Index in K-Theory for Families of Fibred Cusp Operators

A families index theorem in K-theory is given for the setting of Atiyah, Patodi, and Singer of a family of Dirac operators with spectral boundary condition. This result is deduced from such a K-theory index theorem for the calculus of cusp, or more generally fibred-cusp, pseudodifferential operators on the fibres (with boundary) of a fibration; a version of Poincaré duality is also shown in this setting, identifying the stable Fredholm families with elements of a bivariant K-group. (Received: February 2006)     

Lie algebraic characterization of manifolds

Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds. The research of Janusz Grabowski supported by the Polish Ministry of Scientific Research and Information Technology under the grant No. 2 P03A 020 24, that of Norbert Poncin by grant C.U.L./02/010.     

Relaxed Quasimonotone Operators and Relaxed Quasiconvex Functions

In this paper, we introduce the class of multivalued relaxed μ quasimonotone operators and establish the existence of solutions of variational inequalities for such operators. This result is compared with a recent result of Bai et al. on densely relaxed pseudomonotone operators. A similar comparison regarding an existence result of Luc on densely pseudomonotone operators is provided. Also, we introduce a broad class of functions, called relaxed quasiconvex functions, and show that they are characterized by the relaxed μ quasimonotonicity of their subdifferentials. The results strengthen a variety of other results in the literature. This work is supported by NNSF of China (10571046) and by the GSRT of Greece (06FR-062).     

Differential operators associated with holomorphic mappings

Two differential operators which act on holomorphic mappings to complex projective space are studied. One operator is of second order and characterizes projective linear mappings. The other operator is of third order and may be viewed as a curvature. The two operators together play a role analogous to the Schwarzian derivative.A canonical approximation to a holomorphic mapping is defined, and a relationship between the approximation and the operators is derived. In the one variable case, this reduces to a classical result relating the Schwarzian derivative and the best Möbius approximation to a holomorphic function.     

Dense Linear Manifolds of Monsters

In this paper the new concept of totally omnipresent operators is introduced. These operators act on the space of holomorphic functions of a domain in the complex plane. The concept is more restrictive than that of strongly omnipresent operators, also introduced by the authors in an earlier work, and both of them are related to the existence of functions whose images under such operators exhibit an extremely wild behaviour near the boundary. Sufficient conditions for an operator to be totally omnipresent as well as several outstanding examples are provided. After extending a statement of the first author about the existence of large linear manifolds of hypercyclic vectors for a sequence of suitable continuous linear mappings, it is shown that there is a dense linear manifold of holomorphic monsters in the sense of Luh, so completing earlier nice results due to Luh and Grosse-Erdmann.     

Index theory of geometric Fredholm operators on varieties with isolated singularities

It sometimes happens that geometric elliptic differential operators on a noncompact Riemannian manifold are Fredholm. The smooth parts of singular algebraic varieties provide examples of complete and incomplete manifolds where this can happen. The indices of such operators often provide topological or geometric information about the singular variety. This paper shows that the operators of the title represent K homology elements and solves the index problem for these operators by exhibiting equivalent K homology cycles in topological form.This material is based upon work supported by the National Science Foundation under Grant No. DMS-8501513.     

On some properties of a class of degenerate pseudodifferential operators

A new class of pseudodifferential operators with degeneration is considered. The operators are constructed using a special integral transform mapping a weighted differentiation operator to a multiplication operator. The composition and boundedness properties of such operators in special weighted spaces are examined. Theorems on commutation of such operators with differentiation operators are obtained. The behavior of these operators as t → 0and t → +∞ is investigated. The properties of adjoint operators are studied, and an analogue of Gårding’s inequality is proved.     

Multidimensional integral operators with bihomogeneous kernels: A projection method and pseudospectra

We consider multidimensional integral operators with bihomogeneous and rotation invariant kernels. For such operators we obtain a criterion for applicability of a projection method in the scalar and matrix cases and describe the limit behavior of the ε-pseudospectra of the truncated operators $A_{\tau _1 , \tau _2 } $ as τ ₁ → 0 and τ ₂ → 0.     

Matrix finite-zone operators

A survey is given of the spectral properties of matrix finite-zone operators. Conditions of the type of J-self-adjointness for such operators and explicit formulas expressing the coefficients of such operators in terms of theta functions are obtained. The simplest examples of such J-self-adjoint, finite-zone operators turn out to be connected with the theory of ovals of plane, real, algebraic curves.Translated from Itogi Nauki i Tekhniki, Seriya Sovremennye Problemy Matematiki, Vol. 23, pp. 33–78, 1983.     

τ-可测算子迹的不等式

设m是具有忠实正规半有限迹τ的Hilbert空间上的一个半有限von Neumann代数.隶属于m的—个闭稠定算子x称为τ可测,如果存在常数λ≥0使得τ（e｜x｜（λ,∞））〈∞.将一些很有用的已知的Hilbert空间算子迹的不等式推广到τ-可测算子迹.特别是这些不等式蕴涵了n-元τ-可测算子的Clarkson不等式.同时还给出了τ-可测算子的广义平行四边形法则.     

Continuation of a linear operator to an involution operator

A bounded linear operatorA:X→X in a linear topological spaceX is called ap-involution operator,p≥2, ifA ^p=I, whereI is the identity operator. In this paper, we describe linearp-involution operators in a linear topological space over the field ℂ and prove that linear operators can be continued to involution operators. Translated fromMatematicheskie Zametki, Vol. 61, No. 5, pp. 671–676, May, 1997. Translated by M. A. Shishkova     

Invertible sequences of bounded linear operators

In this paper, we study the invertibility of sequences consisting of finitely many bounded linear operators from a Hilbert space to others. We show that a sequence of operators is left invertible if and only if it is a g-frame. Therefore, our result connects the invertibility of operator sequences with frame theory.     

Interior and closure operators on bounded residuated lattice ordered monoids

GMV-algebras endowed with additive closure operators or with its duals-multiplicative interior operators (closure or interior GMV-algebras) were introduced as a non-commutative generalization of topological Boolean algebras. In the paper, the multiplicative interior and additive closure operators on DRl-monoids are introduced as natural generalizations of the multiplicative interior and additive closure operators on GMV-algebras.     

Nonhomogeneous markov chains with desired properties - the nearest markovian model

A new class of operators performing an optimization (optimization operators or, simply, optimators) which generate transition matrices with required properties such as ergodicity, recurrence etc., is considered and their fundamental features are described. Some criteria for comparing such operators by taking into account their strenght are given and sufficient conditions for both weak and strong ergodicity are derived. The nearest Markovian model with respect to a given set of observed probability vectors is then defined as a sequence of transition matrices satisfying certain constraints that express our prior knowledge about the system. Finally, sufficient conditions for the existence of such a model are given and the related algorithm is illustrated by an example.     

Factorization of singular integral operators with a Carleman backward shift: The case of bounded measurable coefficients

In this paper, we generalize our recent results concerning scalar singular integral operators with a Carleman backward shift, allowing more general coefficients, bounded measurable functions on the unit circle. Our aim is to obtain an operator factorization for singular integral operators with a backward shift and bounded measurable coefficients, from which such Fredholm characteristics as the kernel and the cokernel can be described. The main tool is the factorization of matrix functions. In the course of the analysis performed, we obtain several useful representations, which allow us to characterize completely the set of invertible operators in that class, thus providing explicit examples of such operators Dedicated to Professor A. Ferreira dos Santos on the occasion of his seventieth birthday     

p-Adic Pseudodifferential Operators and p-Adic Wavelets

We introduce a new wide class of p-adic pseudodifferential operators. We show that the basis of p-adic wavelets is the basis of eigenvectors for the introduced operators.     

Nonself-adjoint operators with almost Hermitian spectrum: Cayley identity and some questions of spectral structure

Nonself-adjoint, non-dissipative perturbations of possibly unbounded self-adjoint operators with real purely singular spectrum are considered under an additional assumption that the characteristic function of the operator possesses a scalar multiple. Using a functional model of a nonself-adjoint operator (a generalization of a Sz.-Nagy–Foiaş model for dissipative operators) as a principle tool, spectral properties of such operators are investigated. A class of operators with almost Hermitian spectrum (the latter being a part of the real singular spectrum) is characterized in terms of existence of the so-called weak outer annihilator which generalizes the classical Cayley identity to the case of nonself-adjoint operators in Hilbert space. A similar result is proved in the self-adjoint case, characterizing the condition of absence of the absolutely continuous spectral subspace in terms of the existence of weak outer annihilation. An application to the rank-one nonself-adjoint Friedrichs model is given.     

Regular boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces

We consider coerciveness and Fredholmness of nonlocal boundary value problems for complete second order elliptic differential-operator equations in UMD Banach spaces. In some special cases, the main coefficients of the boundary conditions may be bounded operators and not only complex numbers. Then, we prove an isomorphism, in particular, maximal L _p -regularity, of the problem with a linear parameter in the equation. In both cases, the boundary conditions may also contain unbounded operators in perturbation terms. Finally, application to regular nonlocal boundary value problems for elliptic equations of the second order in non-smooth domains are presented. Equations and boundary conditions may contain differential-integral parts. The spaces of solvability are Sobolev type spaces W _p,q ^2,2. The first author is a member of G.N.A.M.P.A. and the paper fits the 60% research program of G.N.A.M.P.A.-I.N.D.A.M.; The third author was supported by the Israel Ministry of Absorption.     

λ-central BMO estimates for commutators of singular integral operators with rough kernels

The authors establish λ-central BMO estimates for commutators of singular integral operators with rough kernels on central Morrey spaces. Moreover, the boundedness of a class of multisublinear operators on the product of central Morrey spaces is discussed. As its special cases, the corresponding results of multilinear Calderon-Zygmund operators and multilinear fractional integral operators can be deduced, resDectivelv.