Given a bounded linear operator T on a normed linear space X, we may regard T as an element of the unital normed algebra B(X) of all bounded linear operators on X, and so have available the numerical range V(B(X), T) and the results of chapters 1 and 2. However, a more natural numerical range of T is also available, defined directly in terms of the space X and its dual space, without intervention of the algebra B(X). We denote this ‘spatial’ numerical range by V(T).

In §9 we compare V(T) with V(B(X),T) and with the numerical range W(T) corresponding to a semi-inner-product on X. §10 is concerned with spectral properties of V(T). The principal result is a theorem of Williams that gives

Sp(T) ⊂ v(T)-

when X is a complex Banach space. Theorems of Lumer and of Nirschl and Schneider give interesting spectral properties of boundary points of V(T).

In general, V(T) is not convex and it is therefore of interest to study its topological or geometrical properties. In §11 we show that V(T) is connected, and that, except when X≅R, this result holds, without assumption of linearity, for any continuous mapping T of the unit sphere of X into X.

THE SPATIAL NUMERICAL RANGE

Let X denote a normed linear space over F, S(X) its unit sphere {x ∈ X : ||x|| = 1}, and X′ it s dual space.