Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Monotone unitary families

Author: Daniel Grieser
Journal: Proc. Amer. Math. Soc. 141 (2013), 997-1005
MSC (2010): Primary 47A55; Secondary 47A56, 15A22
Published electronically: July 31, 2012
MathSciNet review: 3003691
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A unitary family is a family of unitary operators $ U(x)$ acting on a finite-dimensional Hermitian vector space, depending analytically on a real parameter $ x$. It is monotone if $ \frac 1i U'(x)U(x)^{-1}$ is a positive operator for each $ x$. We prove a number of results generalizing standard theorems on the spectral theory of a single unitary operator $ U_0$, which correspond to the `commutative' case $ U(x)=e^{ix}U_0$. So these may be viewed as a noncommutative generalization of the spectral theory of $ U_0$. Also, for a two-parameter unitary family (for which there is no analytic perturbation theory) we prove an implicit function type theorem for the spectral data under the assumption that the family is monotone in one argument.

References [Enhancements On Off] (What's this?)

  • 1. S. Gnutzmann and U. Smilansky, Quantum graphs: Applications to quantum chaos and universal spectral statistics, Advances in Physics 55 (2006), no. 5-6, 527-625.
  • 2. Daniel Grieser, Spectra of graph neighborhoods and scattering, Proceedings of the London Mathematical Society 97 (2008), 718-752. MR 2448245 (2011c:58059)
  • 3. Tosio Kato, Perturbation theory for linear operators. 2nd ed., Grundlehren der mathematischen Wissenschaften. 132. Berlin-Heidelberg-New York: Springer-Verlag, 1976. MR 0407617 (53:11389)
  • 4. Franz Rellich, Störungstheorie der Spektralzerlegung. I. Analytische Störung der isolierten Punkteigenwerte eines beschränkten Operators, Math. Ann. 113 (1937), 600-619 (German). MR 1513109

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47A55, 47A56, 15A22

Retrieve articles in all journals with MSC (2010): 47A55, 47A56, 15A22

Additional Information

Daniel Grieser
Affiliation: Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, D-26111 Oldenburg, Germany

Keywords: Perturbation theory, spectral theory
Received by editor(s): August 2, 2011
Published electronically: July 31, 2012
Communicated by: Richard Rochberg
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society