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 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****[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**0338803****[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**0296738****[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**0236743**, 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