Differential equations in operator algebras. II: Invariance of the order cone (Q1815652)

Operational differential equations such as the matrix Riccati equation \[ u'=a+bu+ud+ucu \] play a prominent role in the theory of scattering and transport and in other areas of technology. Especially important are conditions for invariance of the unit ball and the order cone. In the finite-dimensional case an outline of the theory from the point of view of this paper was set forth in [the first author, Bull. Am. Math. Soc. 81, 485-488 (1975; Zbl 0322.34005)] and developed more fully in [the authors, Linear Algebra, Appl. 118, 77-82 (1989; Zbl 0679.15015) and the first author and \textit{W. Walter}, Monatshefte Math. 102, 237-249 (1986; Zbl 0597.34063)]. Conditions for invariance of the unit ball are extended to the infinite-dimensional case in part I [Math. Ann. 239, 97-110 (1979; Zbl 0417.34097)]. As a sequel to the last cited paper, we now give a corresponding extension for invariance of the order cone. This justifies the main theorem in [the authors, J. Differ. Equations 78, No. 1, 136-143 (1989; Zbl 0671.34013)], which was stated without proof. It turns out that the analysis depends on some rather subtle results in the theory of operator algebras. Since these have only a tenuous connection with differential equations they are presented, under a separate title, in [ibid. 130, 368-376 (1996; review above)].