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It is shown that the equation of the title is valid for certain classes of not necessarily normal operators (including Toeplitz operators, and operators whose spectrum is a spectral set), and a new proof is given of C. R. Putnam's theorem that it is valid for seminormal operators. If Fis any operator (bounded linear, in Hubert space) then f(a(T)) = o(fi(T)) for every rational function/(A) without poles in ct(F). On the other hand, if Fis normal then for every polynomial/(A, A*) one has <r(/(F)) =/(a(F)) = {/(A, A*) : A e o'T)} (see [16] for a recent elegant proof). In the present paper, we are concerned with the polynomial/(A, A*) = ^(A + A*) = Re A for not necessarily normal operators F, i.e., we consider conditions on an operator F such that (*) Re o<T) = cr(Re F). This is a very humble "spectral mapping theorem", but it is a considerable achievement for a nonnormal operator. For example, the fact that (*) holds for F seminormal, which is due to C. R. Putnam [10], plays a role in his recent proof that a seminormal operator whose spectrum has zero area is normal [13]. In Theorem 1 we give a simple new proof of Putnam's theorem, and condition (*) is verified for some classes of not necessarily seminormal operators in Theorems 2-4. Specifically, we prove Theorem 1 (Putnam). If T is a seminormal operator, then (*) holds. Theorem 2. If T is a Toeplitz operator, then (*) holds. Theorem 3. IfT is an operator such that oiT) is a spectral set for T, then (*) holds. Theorem A.IfT satisfies the growth condition iGy) and <r(F) is connected, then (*) holds. All operators contemplated in Theorems 1-4 are convexoid, i.e., conv a'T) = C1 WiT). (Here conv denotes convex hull, Cl denotes closure, and W'T) ={(Tx\x) : \x\ = 1} is the numerical range of F.) Lemmas 4-6 give results weaker than (*) for certain convexoid operators. Received by the editors March 18, 1970. A MS 1969 subject classifications. Primary 4730, 4740.

Conditions on an operator implying ${\rm Re}\,\sigma (T)=\sigma ({\rm Re}\,T)$