A remark on commuting operator exponentials
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- by Edgar M. E. Wermuth PDF
- Proc. Amer. Math. Soc. 125 (1997), 1685-1688 Request permission
Abstract:
In a previous paper the author proved that for square matrices with algebraic entries exp$(A)$exp$(B)=$exp$(B)$exp$(A)$ if and only if $AB=BA$. This result is extended here to bounded operators on an arbitrary Banach space.References
- Dennis S. Bernstein, Problem 88-1, Commuting Matrix Exponentials, SIAM Rev. 30 (1988), 123.
- D. S. Bernstein, Erratum: “Some open problems in matrix theory arising in linear systems and control” [Linear Algebra Appl. 162/164 (1992), 409–432; MR1148412 (92k:93040)], Linear Algebra Appl. 180 (1993), 5–6. MR 1206405
- Robert B. Burckel, An introduction to classical complex analysis. Vol. 1, Pure and Applied Mathematics, vol. 82, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. MR 555733, DOI 10.1007/978-3-0348-9374-9
- Einar Hille, Methods in classical and functional analysis, Addison-Wesley Series in Mathematics, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. MR 0463863
- Roger A. Horn and Charles R. Johnson, Topics in matrix analysis, Cambridge University Press, Cambridge, 1991. MR 1091716, DOI 10.1017/CBO9780511840371
- Edgar Raymond Lorch, Spectral theory, University Texts in the Mathematical Sciences, Oxford University Press, New York, 1962. MR 0136967
- Charles E. Rickart, General theory of Banach algebras, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0115101
- Frédéric Riesz et Béla Sz.-Nagy, Leçons d’Analyse Fonctionelle, Budapest, 1952.
- Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
- Walter Rudin, Functional analysis, 2nd ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991. MR 1157815
- Wasin So, Equality cases in matrix exponential inequalities, SIAM J. Matrix Anal. Appl. 13 (1992), no. 4, 1154–1158. MR 1182719, DOI 10.1137/0613070
- Robert C. Thompson, High, low, and quantitative roads in linear algebra, Linear Algebra Appl. 162/164 (1992), 23–64. Directions in matrix theory (Auburn, AL, 1990). MR 1148395, DOI 10.1016/0024-3795(92)90371-G
- Edgar M. E. Wermuth, Solution to Problem 88-1, SIAM Rev. 31 (1989), 125–126.
- Edgar M. E. Wermuth, Two remarks on matrix exponentials, Linear Algebra Appl. 117 (1989), 127–132. MR 993038, DOI 10.1016/0024-3795(89)90554-5
- Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913
- Nicholas Young, An introduction to Hilbert space, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1988. MR 949693, DOI 10.1017/CBO9781139172011
Additional Information
- Edgar M. E. Wermuth
- Affiliation: Zentralinstitut für Angewandte Mathematik, Forschungszentrum Jülich GmbH, Postfach 1913, D-52425 Jülich, Germany
- Address at time of publication: FB Allgemeinwissenschaften und Informatik, Georg-Simon-Ohm-FH Nürnberg, Postfach 210320, D-90121 Nürnberg, Germany
- Email: e.m.e.wermuth@kfa-juelich.de, edgar.wermuth@ai.fh-nuernberg.de
- Received by editor(s): May 9, 1995
- Received by editor(s) in revised form: September 6, 1995
- Additional Notes: For valuable comments I thank Heinrich Bock, Robert B. Burckel, Hans-Günter Meier, and the referee.
- Communicated by: Theodore W. Gamelin
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 1685-1688
- MSC (1991): Primary 39B42, 47A60, 30E10
- DOI: https://doi.org/10.1090/S0002-9939-97-03643-5
- MathSciNet review: 1353407