By Idun Reiten, Sverre O. Smalø, Øyvind Solberg

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**Extra info for Algebras and Modules One**

**Sample text**

Z, Proof. Let gn = fn - f, gn ~ 0, and IIgnll :S 2. Hence, by considering subsequences, we may assume that IIgnll -+ a E JR. ) -+ z. ). ) = adlfl1 2 and (Tgn, gn) - (3 IIgnll 2 - n, -'- 0 gn r . Then f3n -+ A E W(T), and we have (Tgn , gn) = Aa2 and z = allfl1 2+ Aa2. Since a2 + IIfl12 = 1, we have that z lies on the segment joining a and A, both belonging to W(T). So z cannot be an extreme point. Thus, either A = z or a = z. ) = zllfl12. So Un} has the property (P). D A point z E W(T) is called a corner if W(T) is contained in a half-cone with vertex at z, and the semivertical angle of the cone is less than l 20 1.

Since A 2 = aI, we have ([1 + Icl 2 + Icl 4 + .. 'lABh, h) :::; ([1 + Icl 2 + Icl 4 + .. 'lh, h). 0 40 2. 5 In the commuting case it was observed by K. Gustafson (1968). "The Angle of an Operator and Positive Operator Products," Bull. Amer. Math. Soc. 74,488-492. that for positive selfadjoint A and bounded accretive B, BA is accretive. Later this was extended to W(AB) C W(A)W(B) for positive selfadjoint A and bounded commuting A and B, in R. Bouldin (1970). "The Numerical Range of a Product," J.

38, 177-216. The proof requires analytic continuation into the upper half-plane. Simpler proofs have been given for the case p = 1/2. For related results see E. Heinz (1951). "Beitrage zur St6rungstheorie der Spektralzerlegung," Math. Ann. 123, 415-438. C. Davis (1963). "Notions Generalizing Convexity for Function Defined on Spaces of Matrices," in Convexity (V. ), Amer. Math. , 187-20l. K. Bhagwat and A. Subramanian (1978). "Inequalities Between Means of Positive Operators," Math Proc Camb. Phil.