CHAPTER 1

Prerequisites on Operators Acting into Finite

Dimensional Spaces

1.1. Introduction

The present chapter is auxiliary. The material collected here will be applied to

special integral operators throughout the book.

Section 1.2 is devoted to functionals and operators acting on n-component

continuous vector-valued functions which are given on a locally compact Hausdorff

space X and vanish at infinity. The space of such functions, endowed with the norm

f = sup{|f(x)| : x ∈ X}, is denoted by

[Cv(X)]n

in the case of real vector-valued

functions and by

[Cv(X)]n

in the case of complex vector-valued functions. From

the Riesz theorem it follows that any linear bounded functional L on

[Cv(X)]n

admits the representation

L(f) =

X

dμ(x)f(x) =

n

j=1

X

fj (x)dμj(x),

where μ = (μ1,...,μn) is a vector-valued measure whose elements are signed finite

regular measures on the σ-algebra BX of Borel subsets of X and f = (f1,...,fn).

A similar representation is valid for a linear bounded functional on [Cv(X)]n with

μ = (μ1,...,μn) being complex regular measures on BX . We show that

L = |μ|, | |

where | |·| | is the variation of the measure. We also consider a linear bounded operator

T mapping

[Cv(X)]n

into the m-dimensional Euclidean space

Rm

(and

[Cv(X)]n

into the unitary space

Cm)

which is defined by

T (f) =

X

dM(x)f(x),

where M = ((μij)) is a (m × n)-matrix whose elements are finite signed (complex)

regular measures on BX . We derive representations for the norms of the operator

T and the functional Tz = (z,T (·)) in the real and complex cases, i.e. for T :

[Cv(X)]n

→

Rm,

z ∈

Rm

and T :

[Cv(X)]n

→

Cm,

z ∈

Cm.

These representations

remain valid if the operators and functionals are defined on the spaces

[Cb(X)]n

and

[Cb(X)]n

of continuous and bounded n-component vector-valued functions on

X supplied with the norm f = sup{|f(x)| : x ∈ X}.

In Section 1.3 we consider the operator

S(f) =

X

G(x)f(x)dμ(x),

acting from the space

[Lp(X

,

A,μ)]n

([Lp(X

,

A,μ)]n)

of real (complex) vector-

valued n-component functions into

Rm (Cm).

Here (X , A,μ) is the space with

9

http://dx.doi.org/10.1090/surv/183/02