An Approximation Theorem for a Class of Operators
GARY K. LEAF · Proceedings of the American Mathematical Society · 1965
It is well known that if Uis a unitary operator in a Hilbert space H, then the following approximation theorem is an immediate consequence of the spectral representation for the operator U. Theorem A. (a) Let {Et:0^t^2w} denote the family of spectral projections associated with U; then if e>0 and a, 0 = a = 27r, are given and if x is any element in the range of the projection Ea+t-Ea, we have: ||(U -e«'")x|| = e||s||; (b) moreover, for this same e, there exists a finite collection of closed linear manifolds in H such that H is the direct sum of these manifolds ; and in each subspace, U behaves as in part (a).E. R. Lorch [7] extended this theorem to certain classes of operators in a reflexive Banach space.The class considered by Lorch consists of those bounded, invertible operators V for which the norms of their iterates are restricted by the condition, || F"|| =0(1) as |w| tends to infinity.The author [ó] extended the results or Lorch to a somewhat larger class restricted by the condition, || Fn|| = 0(| n\).This extension was made through the use of methods developed by N. Dunford [4], [5].In the present paper, the result is extended to a much larger class of operators by using the methods of Harmonic analysis as developed by A. Beurling [l], J. Wermer [8], Y. Domar [3], and others.It should be mentioned that this extension might have been possible through the use of methods developed by F. Wolf [9] in his spectral theory for operators based on the generalized trigonometric integrals of S. Bochner.The present class is restricted by the condition, (i) || Fn|| =0(|«| *) as \ tends to infinity, for some g>0.An operator V, defined in a Banach space B, which satisfies condition (i) is easily seen to have it spectrum on the circumference of the unit circle.Moreover, the usual operational calculus may be extended by introducing a certain weighted algebra associated with the sequence {| Fn||:« = 0, ±1, • • • }.Such algebras were introduced by Beurling [l], and later generalized by Wermer and Domar.