The science or art of exact reasoning, or of pure and formal thought, or of the laws according to which the processes of pure thinking should be conducted

Mathematics document containing theorems and formulas.

Excerpt: This chapter is intended to give a brief review of the basic notions in the theory of abstract spaces which will be needed in what follows. The material is grouped under four paragraph headings: Topological Concepts, Additive Spaces, Linear Spaces, and Algebraic Spaces. The abstract spaces occurring in this treatise normally have a definite algebraic structure in addition to being topological spaces of one type or another. This fact underlies the choice of the h...

Mathematics document containing theorems and formulas.

Excerpt: In this chapter we shall give a survey of the theory of linear transformations. The presentation is self-contained for those parts of the theory which have a bearing on our main problem. Although the material is far from complete in other respects, the reader will find some indications of other directions of research; and the References are recommended for supplementary reading. The chapter is grouped under four paragraph headings: Additive Transformations, Line...

Mathematics document containing theorems and formulas.

Excerpt: We shall now prepare the frame-work for a theory of functions having values in a Banach space by generalizing the basic notions from real and complex variable theory. We begin by studying vector-valued functions defined on the real numbers and develop a Riemann integral for such functions. We next treat vector-valued functions on an abstract measure space, extending the theory of the Lebesgue integral. The consideration of vector-valued functions on the complex ...

Mathematics document containing theorems and formulas.

Excerpt: This chapter is devoted to a study of Banach algebras, at present one of the most active fields in functional analysis. Banach algebras were unknown before 1935; nevertheless by 1940 their importance had already been established due principally to the independent efforts of M. H. Stone and I. Gelfand. Stone was motivated in this direction by the Hilbert space spectral theory, whereas Gelfand seems to have been motivated by the underlying algebraic nature of Wiener's Tauberian theorems.

Mathematics document containing theorems and formulas.

Excerpt: We shall now take up the main theme of these Lectures; the theory of one-parameter semi-groups of endomorphisms and the various ramifications and applications of this theory. The point of departure is the following problem : Let X be a complex Banach space, G(5) the corresponding Banach algebra of endomorphisms of X. Further let 2 be a given spinal semi-module of real or complex numbers...

Mathematics document containing theorems and formulas.

Excerpt: In the present chapter we continue our study of Banach algebras. Whereas the previous chapter was mainly concerned with algebraic aspects, the emphasis here is on the analysis, both with regard to problems and method. The central idea of the chapter is the operational calculus defined for any (B)algebra, a. The operational calculus may be described as follows: Let A be an open subset of the complex plane and let @(A) be the set of all x E $3 such that - ( xC) A....

Mathematics document containing theorems and formulas.

Excerpt: An important part of classical analysis is concerned with functions which are holomorphic in a half-plane. Such a function may be representable by a Cauchy or Poisson integral in terms of the boundary values on the line bounding the half-plane, or it may be representable by one of the several forms of the Laplace integral, or by a suitable interpolation series such as the binomial series, to mention just a few alternatives. The rate of growth of the function on ...

Mathematics document containing theorems and formulas.

Excerpt: The present chapter is concerned with real-valued functions defined on a subset 2 of En and satisfying the condition (7.1.1) f (tl + t2) 5 f (tl) + f(t2), tr , t2 E 2. Obviously with tl, tz E Z we must require that tl + t2 also belong to Z. Functions of this kind are called subadditive and their domains are called additive semi-groups or semi-modules. We have already encountered subadditive functions in section 2.5 and in the theory of S(p) algebras described in...

Mathematics document containing theorems and formulas.

Excerpt: The present chapter serves several different purposes. It contains the basic definitions in the theory of abstract semi-groups; it gives the elements of a theory of semi-modules ( = additive abelian semi-gr ups) with special reference to semi-modules in a Euclidean space; and finally it offers a discussion of a special semi-group related to the theory of relativity. Semimodules of real or complex numbers form the parameter manifolds of oneparameter semi-groups o...

Mathematics document containing theorems and formulas.

Excerpt: In the present chapter we deal with the strong case analogues of Problems A and B of the preceding chapter. The first problem is that of a one-parameter semi-group t5 = [T(t); E > 01 of linear bounded transformations on a complex (B)-space% to itself with the property that T(h + t2)x = T(&)[T(tz)xl for all h , 52 > 0 and all x E X. We assume throughout the remainder of this treatise that T(5) is strongly continuous for t > 0. Actually this assumption is not as r...

Mathematics document containing theorems and formulas.

Excerpt: Further study of the semi-group G = [T([)l of linear bounded transformations on a complex (B)-space to itself centers around the properties of the generating operator A and its resolvent R(X; A). The Laplace transform turns out to be the natural intermediary between T(f) and R(X; A). The following simple case will elucidate the situation. We take 5 = 21, the space of complex numbers with the usual metric, and consider linear transformations on 21 to itself. Here...

Mathematics document containing theorems and formulas.

Excerpt: We next consider the converse problem: What properties should an operator U possess in order that it be the infinitesimal generator of a semi-group [T(f)] of linear bounded operators? In order to make this problem more precise the type of convergence at the origin as well as the type of continuity for [ > 0 must be specified. If we require that T([)t end to I in the uniform operator topology as -+O f , then the solution is obvious: U must be bounded and every bo...

Mathematics document containing theorems and formulas.

Excerpt: Perturbation theory has long been a useful tool in the hands of the analyst. It is used to determine the state of a system which is in a certain sense close to a known system. In our ease the known system is a semi-group T(t; Ao) of linear bounded operators with infinitesimal generator A. (not to be confused with the infinitesimal operator A. of Chapter X) and we wish to ascertain that nearby operators A likewise generate semi-groups. Moreover it is desirable th...

Mathematics document containing theorems and formulas.

Excerpt: The concepts of adjoint space and adjoint operator have already been introduced in Chapter 11. From a given (B)-space X, there ensues a hierarchy of adjoint spaces X*, X**, . - . , each space being embeddable in its second adjoint space under the natural mapping. Likewise if U E @(X), then U* E @(X*), U** E @(X**), , each operator being a restriction of its second adjoint in the sense of the above embedding. For closed linear operators U E D(X) with D(U) dense i...

Mathematics document containing theorems and formulas.

Excerpt: The two special classes of semi-groups namely translations and powers, which are to be considered in the present chapter, are among the simplest concrete illustrations of the general theory that can be found. For this reason alone they would merit a place in this treatise. In addition they provide us with a number of examples and counterexamples which enable us to answer several of the existence questions raised in parts two and three. Thus the simple semi-group...

Mathematics document containing theorems and formulas.

Excerpt: In paragraph 5.3 an operational calculus for arbitrary closed operators was developed which can be described in the following way: Given an open set of the extended complex plane, the calculus defines an isomorphic mapping of the algebra of functions f ( X ) holomorphic in A into the algebra of operator valued functions f(A) with domain @(A) = [A; o,(A) c A] and range @(%). In the present chapter we shall limit our considerations to infinitesimal generators of s...

Mathematics document containing theorems and formulas.

Excerpt: Having developed an operational calculus for semi-groups of linear operators, we now consider the relation between the spectrum of *(a; A ) and that of A, where either a E S(p) and A 5 p or a E S(p) and A < 9. In this general setting the following mapping theorem holds: #(a; a(A)) C u[E(a;A ) ] ,w here the inclusion can be proper. If the set function of the infinitesimal generator is sufficiently specialized a more precise mapping theorem can be proved. For inst...

Mathematics document containing theorems and formulas.

Excerpt: The study of holomorphic semi-groups of linear bounded operators introduces several new aspects into semi-group theory which we now proceed to investigate. Ordinarily a semi-group G = [T([); [ > 0] does not admit of an analytic extension. However, if O can be extended analytically into a part of the complex plane, then it becomes pertinent to inquire as to the properties of such an extension. It turns out that the analytic extension of a semi-group of operators ...

Mathematics document containing theorems and formulas.

Excerpt: The original ergodic hypofhesis concerned the long run average behavior of individual trajectories in the phase space 2 = [a] of a dynamical system. Such trajectories define a measure preserving flow p ( -5,) in 2. The first ergodic theorems, which were due to G. D. Birkhoff, T. Carleman, B. 0. Koopman, and J. von Neumann, evolved in 1931-1932. The problem as then formulated dealt with the isometry [T([)x](u) = X[-(U[)I, and consisted in establishing the existen...

Mathematics document containing theorems and formulas.

Excerpt: In the theory of summability of trigonometric Fourier series one encounters factor sequence transformations of the form One of the more familiar instances of this is the case of Abel-Poisson summability where Xn = - I n 1 ; here it is customary to set e-' = r. For a proper choice of f An) these transformations form semi-groups on Lp(-n, A) (cf. E. Hille [7]). Such semi-groups may be characterized by the fact that they are the only measurable semi-groups of linea...