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Let S be a subnormal operator on the Hilbert space H, and let N = f z dE(z) be its minimal normal extension on K. Let ¡i be a scalar spectral measure for N. If / e L°°(ii), define 7} = Pf(N)\H: H -> H, where P: K -> H denotes orthogonal projection. S has the C*-Spectral Inclusion Property (C*-SIP) if o(f(N)) Ç a(7)), for all / G C(o(N)), and S has the rf*-Spectral Inclusion Property ( W*-SIP) if o(f(N)) Q o(Tf), for all/ <E £°°(fi). It is shown that 5 has the C-SIP if and only if o(N) = H(S), the approximate point spectrum of S. This is equivalent to requiring that E(ùt)K have angle 0 with H, for all nonempty, relatively open A C o(N). S has the W * -SIP if this angle condition holds for all proper Borel subsets of o(N). If S is pure and has the C*or W*-SIP, then it is shown that o(f(N)) Ç oe(Tf), for all appropriate /. The spectrum of an operator on a Hilbert space has always been a valuable tool for the operator theorist. In particular, the study of normal operators is quite complete (see, for example, Halmos [1951]). Two classes of operators where knowledge of the spectrum and/or spectral properties has been crucial are the class of Toeplitz operators and the class of subnormal operators. In the former class, a result of Hartman and Wintner states that the spectrum of every Toeplitz operator contains that of its associated Laurent operator (Hartman-Wintner [1950]). In the latter class, a theorem obtained by Halmos states that the spectrum of a subnormal operator contains that of its minimal normal extension (Halmos [1952]). A recent question of Halmos is to describe, if possible, what the relationship is between these two results-perhaps one may even be a consequence of the other? To formulate a setting in which to view the two results, we shall adopt the point of view that Toeplitz operators are, in a reasonable sense, generated by a particular subnormal operator-namely, the unilateral shift (i.e., multiplication by the independent variable on the usual Hardy space). To see what this means, consider that the bilateral shift is the minimal normal extension of the unilateral shift, and that the classical Laurent operators form exactly the von Neumann algebra generated by the bilateral shift. Toeplitz operators are then compressions of operators in some Received by the editors June 22, 1979 and, in revised form, January 11, 1980. AMS (MOS) subject classifications (1970). Primary 47B20, 47A10; Secondary 47B35, 47A20.

Subnormal operators, Toeplitz operators and spectral inclusion