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

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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**0335987 (49:763)****[2]**-,*Inner functions analytic at a point*, Illinois J. Math.**16**(1972), 651-652. MR**0310609 (46:9707)****[3]**-,*Operator valued inner functions*, Dissertation, Northwestern University, Evanston, Ill., 1972.**[4]**-,*Operator valued inner functions analytic on the closed disc*, Pacific J. Math.**40**(1972), 57-62. MR**0338803 (49:3567)****[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**45**#5797. MR**0296738 (45:5797)****[7]**-,*The differential equation of an inner function*. II (preprint).**[8]**-,*Lectures on invariant subspaces*, Academic Press, New York, 1964. MR**30**#1409. MR**0171178 (30:1409)****[9]**Domingo Antonio Herrero,*A geometric approach to inner function-operators and their differential equations*(preprint). MR**0634685 (58:30364)****[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 Math. und ihrer Grenzgebiete, Band 36, Springer-Verlag, New York, 1967. MR**36**#707. MR**0217618 (36:707)****[12]**Malcolm J. Sherman,*A spectral theory for inner functions*, Trans. Amer. Math. Soc.**135**(1969), 387-398. MR**38**#5037. MR**0236743 (38:5037)**

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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