Let H be a self-adjoint operator with spectral measure E(S) over the Borel sets S of the real line. The spectrum of H is said to be strongly concentrated on S if whenever H n converges strongly to H in the generalized sense it is true that E n (S) converges strongly to the identity. Sufficient conditions on H are given for this to occur for a given arbitrary Borel set S and necessary and sufficient conditions when S is the spectrum of H. In addition several more workable sufficient conditions

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... e cited and a few examples illustrating the results are given. Many authors have studied the changes in the spectra of a sequence of self-adjoint operators H n as it converges strongly in some sense to a self-adjoint operator-e.g., [11] . It is known that while as point sets the spectra of H n do not necessarily converge to the spectrum of H, nevertheless in some sense the spectra of H n are concentrated on that of H. This spectral concentration phenomenon is described through the spectral measures E n , E of the operators involved. In particular since E(Σ) is the identity when Σ is the spectrum of H it is reasonable to say that the spectrum of the sequence H n is concentrated on Σ if EJΣ) converges to the identity as n -> °°. Our main results concern necessary and sufficient conditions for this to occur for an arbitrary sequence H n converging strongly to a fixed operator H. We make extensive use of the properties of the spectral measure E(S) over the Borel sets S of the real line for which a general reference is [4] § § X. 2 and XII. 2. l Preliminaries* Throughout this paper the following notation will be adhered to. H will denote a self-adjoint operator over a Hubert space H. Its domain will be denoted by D{H) and its spectrum by Σ (which is always a closed subset of the real line R). The resolution of the identity of H will be denoted by E(X), -°o < \ < oo, and the associated projection-valued spectral measure by E(S) over all Borel subsets S of R. By convention we take E(λ) to be right continuous, i.e., E(X + 0) = 2?(λ). For a sequence of self-ad joint operators H nJ n= 1,2, •••, over H the quantities D(H n ), Σ n , E n (X), and E n (S) are defined accordingly. According to a definition of Rellich (cf. [9] or [7, p. 429]) we