Beurling-Pollard type theorems

We establish a version of the Beurling–Pollard theoremfor operator synthesis and apply it to derive some results onlinear operator equations and to prove a Beurling–Pollardtype theorem for Varopoulos tensor algebras. In addition, weestablish a Beurling–Pollard theorem for weighted Fourieralgebras and use it to obtain ascent estimates for operatorsthat are functions of generalized scalar operators.     

Existence Theorems for Set-valued Operators in Banach Spaces

In this paper, we study the problem of existence of zero points for set-valued operators in Banach spaces. We present a simple condition which can be used to verify that under some conditions, set-valued operators have zero points. Furthermore, we derive the surjectivity result and the Kakutani fixed point theorem by using this condition.      

Sum theorems for monotone operators and convex functions

In this paper, we derive sufficient conditions for the sum of two or more maximal monotone operators on a reflexive Banach space to be maximal monotone, and we achieve this without any renorming theorems or fixed-point-related concepts. In the course of this, we will develop a generalization of the uniform boundedness theorem for (possibly nonreflexive) Banach spaces. We will apply this to obtain the Fenchel Duality Theorem for the sum of two or more proper, convex lower semicontinuous functions under the appropriate constraint qualifications, and also to obtain additional results on the relation between the effective domains of such functions and the domains of their subdifferentials. The other main tool that we use is a standard minimax theorem.

     

New Formulae for the Wave Operators for a Rank One Interaction

We prove new formulae for the wave operators for a Friedrichs scattering system with a rank one perturbation, and we derive a topological version of Levinson’s theorem for this model.     

On fredholm riggings of G-translators

We consider riggings of G-translators on manifolds with multidimensional singularities. We derive a condition for the ellipticity of G-riggings, prove the finiteness theorem, and analyze the relationship between the notions of G-ellipticity and ordinary ellipticity of the riggings. The interest in this class of operators arises in connection with the study of nonlocal pseudodifferential operators.     

Representation for the W-weighted Drazin inverse of linear operators

In this paper we study the W-weighted Drazin inverse of the bounded linear operators between Banach spaces and its representation theorem. Based on this representation, utilizing the spectral theory of Banach space operators, we derive an approximating expression of the W-weighted Drazin inverse and an error bound. Also, a perturbation theorem for the W-weighted Drazin inverse is uniformly obtained from the representation theorem.     

Weyl-Titchmarsh -Function Asymptotics, Local Uniqueness Results, Trace Formulas, and Borg-type Theorems for Dirac Operators

We explicitly determine the high-energy asymptotics for Weyl-Titchmarsh matrices associated with general Dirac-type operators on half-lines and on . We also prove new local uniqueness results for Dirac-type operators in terms of exponentially small differences of Weyl-Titchmarsh matrices. As concrete applications of the asymptotic high-energy expansion we derive a trace formula for Dirac operators and use it to prove a Borg-type theorem.

     

A theorem on iterations of polynomial maps in several variables

We establish a polynomial version of a theorem obtained by Enflo, Gurarii, Lomonosov and Lyubich for linear operators. As a consequence, we also derive a polynomial version of a result due to Pták.

     

Index of nonlocal problems associated with a bundle

We study operators associated with a bundle with compact base and fiber. We construct an algebra of such operators. For elliptic elements of the algebra, we prove the finiteness theorem and derive an index formula.     

An adiabatic theorem for section determinants of spectral projections

We derive an adiabatic‐type theorem that expresses the section determinants of spectral projections of a selfadjoint operator through the solution to an operator‐valued Wiener‐Hopf equation. The solution theory of this equation is developed and for a special case a concrete criterion that ensures uniqueness of the solution is presented. Furthermore, for a special class of operators a dichotomy criterion, which is used in the proof of the adiabatic theorem, is proved. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

ASYMPTOTIC APPROXIMATION BYBERNSTEIN-DURRMEYER OPERATORS AND THEIR DERIVATIVES

The concern of this paper is to study local approximation properties of the Bernstein-Durrmeyer operators M_n. We derive the complete asymptotic expansion of the operators M_n and their derivatives as n tends to infinity. It turns out that the appropriate representation is a series of reciprocal factorials. All coefficients are calculated explicitly in a very concise form. Our main theorem contains several earlier partial results as special cases. Finally, we obtain a Voronovskaja-type formula for simultaneous approximation by linear combinations of M_n,     

Asymptotic approximation by Bernstein-Durrmeyer operators and their derivatives

The concern of this paper is to study local approximation properties of the Bernstein-Durrmeyer operators M_n. We derive the complete asymptotic expansion of the operators M_n and their derivatives as n tends to infinity. It turns out that the appropriate representation is a series of reciprocal factorials. All coefficients are calculated explicitly in a very concise form. Our main theorem contains several earlier partial results as special cases. Finally, we obtain a Voronovskaja-type formula for simultaneous approximation by linear combinations of M_n,     

Sampling in a Hilbert space

An analog of the Whittaker-Shannon-Kotel'nikov sampling theorem is derived for functions with values in a separable Hilbert space. The proof uses the concept of frames and frame operators in a Hilbert space. One of the consequences of this theorem is that it allows us to derive sampling theorems associated with boundary-value problems and some homogeneous integral equations, which in turn gives us a generalization of another sampling theorem by Kramer.

     

Coupled fixed points of multivalued operators and first-order ODEs with state-dependent deviating arguments

We establish a coupled fixed point theorem for a meaningful class of mixed monotone multivalued operators, and then we use it to derive some results on the existence of quasisolutions and unique solutions to first-order functional differential equations with state-dependent deviating arguments. Our results are very general and can be applied to functional equations featuring discontinuities with respect to all of their arguments, but we emphasize that they are new even for differential equations with continuously state-dependent delays.     

Stability conditions of nonlinear distributed processes under parametric excitation

We derive a stability criterion relative to a given measure on a finite time interval for distributed processes under parametric excitation. The corresponding theorem is proved by the comparison method combined with Lyapunov second method. Treating time as a parameter, we use the extremal properties of the Rayleigh quotient for self-adjoint operators in a Hilbert space, which in turn involves solving the eigenvalue problem generated by the linear operators corresponding to the original problem. The results are applied to establish sufficient conditions of technical stability relative to a given measure in the nonlinear problem of a hinged pole under the action of a continuous longitudinal force.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 58, pp. 107–116, 1986.     

A fixed point approach to stability of functional equations

We prove a simple fixed point theorem for some (not necessarily linear) operators and derive from it several quite general results on the stability of a very wide class of functional equations in single variable.     

Irreducible subalgebras of quasi-normed operator ideals

It is known that the only irreducible -closed subalgebra of is itself. A generalization of this theorem is established and some extensions of recent reducibility results are obtained based on this generalization. We then apply these extensions to derive some decomposability results concerning semigroup of compact quasinilpotent positive operators on Banach lattices. Received June 18, 1996; February 13, 1997     

Bernstein-Sikkema-Bézier算子的点态逼近

讨论了Bernstein-Sikkema-Bézier算子点态逼近的等价定理,首先利用插项的的方法证明了正定理,然后应用讨论算子逼近的常规方法给出了其逼近的逆定理.     

Singular functions and relative index for elliptic corner operators

We show an explicit link between the nature of a singular point and the behaviour of the coefficients of the equation, under which formal asymptotic expansions are still available. We also derive a general relative index theorem for elliptic operators. To the memory of Lamberto Cattabriga     

A general algorithm for multiple-sets split feasibility problem involving resolvents and Bregman mappings

In this paper, by using Bregman distance, we introduce a new iterative process involving products of resolvents of maximal monotone operators for approximating a common element of the set of common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings and the solution set of the multiple-sets split feasibility problem and common zeros of maximal monotone operators. We derive a strong convergence theorem of the proposed iterative algorithm under appropriate situations. Finally, we mention several corollaries and two applications of our algorithm.