Some results on almost L-weakly and almost M-weakly compact operators

Positivity - We introduce and study a new topology on trees, that we call the countably coarse wedge topology. Such a topology is strictly finer than the coarse wedge topology and it turns every...     

Some properties of almost L-weakly and almost M-weakly compact operators

Positivity - In this paper, we investigate necessary and sufficient conditions under which compact operators between Banach lattices must be almost L-weakly compact (resp. almost M-weakly compact)....     

Fuzzy compact linear operators

In this paper, we introduce fuzzy compact linear operators between fuzzy normed spaces and investigate some important general properties of them.     

On the class of almost L-weakly and almost M-weakly compact operators

In this paper, we introduce and study new concepts of almost L-weakly and almost M-weakly compact operators.     

Collectively compact sets of operators and almost periodic functions

A characterization of collectively compact sets of linear operators on the Banach algebra of almost periodic functions is obtained by using Gelfand-Naimark theory. Properties of collectively compact sets are then used to prove that the Fourier coefficients of an almost periodic function are approximately the eigenvalues of Fredholm type integral operators.     

Measure compact,almost compact,and integral operators of the first,second, and third kind

Under study are the metric, algebraic, spectral, and characteristic properties of measure compact, almost compact, and integral operators of the first, second, and third kind in spaces of integrable functions.     

Holomorphic functional calculus of Hodge-Dirac operators in L p

We study the boundedness of the H ^∞ functional calculus for differential operators acting in L ^p (R ⁿ ; C ^N ). For constant coefficients, we give simple conditions on the symbols implying such boundedness. For non-constant coefficients, we extend our recent results for the L ^p theory of the Kato square root problem to the more general framework of Hodge-Dirac operators with variable coefficients Π _B as treated in L ²(R ⁿ ; C ^N ) by Axelsson, Keith, and McIntosh. We obtain a characterization of the property that Π _B has a bounded H ^∞ functional calculus, in terms of randomized boundedness conditions of its resolvent. This allows us to deduce stability under small perturbations of this functional calculus.     

Differential equations in spaces of compact operators

We study a system of differential equations in C(H), the space of all compact operators on a separable complex Hilbert space, H. The systems considered are infinite-dimensional generalizations of mathematical models of learning, implementable as artificial neural networks. In this new setting, in addition to the usual questions of existence and uniqueness of solutions, we discuss issues which are operator theoretic in nature. Under some restrictions on the initial condition, we explicitly solve the system and represent the solution in terms of the spectral representation of the initial condition. We also discuss the stability of those solutions, and describe the weak, strong, and uniform limit sets in terms of their respective spectral properties.     

Integrated semigroups and parabolic equations. Part I: linear perburbation of almost sectorial operators

The paper deals with linear abstract Cauchy problem with non-densely defined and almost sectorial operators, whenever the part of this operator in the closure of its domain is sectorial. This kind of problem naturally arises for parabolic equations with non-homogeneous boundary conditions. Using the integrated semigroup theory, we prove an existence and uniqueness result for integrated solutions. Moreover, we study the linear perturbation problem.     

Normal and compact solvability of linear operators

Dedicated to Yu. G. Reshetnyak on his sixtieth birthday.     

Non-autonomous semilinear evolution equations with almost sectorial operators

Inspired by the theory of semigroups of growth α, we construct an evolution process of growth α. The abstract theory is applied to study semilinear singular non-autonomous parabolic problems. We prove that, under natural assumptions, a reasonable concept of solution can be given to such semilinear singularly non-autonomous problems. Applications are considered to non-autonomous parabolic problems in space of H?lder continuous functions and to a parabolic problem in a domain with a one dimensional handle. Marcelo J. D. Nascimento: Research partially supported by FAPESP # 02/11855-2, Brazil.     

Scattering theory for almost unitary operators,and a functional model

On the basis of a functional model one considers the scattering for operators that are close to unitary. In particular cases the presented scheme contains a series of results of L. de Branges and L. Shulman, S. N. Naboko, L. A. Sakhnovich, and H. Neidhardt. The fundamental new result is: the existence of complete local wave operators for a unitary operator and its nuclear perturbation, where the spectrum of the latter does not fill out the unit circle. In this same situation an invariance principle is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova Akademii Nauk SSSR, Vol. 178, pp. 92–119, 1989.     

An estimate for some positive linear operators in L p spaces

In this note we prove that an estimate due to Ivanov and Pichugov, for positive linear convolution operators in L _p spaces, can be extended to a class of positive linear operators which are not of convolution type.