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Biographical Memoirs: Volume 63
transition operators satisfying both Kolmogoroff differential equations. Another result (1954, 6) has to do with null solutions: suppose that Z(t), satisfying the second Kolmogoroff differential equation, is differentiable in the norm topology for t > 0, that limt → o 1Z(t)1 = 0 and that Z(t) is of normal type. Then for A triangular and arbitrary B in M, the equation Q′(t) = Q(t) (A + B) has a solution with the same properties.
Hille followed this up with a series of papers (1961, 2; 1968, 2; 1966, 2) on differential equations in a Banach algebra of the form
Here z is a complex variable while f(z) and w(z) belong to a complex noncommutative Banach algebra B with unit f(z) being analytic in z. He studied the solutions of this equation in a simply connected domain of holomorphism of f(z), in a neighborhood of a simple pole of f(z), and in a partial neighborhood of a multiple pole.
Hille also became interested in transfinite diameters at about the same time, that is, 1961, and pursued this off and on for another five years. To understand the problem, we go back to Kolmogoroff's abstract definition of an averaging process A:
(i) A assigns to any finite set of positive numbers (x1,. . . , xm ) positive average A(x1, . . . , xm);
(ii) A(x1, . . . , xm) is continuous, symmetric, and strictly increasing in each argument;
(iii) A(x, . . . , x) = x;
(iv) If A(x1, . . . , xk) = y, then
Given a compact metric space E, consider sets of n points,