On operators with rational resolvent
Author: Anthony F. Ruston
Journal: Proc. Amer. Math. Soc. 32 (1972), 329-330
MSC: Primary 47A10
MathSciNet review: 0301527
Full-text PDF Free Access
Abstract: It is shown that a bounded linear operator T on a complex Banach space into itself has a rational resolvent if and only if every bounded linear operator which commutes with every bounded linear operator that commutes with T can be expressed as a polynomial in T.
-  S. R. Caradus, On meromorphic operators. I, II, Canad. J. Math. 19 (1967), 723–736; 737–748. MR 0215123, https://doi.org/10.4153/CJM-1967-066-5
-  -, Review of , Math. Rev. 39 (1970), 371, #1999.
-  Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, With the assistance of W. G. Bade and R. G. Bartle. Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. MR 0117523
-  A. F. Ruston, A note on the Caradus class 𝔉 of bounded linear operators on a complex Banach space, Canad. J. Math. 21 (1969), 592–594. MR 0240653, https://doi.org/10.4153/CJM-1969-066-6
-  Angus E. Taylor, Introduction to functional analysis, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 0098966
- S. R. Caradus, On meromorphic operators. I, Canad. J. Math. 19 (1967), 723-736. MR 35 #5966. MR 0215123 (35:5966)
- -, Review of , Math. Rev. 39 (1970), 371, #1999.
- N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol. 7, Interscience, New York and London, 1958. MR 22 #8302. MR 0117523 (22:8302)
- A. F. Ruston, A note on the Caradus class of bounded linear operators on a complex Banach space, Canad. J. Math. 21 (1969), 592-594. MR 39 #1999. MR 0240653 (39:1999)
- A. E. Taylor, Introduction to functional analysis, Wiley, New York; Chapman and Hall, London, 1958. MR 20 #5411. MR 0098966 (20:5411)
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A10
Retrieve articles in all journals with MSC: 47A10
Keywords: Bounded linear operator, rational resolvent, Caradus class, pole of the resolvent, spectrum, minimal equation theorem
Article copyright: © Copyright 1972 American Mathematical Society