# Hilbert Space, Boundary Value Problems and Orthogonal by Allan M. Krall

oo lim fn =f. Thus (A - >'I)-1 is defined on all of Hand>. is not in a(A), but is instead in p(A). D II. 4. 5. Corollary. Let H be a Hilbert space. c. Then areA) is empty. Proof. If A is in areA), then there is an element z orthogonal to (A - AI)H. Then for all x E H, (A - AI)x, z) = 0, and (x, (A - AI)z) = Let x o.

Additional references to their works may be found in the papers of Birkhoff and Langer , and in the book  by Coddington and Levinson. After World War II, W. T. Reid  continued the work of Bliss, and F. V. Atkinson wrote his classic book  on self-adjoint differential systems. It was in Atkinson's book that the linear Hamiltonian system format seems to have first appeared. Further motivation to study Hamiltonian systms began in the 1970's with the books of B. M. Levitan and I. S. Sargsjan , , and the papers mentioned there which discussed Dirac systems, both regular and singular.

If A is in areA), then there is an element z orthogonal to (A - AI)H. Then for all x E H, (A - AI)x, z) = 0, and (x, (A - AI)z) = Let x o. = (A - AI)Z to see that A is in ap(A). This is impossible. 6. Theorem. Let U be a unitary operator on a Hilbert space H. If A E a(U), then IAI = l. Proof. If IAI > 1, let (AI - U)x = f. This has a solution and A is in p(U). If IAI < 1, then 00 n=O and A is in p(U). Thus a(U) is on the unit circle. D We conclude with yet another characterization of the operator norm.