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Differential Equations in Operator Algebras: II. Invariance of the Order Cone

✍ Scribed by Ray Redheffer; Peter Volkmann

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 340 KB
Volume: 130
Category: Article
ISSN: 0022-0396
DOI: 10.1006/jdeq.1996.0148

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✦ Synopsis

Operator 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; see, for example, [4,5,7] and the references cited there. 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 [6] and developed more fully in [10,12]. Conditions for invariance of the unit ball are extended to the infinite-dimensional case in [8].

As a sequel to [8], we now give a corresponding extension for invariance of the order cone. This justifies the main theorem in [9], 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 the following paper [11].

1. NOTATION

The sets of reals, nonnegative reals and complex numbers are denoted respectively by R, R + , and C, and X is a real or complex Hilbert space of dimension dim X 1. The set of bounded linear operators x: X Ä X is article no.

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