Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Commutation properties of the coefficient matrix in the differential equation of an inner function

Author: Stephen L. Campbell
Journal: Proc. Amer. Math. Soc. 42 (1974), 507-512
MSC: Primary 47A65
MathSciNet review: 0348539
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A(x)$ be an operator valued function that is analytic on the real axis. Assume that $ A(x)$ is selfadjoint for each real x. It is shown that $ A(x)$ and $ \smallint _0^xA(s)$ ds commute for all real x iff $ A(x)$ and $ A(y)$ commute for all real x and y. This result is then used to establish several new characterizations of the Potapov inner functions of normal operators T such that $ \left\Vert T \right\Vert < 1$. The case where $ \left\Vert T \right\Vert = 1, r(T) < 1$ and $ {A_T}(x)$ and $ {A_T}(y)$ commute for real x and y is discussed. Here $ {A_T}(x) = - i{U'_T}(x){U_T}{(x)^\ast}$ and $ {U_T}(x)$ is the Potapov inner function for T.

References [Enhancements On Off] (What's this?)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A65

Retrieve articles in all journals with MSC: 47A65

Additional Information

Keywords: Operator valued inner function, Potapov inner function, exponential representation, normal operators, commutation properties
Article copyright: © Copyright 1974 American Mathematical Society