch. VIII and Sec. 2 Ch. XI of Treves in Introduction to the theory of pseudodifferential and Fourier integral operator, vols. 1 & 2, Plenum Press, New York, 1980). An application of the exponential function to the fundamental solution of the Cauchy problem for the hyperbolic operators with the characteristics of variable multiplicity is given.     

Fredholm Indices of Band-Dominated Operators

The Fredholmness of a band-dominated operator on is closely related with the invertibility of its limit operators: the operator is Fredholm if and only if each of its limit operators is invertible and if the norms of their inverses are uniformly bounded. The goal of the present note is to show how the Fredholm index of a Fredholm band-dominated operator can be determined in terms of its limit operators.     

Spectra of Pseudo-Differential Operators on the Schwartz Space

We initiate the study of the spectrum, point spectrum, continuous spectrum and residual spectrum of a constant coefficient pseudo-differential operator on the Schwartz space. The results are in sharp contrast with the corresponding ones for the Banach space L^p(Rⁿ), 1 ≤ p ≤ ∞. An application to the study of spectra of constant coefficient pseudo-differential operators on the space of Schwartz distributions is given.     

Fredholm Operators and Riesz Theory for Polynomially Compact Operators

In this article, we give a characterization of a class of bounded Fredholm operators on a Banach space which is developed to present some general existence results of the operators equation of the second kind. The obtained results are used to describe the Riesz–Schauder theory of compact operators in the more general setting of polynomially compact operators.     

Band-dominated Fredholm Operators on Discrete Groups

No Abstract.  .     

Eigenvalue Problems for Fredholm Integral Operators

The calculation of an approximation to the eigenelements ofa Fredholm integral operator can be reduced, using the Nyströmmethod, to the solution of an algebraic eigenvalue problem.To reduce the size of the system of equations to be solved,we propose two iterative variants. In the case of a self-adjointoperator, a special integration rule can be used to regularizethe system of equations. In this case, orthogonality relationscan be introduced for close or multiple eigenvalues.     

Boundedness of Toeplitz Type Operators Associated to Fractional and Pseudo-Differential Operators on Orlicz Space

In this paper, the boundedness from Lebesgue space to Orlicz space of certain Toeplitz type operator related to the fractional and pseudo-differential operators is obtained.     

Fredholm Composition Operators on Riemann Surfaces

It is proved that the invertibility of a composition operator on the differential form space for a Riemann surface is equivalent to its Fredholmness. In addition, the Fredholmness of weighted composition operators is discussed.     

Pseudo-Differential Operators in Algebras of Generalized Functions and Global Hypoellipticity

The aim of this work is to develop a global calculus for pseudo-differential operators acting on suitable algebras of generalized functions. In particular, a condition of global hypoellipticity of the symbols gives a result of regularity for the corresponding pseudo-differential equations. This calculus and this frame are proposed as tools for the study in Colombeau algebras of partial differential equations globally defined on R ⁿ .     

A Fredholm Determinant Formula for Section Determinants of Bounded Operators

We show that the section determinant of e^A can be expressed, under certain conditions, by the Fredholm determinant of an integral operator. The kernel function of this integral operator is computed explicitly in terms of the operator A. As a simple consequence we derive a Weierstrass type product expansion for the section determinant.