Convolution Operators (Vector-Valued Case)

Abstract

In Chapters 6 and 7 we shall need analogues of the results in Chapter 2 for operators of the type

$${L_K}f\left( x \right) = \int K \left( {x - y} \right)f\left( y \right)dy $$

where now f is a function on G taking values in a Hilbert space ℋ (i.e., a vector-valued function) and K is a function on G taking values in B(ℋ₁, ℋ₂), the space of bounded linear mappings of ℋ into a second Hilbert space ℋ₂ (i.e., an operator-valued kernel).