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
DOI:
https://doi.org/10.1090/S0002-9939-1974-0348539-6
MathSciNet review:
0348539
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: Let be an operator valued function that is analytic on the real axis. Assume that
is selfadjoint for each real x. It is shown that
and
ds commute for all real x iff
and
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
. The case where
and
and
commute for real x and y is discussed. Here
and
is the Potapov inner function for T.
- [1] Stephen L. Campbell, The exponential representation of operator valued, differentiable inner functions, J. Differential Equations 12 (1972), 455–461. MR 335987, https://doi.org/10.1016/0022-0396(72)90018-6
- [2] Stephen L. Campbell, Inner functions analytic at a point, Illinois J. Math. 16 (1972), 651–652. MR 0310609
- [3] -, Operator valued inner functions, Dissertation, Northwestern University, Evanston, Ill., 1972.
- [4] Stephen L. Campbell, Operator-valued inner functions analytic on the closed disc, Pacific J. Math. 41 (1972), 57–62. MR 338803
- [5] Morton J. Hellman, Matrix commutator relationships arising from systems of linear ordinary differential equations, Dissertation, New York University, New York, 1954.
- [6] Henry Helson, The differential equation of an inner function, Studia Math. 35 (1970), 311–321. MR 296738, https://doi.org/10.4064/sm-35-3-311-321
- [7] -, The differential equation of an inner function. II (preprint).
- [8] Henry Helson, Lectures on invariant subspaces, Academic Press, New York-London, 1964. MR 0171178
- [9] Domingo Antonio Herrero, Inner function operators, Notas Ci. 9 (1971), no. 1, 7–41. MR 0634685
- [10] S. Lynn Jackson, Operators and inner functions. Dissertation, University of California, Berkeley, Calif., 1967.
- [11] C. R. Putnam, Commutation properties of Hilbert space operators and related topics, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 36, Springer-Verlag New York, Inc., New York, 1967. MR 0217618
- [12] Malcolm J. Sherman, A spectral theory for inner functions, Trans. Amer. Math. Soc. 135 (1969), 387–398. MR 236743, https://doi.org/10.1090/S0002-9947-1969-0236743-X
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A65
Retrieve articles in all journals with MSC: 47A65
Additional Information
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