Let H be a separable infinite
dimensional complex Hilbert space, B(H) the set of bounded linear operators on H.
Consider a holomorphic map z → K(z) from a complex neighborhood of some
interval of the real axis to B(H), such that for z real K(z) is hermitian. These
conditions are satisfied by K(z) = H − z1, with H hermitian. In this special case
K(z) has a bounded inverse S(z) (the resolvent of H), for z not on the real axis, and
S(z) can be represented as the Hilbert transform of a measure whose values
are bounded positive operators (the spectral measure of H); for z on the
real axis K(z) has a (generally unbounded) inverse for z not in the point
spectrum of H; closely related to the spectral representation of S(z) is an
approximation theorem which asserts roughly that for most real values of
z,[K(z)]^{−1} can be approximated by operators of finite rank obtained by
taking the orthogonal projector P onto a finite dimensional subspace D
and inverting PK(z)P on D. The object of this paper is to give conditions
on K(z) sufficient to imply the conlusions just noted in the special case
K(z) = H − z1.
The main theorems are Theorem 1 in §1, and Theorem 4 in §4; each has two
parts—a representation for [K(z)]^{−1} for z complex, and an approximation theorem
for [K(z)]^{−1} for z real. Theorem 4 is used in §5 to prove a convergence theorem
(Theorem 5) for the Kohn variational method in quantum mechanical potential
scattering (the relevant terms are defined in that section). This application motivated
the writing of the paper.
