1. A bounded operator T on a Hilbert space e is said to be hyponormal if (1. 1) T*T -TT* ==D_O. For a survey of some of the properties of such operators, see Putnam [6]. ilyponormal and normal operators have certain common properties. In particular, if T has the rectangular representation T = H + iJ, so that (1. 1) becomes (1.2) HJ-JH -iC, C-=D?_O, then the spectra of the real and imaginary parts H and J are precisely the (real) sets obtained by projecting sp(T) onto the corresponding axes; see [6], p. 46. Also, (1. 3) (T-zl)x _ (T* zI) x dist(z,sp(T) x x C. Let the real part, H, of T have the spectral resolution (1.4) H f A dEx. For any bounded operator A on g and any open interval A, consider the operator AA= -E(A)AE(/) on the Hilbert space E(A) ) and with spectrum sp (AA). It follows from (1. 2) that if T is hyponormal, then (1.5) HAJA -JAT =--i-C'A, CA E(A)CE(A) >0O so that TA== E (A) TE (A) is hyponormal on E(A) . It was proved in Putnam [7] that if T is hyponormal, then (1.6) sp(TA) Csp(T) and that (cf. (1.1)) (1. ') ?1 D meas2(sp(T)),