Operator Radii of Commuting Products

Operator radii wp(T) for a bounded linear operator T on a Hubert space were introduced in connection with unitary p-dilations. We shall be concerned with universal estimates for the ratios wp(ST)/(wa(S)wl,(T)) for commuting operators S, T and o, p > 0. 1. All operators in this paper are bounded linear operators on a complex Hubert space §. We say an operator T belongs to the class G (0 < p < oo) if there exists a unitary operator U on some Hubert space sí such that H contains § as a subspace and such that T*h = pPU"h for h G $ and n = 1, 2, . . . , where P is the orthogonal projection of © onto Q. The classes G were defined by Sz.-Nagy and Foias, [7] while Holbrook [4] introduced the operator radii wp(T) of an operator T, relative to Gp, by the formula: wp(T) = inf{y; y >0,y~lT G Gp). The family of operator radii includes the familiar quantities in operator theory: wx(T) =||7]| (norm of T), w2(T) = w(T) := sup{|(77i, h)\; \\h\\ = 1} (numerical radius of T), and limp_>aowp(T) = r(T) (spectral radius of T). For each p > 0 the operator radius wp(-) is a pseudonorm on 93(D), the space of all operators, in the sense that wp(aT) =|«K(T), wp(T+ S) < yp{wp(T) + wp(S)} where y is a positive constant depending only on p. The constant yp can be equal to 1 or p according as 0 0 the Lipschitz constant of the map ¿. with respect to this pseudonorm wp(-) is majorated by a(2 p) • wa(S) or op ■ wa(S) according as 0<p<lorl<p<oo (see the next section): o(2-p)-wa(S)-wp(T) for0<P< 1, p Wp(Sr) 1 op ■ wa(S)w(T) for 1 < p < oo. Received by the editors August 12, 1974 and, in revised form, March 3, 1975. A MS (MOS) subject classifications (1970). Primary 47 A10, 47A20, 47A30.