Operator Norms

Abstract

We consider the algebra End (E) of all linear operators in the real vector space E [beginning with this section we shall assume that the underlying space E is real in order to simplify, to some extent, the notation; however, all results discussed below extend to the complex case with minor modifications]. End (E) can be endowed with a norm in infinitely many ways which are all topologically equivalent (thanks to the finite-dimensionality of the space End (E)), but which exhibit essential differences from the algebraic and geometric points of view. At any rate, every norm compatible with the algebra structure of End (E) must be a ring norm, i.e., possessthe ring property ∖AB∖ ≤ ∖A∖ ∖B∖, as well as unit-preserving,i.e., such that ∖I∖ = 1 (from the ring property it follows that ∖I∖ ≥ 1). As we know from §1 of Chapter 1, the indicated properties are enjoyed by every norm on End (E) that is defined by some norm ∖· ∖ on E in the standard manner:

$$||~A~||~=~\sup ~||~Ax~||~(=~\underset{x\ne 0}{\mathop{\sup }}\,~\frac{||Ax||}{||x||}~=~\underset{||x||\le 1}{\mathop{\sup }}\,~||Ax||)$$

(in which case we say that the norm ∖· ∖ is subondinate to the norm ∖· ∖ on E, or induced by the latter; translator’s note).