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We present some sufficient conditions for a function from an open set in C into a Hilbert space H such that (T z)f(z) = x (T E B(H) and x e H) to be analytic. As an application we show that hyperinvariant subspaces exist for certain class of operators. Let T be a bounded operator on a Hilbert space H. Suppose f is a vectorvalued mapping from an open set U in the complex plane C into H, y is a vector in H, and (T z)f(z) = y for all z in U. We ask what additional conditions force f to be analytic. For example, a recent work of Stampfli and Wadhwa [6] showed that if T is dominant and f is bounded, then f is analytic. (Also see [5].) In this note, we present some circumstances under which f is analytic. As an application we give a sufficient condition for the existence of hyperinvariant subspaces. For a Hilbert space H, we shall write B(H) for the set of all bounded operators on H. Let T E B(H) and F be a compact set in C. We shall write XT(F) for the linear manifold consisting of those x in H such that (T z)f(z) x for some analytic vector-valued function f from C\F into H. For convenience, we call the closure of XT(F) a spectral subspace of T. Obviously, a spectral subspace of T is always hyperinvariant for T; that is, it is invariant for every operator commuting with T. For basic properties of spectral manifolds XT(F) we refer to [1]. We shall write Sp(T) for the spectrum of T and II(T) for the approximate point spectrum of T. For the definition and basic properties of approximate point spectra, see Chapter 8 in [2]. For F C C, we write F* for {z: z E F). PROPOSITION 1. If T E B(H) andy E nzeu(T z)H where U is an open set in C such that U n I(T) = 0, then z ( T z)y is an analytic vectorvalued function. PROOF. For convenience, write f(z) = (T z) y (z E U). (This function is uniquely defined, by the hypothesis on y.) First we show that f is bounded on compacta. If not, there exists a convergent sequence (z,,} in U such that Zo= limnzn E U and limnIIf(zO)II = oo. Let xn = If(zn)II 'f(zn). Then Received by the editors September 15, 1976 and, in revised form, October 8, 1976. AMS (MOS) subject classifications (1970). Primary 47A 10, 47A15; Secondary 47B20.

On analyticity of local resolvents and existence of spectral subspaces