摘要:

This paper is concerned with the stability of certain properties of linear operators in locally convex topological vector spaces under perturbations by operators which aresmallin some sense. Section 3 deals with the very useful concept of Banach balls which was introduced by Raĭkov [9]. Some properties are discussed. The following section investigates the invertibility of certain operators generalizing results of Robert [10] and de Bruyn [2],[3]. These results are used extensively in the sequel. We go on to discuss Riesz operators. We obtain results stronger than those of de Bruyn [1] with regard to asymptotically quasi-compact operators in locally convex spaces. The proofs are basically adaptations of those from [1]. In the final section we observe some results concerning the range ad null space of an operator perturbed byboundedoperators. We obtain a result very similar to an unproved theorem of Vladimirskiĭ [a] and point out their differences. MOS codes 4601, 4710, 4745, 4768, 4755.

ISSN:1607-3606    

DOI:10.1080/16073606.1976.9632512

出版商:Taylor & Francis Group    

年代:1976

数据来源: Taylor

摘要:

A generalization is given of Finsler's theorem on the positivity of a quadratic form subject to a quadratic equality. The generalized result is, under a certain assumption, a set of necessary and sufficient conditions for non-negativity of a quadratic form subject to an arbitrary number of quadratic inequality and equality constraints. Certain properties of matrix inverses are deduced using the conditions.

ISSN:1607-3606    

DOI:10.1080/16073606.1976.9632513

出版商:Taylor & Francis Group    

年代:1976

数据来源: Taylor

摘要:

A general theory of adjoint variational problems is formulated for essentially arbitrary Lagrangians involving m independent and n dependent variables, together with the first derivatives of the latter, This approach contains as a special case the theory of Haar [4], in which the Lagrangian may depend solely on the derivatives of a single dependent function of two arguments. Because of the eventual occurrence of possibly incompatible sets of integrability conditions, the basic theory is developed against the background of non-integrable m-dimensional subspaces, which is in sharp contrast to the traditional approach to the calculus of variations. Relatively self-adjoint Lagrangians are defined and completely characterized in terms of an arbitrary Riemannian metric. In the course of the general theory certain geometric object fields are encountered in a very natural manner, some of which had arisen previously in the canonical formalism proposed by Caratheodory [2]. Accordingly the analysis of the present paper may serve to shed some light on this conceptually extremely difficult formalism.

ISSN:1607-3606    

DOI:10.1080/16073606.1976.9632514

出版商:Taylor & Francis Group    

年代:1976

数据来源: Taylor

摘要:

In recent years many authors have studied properties of powers of symmetric (formally self-adjoint) ordinary linear differential expressions. So that these powers can be formed in the classical way these authors have placed heavy smoothness assumptions on the coefficients. Here we show that no differentiability conditions whatsoever are needed on the coefficients in order to form powers of a given expression-provided these powers are formed in the quasi-differential sense rather than the classical one.

ISSN:1607-3606    

DOI:10.1080/16073606.1976.9632515

出版商:Taylor & Francis Group    

年代:1976

数据来源: Taylor

ISSN:1607-3606    

DOI:10.1080/16073606.1976.9632516

出版商:Taylor & Francis Group    

年代:1976

数据来源: Taylor

ISSN:1607-3606    

DOI:10.1080/16073606.1976.9632511

出版商:Taylor & Francis Group    

年代:1976

数据来源: Taylor