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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
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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.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0348539-6
Keywords: Operator valued inner function, Potapov inner function, exponential representation, normal operators, commutation properties
Article copyright: © Copyright 1974 American Mathematical Society