On a Multivariable Class of Mittag-Leffler Type Functions

The present paper studies and investigates a class of Mittag-Leffler type multivariable functions. We derive the necessary convergence conditions and establish several properties associated with this class and those related with the corresponding class of fractional integral operators. New extensions of the introduced definitions and special cases of some of the results are also pointed out.     

On the spectral theory of the Bessel operator on a finite interval and the half-line

We study the minimal and maximal operators generated by the Bessel differential expression on a finite interval and the half-line. We describe the domains of the Friedrichs and Krein extensions of the minimal operator and all nonnegative self-adjoint extensions of the minimal operator.     

Dissipative extensions of a symmetric relation generated by a system of integral equations with operator measures

With the help of boundary conditions we describe dissipative and accumulative extensions of the minimal relation generated by a system of integral equations with operator measures in an infinite-dimensional case.     

Triplet extensions I: Semibounded operators in the scale of Hilbert spaces

The extension theory for semibounded symmetric operators is generalized by including operators acting in a triplet of Hilbert spaces. We concentrate our attention on the case where the minimal operator is essentially self-adjoint in the basic Hilbert space and construct a family of its self-adjoint extensions inside the triplet. All such extensions can be described by certain boundary conditions, and a natural counterpart of Krein’s resolvent formula is obtained.     

On invertible contractions of quotients generated by a differential expression and by a nonnegative operator function

In the present paper, we describe invertible contractions of the maximal quotient generated by a differential expression with bounded operator coefficients and by a nonnegative operator function. We show that the operators inverse to such contractions are integral operators and prove a criterion for such operators to be holomorphic. Using the results obtained, we describe the generalized resolvents of symmetric quotients.     

Discrete Choquet integral and some of its symmetric extensions

Classical extensions of the Choquet integral (defined on [0,1]) to [−1,1] are the asymmetric and the symmetric Choquet integral, the second one being called also the Šipoš integral. Recently, the balancing Choquet integral was introduced as another kind of a symmetric extension of the discrete Choquet integral. We introduce and discuss a new type of such extension, the fusion Choquet integral, and discuss its properties and relationship to the balancing and the symmetric Choquet integral. The symmetric maximum introduced by Grabisch is shown to be a special case of the fusion and the balancing Choquet integral. Several extensions of OWA operators are also discussed.     

Invertible linear relations generated by an integral equation with Nevanlinna measure

We define families of maximal and minimal relations generated by integral equations with Nevanlinna operator measure and non-selfadjoint operator measure. We prove that if a restriction of a maximal relation is continuously invertible, then the inverse operator is integral. We study the case when the convergence of non-selfadjoint operator measures implies the convergence of the corresponding integral operators inverse to restrictions of maximal relations, and establish a sufficient condition for the validity of this implication. The obtained results are applicable to the study of differential equations with singular potentials.     

Modular Convergence Theorems for Integral Operators in the Context of Filter Exhaustiveness and Applications

We prove some modular convergence theorems for nonlinear Urysohn-type integral operators, applying filter convergence of sequences of functions. We give some applications to Mellin operators, including moment, Mellin-Poisson-Cauchy and Mellin-Gauss-Weierstrass operators. We show that our results are proper extensions of the classical ones, and we pose an open problem.     

Modular filter convergence theorems for Urysohn integral operators and applications

We prove some versions of modular convergence theorems for nonlinear Urysohn-type integral operators with respect to filter convergence. We consider pointwise filter convergence of functions giving also some applications to linear and nonlinear Mellin operators. We show that our results are strict extensions of the classical ones.     

Singularly perturbed normal operators

We present a generalization of the definition of singularly perturbed operators to the case of normal operators. To do this, we use the idea of normal extensions of a prenormal operator and prove the relation for resolvents of normal extensions similar to the M. Krein relation for resolvents of self-adjoint extensions. In addition, we establish a one-to-one correspondence between the set of singular perturbations of rank one and the set of singularly perturbed (of rank one) operators. Kiev Polytechnic Institute, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 8, pp. 1045–1053, August, 1999.     

带粗糙核的奇异积分交换子的有界性

本文研究了带粗糙核的奇异积分算子与BMO函数生成的交换子的有界性问题.利用原子分解的方法,获得了带粗糙核的奇异积分交换子TΩb在Lp空间、Hardy空间、弱Hardy空间上的有界性结果,推广了交换子Tb在各类函数空间上有界性的结果.     

具有三角分解李代数的积分元和中心扩张

本文将应用广义限制李代数的概念来研究具有三角分解李代数的积分元和中心扩张的关系.对于给定的广义限制普遍包络代数,我们确定了它的积分元并且提供了一种计算dim H2(L,F)的方法.     

On regular linear relations

For a closed linear relation in a Banach space the concept of regularity is introduced and studied. It is shown that many of the results of Mbekhta and other authors for operators remain valid in the context of multivalued linear operators. We also extend the punctured neighbourhood theorem for operators to linear relations and as an application we obtain a characterization of semiFredholm linear relations which are regular.     

Integral Criteria for the Quality of the Dichotomy of a Matrix Spectrum by a Closed Contour

A well-known method for estimating the sensitivity of invariant projection operators, which is based on the dichotomy quality integral criteria and is oriented to perturbations of general form, is extended to the case of regularly structured perturbations. For finite-dimensional approximations of differential operators, this allows us, in particular, to obtain significantly more precise estimates of the sensitivity of invariant projection operators (corresponding to the eigenvalues minimal in absolute value) to perturbations of the coefficients of these operators. We give a new definition of the dichotomy quality integral criteria as the L _p -norm of the resolvent, which allows us to simplify the proofs and to formulate the final results in a more general and simpler form.     

On the Existence of a Classical Optimal Solution and of an Almost Strongly Optimal Solution for an Infinite-Horizon Control Problem

We consider an infinite-horizon optimal control problem with the cost functional described either by an integral over an unbounded interval (a Lebesgue integral) or by a limit of integrals (an improper Lebesgue integral). We prove some theorems on the existence of solutions to such problems. The proofs are based on appropriate lower closure theorems and some extensions of Olech’s theorem on the lower semicontinuity of an integral functional; these extensions cover the cases of functionals described by an integral over an unbounded interval and by a limit of integrals.     

Spectral analysis of the Direct Sum Hamiltonian Operators

In this paper we investigate the deficiency indices theory and the selfad-joint and nonselfadjoint (dissipative, accumulative) extensions of the minimal symmetric direct sum Hamiltonian operators. In particular using the equivalence of the Lax-Phillips scattering matrix and the Sz.-Nagy-Foia¸s characteristic function, we prove that all root (eigen and associated) vectors of the maximal dissipative extensions of the minimal symmetric direct sum Hamiltonian operators are complete in the Hilbert spaces.     

M ‐elliptic pseudo‐differential operators on L p (ℝn )

We construct the minimal and maximal extensions in L ^p (?ⁿ ), 1 < p < ∞, for M ‐elliptic pseudo‐differential operators initiated by Garello and Morando. We prove that they are equal and determine the domains of the minimal, and hence maximal, extensions of M ‐elliptic pseudo‐differential operators. For M ‐elliptic pseudodifferential operators with constant coefficients, the spectra and essential spectra are computed. An application to quantization is given. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

L p-Theory of elliptic differential operators on manifolds of bounded geometry

This paper is devoted to some of the properties of uniformly elliptic differential operators with bounded coefficients on manifolds of bounded geometry in L ^pspaces. We prove the coincidence of minimal and maximal extensions of an operator of a considered type with a positive principal symbol, the existence of holomorphic semigroup, generated by it, and the estimates of L ^p-norms of the operators of this semigroup. Some spectral properties of such operators in L ^pspaces are also studied.     

New ring of a class of Bessel integral operators

In this paper, we introduce a new ring of ponderation functions which allowed us to describe and classify a large class of ring of Bessel integral operators. The main result is to give an explicit form of some ideals of such a ring of operators.