Kôsaku Yosida

The Hahn-Banach Theorems

In a Hilbert space, we can introduce the notion of orthogonal coordinates through an orthogonal base, and these coordinates are the values of bounded linear functionals defined by the vectors of the base. This suggests that we consider continuous linear functionals, in a linear topological space, as generalized coordinates of the space. To ensure the existence of non-trivial continuous linear functionals in a general locally convex linear topological space, we must rely upon the Hahn-Banach extension theorems.

Ergodic Theory and Diffusion Theory

These theories constitute fascinating fields of application of the analytical theory of semi-groups. Mathematically speaking, the ergodic theory is concerned with the “time average” \(\mathop {\lim }\limits_{t \uparrow \infty } {t^{ - 1}}\int\limits_0^t {{T_s}} ds\) of a semigroup T t , and the diffusion theory is concerned with the investigation of a stochastic process in terms of the infinitesimal generator of the semi-group intrinsically associated with the stochastic process.

Resolvent and Spectrum

Let T be a linear operator whose domain D(T) and range R(T) both lie in the same complex linear topological space X. We consider the linear operator

Strong Convergence and Weak Convergence

{T_\lambda } = \lambda I - T,

On the Differentiability of Semi-groups of Linear Operators

, where λ is a complex number and I the identity operator. The distribution of the values of λ for which T λ has an inverse and the properties of the inverse when it exists, are called the spectral theory for the operator T. We shall thus discuss the general theory of the inverse of T λ .

An operator-theoretical integration of the wave equation.

In this chapter, we shall prove three representation theorems in linear spaces. The first one, the Krein-Milman theorem says that a non-void convex compact subset K of a locally convex linear topological space is equal to the closure of the convex hull of the extremal points of K. The other two theorems concern the representations of a vector lattice as point functions and as set functions.

An abstract analyticity in time for solutions of a diffusion equation

In this chapter, we shall be concerned with certain basic facts pertaining to strong-, weak- and weak* convergences, including the comparison of the strong notion with the weak notion, e.g., strong- and weak measurability, and strong- and weak analyticity. We also discuss the integration of B-space-valued functions, that is, the theory of Bochner’s integrals. The general theory of weak topologies and duality in locally convex linear topological spaces will be given in the Appendix.