Appendix

Summary

In this appendix we present a number of auxiliary results about regulated functions, the positive square root and polar decomposition of a closed unbounded operator convolutions, and inversion of block matrices.

Regulated functions

Definition A.1.1 Let X be a Banach space, and let I = [a, b] ⊂ ℝ be a closed bounded interval.

1. function f: I → X is a step function if f is constant on successive intervals I_k = [a_k, a_k+1), 0 ≤ k < n, where a₀ = a and a_n = b.

2. function f : I → X is regulated if f is right-continuous and has a left-hand limit at every point of [a, b]. We denote the class of all regulated X-valued functions on I = [a, b] by Reg(I; X).

3. an X-valued step function on a closed unbounded subinterval J of ℝ we mean a function whose restriction to any closed bounded interval is a step function in the sense of (i).

4. an X-valued regulated function on a closed unbounded subinterval J of ℝ we mean a function whose restriction to any closed bounded interval is regulated in the sense of (ii). We denote this class of functions by Reg_loc(J; X), and let Reg(J; X) be the space of all functions which are both regulated and bounded.

Proposition A.1.2Let X be a Banach space, and let I = [a, b] be a closed bounded interval.