On approximation spectrum of bounded self-adjoint operators

Abstract

In this formula i t is agreed that whenever the transformation (PnTPn-XI)IHn does not have an inverse we write H((PnTPn-XI)IHn)'III = +~ . Denote by ~a(T) the approximation spectrum of T and by o(T) the spectrum of T . I t is easily seen that Oa(T) is compact. In fact i t follows from the definit ion that X belongs to the complement of Oa(T) i f f there exists an in f in i te sequence of integers n I < n 2 < . . such that Hnk) II I supll((P. TP,~ X I ) I < ~o k "k '~ I f T is self-adjoint i t is clear that ~(T) C~a(T) . In this case we wil l use the following property of ~a(T) , ~ is in ~a(T) i f and }, ~ E O(PnTPnlH n) for each only i f there is a sequence { n}n=i such that 1n n and lim ~'n = X . In this paper we consider the following problem. When does the equality Oa(T) = ~(T) holds, T being a self-adjoint operator with a f in i te number of nonzero diagonals. We say that T is a 2n + 1 diagonal operator i f t i j = 0 for