Two-sided estimates on singular values for a class of integral operators on the semi-axis

(Quasi)-norms inC _p andC _{p, w} of weighted operators of the integration of (fractional) order are estimated. It is shown that, in most cases, the estimates obtained are sharp both in order and in function classes for the weight function involved.     

Eigenvalues of integral operators onL 2(I) given by analytic kernels

Let {_n} _n=0 be the eigenvalue sequence of a symmetric Hilbert-Schmidt operator onL ₂(I). WhenI is an open interval, a necessary condition for {_n} _n=0 to be in the sequence space is obtained. WhenI is a closed bounded interval, sufficient conditions for {_n} _n=0 to be in the sequence space ^– are obtained.     

On the Riesz representation theorem and integral operators

We present a Riesz representation theorem in the setting of extended integration theory as introduced in [6]. The result is used to obtain boundedness theorems for integral operators in the more general setting of spaces of vector valued extended integrable functions.     

A shift-invariant algebra of singular integral operators with oscillating coefficients

LetB be the Banach algebra of all bounded linear operators on the weighted Lebesgue spaceL ^p (T, ) with an arbitrary Muckenhoupt weight on the unit circleT, and the Banach subalgebra ofB generated by the operators of multiplication by piecewise continuous coefficients and the operatorse _h,S _T e _h, ^–1 I (hR, T) whereS _T is the Cauchy singular integral operator ande _h,(t)=exp(h(t+)/(t–)),tT. The paper is devoted to a symbol calculus, Fredholm criteria and an index formula for the operators in the algebra and its matrix analogue . These shift-invariant algebras arise naturally in studying the algebras of singular integral operators with coefficients admitting semi-almost periodic discontinuities and shifts being diffeomorphisms ofT onto itself with second Taylor derivatives.Partially supported by CONACYT grant, Cátedra Patrimonial, No. 990017-EX and by CONACYT project 32726-E, México.     

Weakly singular integral equations with operator-valued kernels and an application to radiation transfer problems

The standard problem of radiation transfer in a bounded regionG ⁿ can be reformulated as a weakly singular integral equation with an unknown functionu: GC(S ^n–1) and a kernelK: ((G × G }x=y}, which is continuously differentiable with respect to the operator strong convergence topology. We take these observations into the basis of an abstract treatment of weakly singular integral equations with (E)-valued kernels, whereE is a Banach space. Our purpose is to characterize the smoothness of the solution by proving that it belongs to special weighted spaces of smooth functions. On the way, realizing the proof techniques, we establish the compactness of the integral operator or its square inL _p (G,E),BC(G,E), and other spaces of interest in numerical analysis as well as in weighted spaces of smooth functions. The smoothness results are specified for the standard problem of radiation transfer as well as for the corresponding eigenvalue problem.     

Trace class norm inequalities for localization operators

We give sharp estimates on the norms in the trace class of localization operators in terms of their symbols.     

Parabolic boundary integral operators symbolic representation and basic properties

In this paper we develop a theory of parabolic pseudodifferential operators in anisotropic spaces. We construct a symbolic calculus for a class of symbols globally defined on ⁿ⁺¹× ⁿ⁺¹, and then develop a periodisation procedure for the calculus of symbols on the cylinder ×. We show Gårding's inequality for suitable operators and precise estimates for the essential norm in anisotropic Sobolev spaces. These new mapping properties are needed in localization arguments for the analysis of numerical approximation methods.     

Stability of integral operators on a space of homogeneous type

In this paper, we consider integral operators T on compact spaces of homogeneous type with finite diameter, whose kernels have certain Hölder regularity and mild singularity near the diagonal. We show that given any , the ‐stability of the operator is equivalent for different , where I stands for the identity operator.     

Norms of the singular integral operator with Cauchy Kernel along certain contours

The norm of the above-mentioned operatorS is computed on the unions of parallel lines or concentric circles. The upper bound is found for its norm on the ellipse. In case of weighted spaces on the unit circle, the exact norm is found for some rational weights, and necessary and sufficient conditions on the weight are established, under which the essential norm ofS equals 1.     

Hybrid cross approximation of integral operators

The efficient treatment of dense matrices arising, e.g., from the finite element discretisation of integral operators requires special compression techniques. In this article we use the -matrix representation that approximates the dense stiffness matrix in admissible blocks (corresponding to subdomains where the underlying kernel function is smooth) by low-rank matrices. The low-rank matrices are assembled by a new hybrid algorithm (HCA) that has the same proven convergence as standard interpolation but also the same efficiency as the (heuristic) adaptive cross approximation (ACA).     

Asymptotic behavior of one-parameter semigroups of positive operators

In this paper we survey the Perron-Frobenius spectral theory for positive semigroups on Banach lattices and indicate its applications to stability theory of retarded differential equations and quasi-periodic flows.     

Strongly omnipresent integral operators

An operatorT on the spaceH(G) of holomorphic functions on a domainG is strongly omnipresent whenever there is a residual set of functionsfH(G) such thatT f exhibits an extremely wild behaviour near the boundary. The concept of strong omnipresence was recently introduced by the first two authors. In this paper it is proved that a large class of integral operators including Volterra operators with or without a perturbation by differential operators has this property, completing earlier work about differential and antidifferential operators.The work of the first two authors has been partially supported by DGES grant PB96-1348 and the Junta de Andalucía.     

On collisionless transport semigroups with boundary operators of norm one

We deal with streaming operators T _H defined in L ¹ spaces by the directional derivative with positive boundary operator H of norm 1 relating the incoming and outgoing fluxes. It is known that T _H need not be a generator but there exists a contraction semigroup generated by an extension A of T _H . This paper deals with the total mass carried by individual trajectories {e ^tA f; t ≥ 0} for nonnegative initial data f and related topics. In particular, our analysis covers the problem of (the lack of) stochasticity of {e ^tA ; t ≥ 0} for conservative boundary operator H.      

Transmission problems and spectral theory for singular integral operators on Lipschitz domains

We prove the well-posedness of the transmission problem for the Laplacian across a Lipschitz interface, with optimal non-tangential maximal function estimates, for data in Lebesgue and Hardy spaces on the boundary. As a corollary, we show that the spectral radius of the (adjoint) harmonic double layer potential K∗ in is less than , whenever is a bounded convex domain and 1<p?2.     

Bound states and scattered states for contraction semigroups

LetA generate a strongly continuous contraction semigroup {T(t)} on a Hilbert space and letL be a bounded operator. IfL(ζI−A)⁻¹ is compact, then the Cesàro limit of ‖LT(t)f‖² (ast→∞) is computed for all vectorsf. This limit is interpreted in terms of bound and scattered states in the context of quantum mechanical and classical wave propagation problems. Partially supported by a NSF grant.     

An algebra of integral operators with fixed singularities in kernels

We continue the study of algebras generated by the Cauchy singular integral operator and integral operators with fixed singularities on the unit interval, started in R.Duduchava, E.Shargorodsky, 1990. Such algebras emerge when one considers singular integral operators with complex conjugation on curves with cusps. As one of possible applications of the obtained results we find an explicit formula for the local norms of the Cauchy singular integral operator on the Lebesgue spaceL ₂ (, ), where is a curve with cusps of arbitrary order and is a power weight. For curves with angles and cusps of order 1 the formula was already known (see R.Avedanio, N.Krupnik, 1988 and R.Duduchava, N.Krupnik, 1995). Dedicated to Professor Israel Gohberg on the occasion of his 70-th birthday Supported by EPSRC grant GR/K01001     

The Hukuhara–Kneser property for a nonlinear integral equation

Applying a structure theorem of Krasnosel’skii and Perov, we show that the solution set of a nonlinear integral equation satisfies the classical Hukuhara–Kneser property.     

Stability inL 1 of some integral operators

A class of Markov operators appearing in biomathematics is investigated. It is proved that these operators are asymptotic stable inL ¹, i.e. lim _n P ⁿ f=0 forfL ¹ and f(x) dx=0.