Commutation properties of the coefficient matrix in the differential equation of an inner function
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- by Stephen L. Campbell
- Proc. Amer. Math. Soc. 42 (1974), 507-512
- DOI: https://doi.org/10.1090/S0002-9939-1974-0348539-6
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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 $\left \| T \right \| < 1$. The case where $\left \| T \right \| = 1, r(T) < 1$ 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
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Bibliographic Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 42 (1974), 507-512
- MSC: Primary 47A65
- DOI: https://doi.org/10.1090/S0002-9939-1974-0348539-6
- MathSciNet review: 0348539