Spectral Operators in a Direct Sum of Hilbert Spaces.

Let E C X be the finite-dimensional subspace spanned by xi, ..., xm. Now, for y E Y and variable x e E, y(x) is a continuous linear functional on E which we denote by y'. Thus, there is defined in a natural way a function f'(x) with domain (E n B) and range in E*, the conjugate space of E. The domain contains and surrounds densely every point of E n K(D), hence every point of K(E n D). It is easy to show that f' is monotonic and hemicontinuous. The set (E n D) surrounds the origin in E, and x e (E n D) implies (x, f'(x)) > 0. All the hypotheses of Lemma 2 are satisfied, so there exists x e K(E n D) with f'(x) = 0. Of course, it does not follow that (x) = 0; however, by the monotonicity of f', (xix, f'(xi) 0) > 0 for i = 1, . . ., m, and from this (5) follows, since all the (x1 x) are elements of E. The proof is complete.