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)

 
 

 

Nonnegative unitary operators


Authors: K.-H. Förster and B. Nagy
Journal: Proc. Amer. Math. Soc. 132 (2004), 1181-1193
MSC (2000): Primary 47B15, 47B65
DOI: https://doi.org/10.1090/S0002-9939-03-07202-2
Published electronically: October 3, 2003
MathSciNet review: 2045436
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Unitary operators in Hilbert space map an orthonormal basis onto another. In this paper we study those that map an orthonormal basis onto itself. We show that a sequence of cardinal numbers is a complete set of unitary invariants for such an operator. We obtain a characterization of these operators in terms of their spectral properties. We show how much simpler the structure is in finite-dimensional space, and also describe the structure of certain isometries in Hilbert space.


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

  • [Bro] A. Brown, A version of multiplicity theory, Math. Surveys, Vol. 13, Amer. Math. Soc., Providence, RI, 1974, pp. 129-159. MR 54:8336
  • [BCRS] R. Bru, C. Coll, S. Romero, and E. Sanchez, Positively similar linear systems, Abstract of 10th ILAS Conference, Auburn, Alabama, 2002.
  • [CaL] S. Chang and C. K. Li, Certain isometries on ${\mathbf R}^n$, Linear Algebra Appl., 165 (1992), 251-265. MR 92k:15059
  • [CL] Z. Chen and W. Li, A note on nonnegative normal matrices, Linear Algebra Appl., 279 (1998), 281-283. MR 99c:15037
  • [Conw] J. B. Conway, A course in functional analysis, Graduate Texts in Mathematics, Vol. 96, Springer-Verlag, New York, 1985. MR 86h:46001
  • [DS3] N. Dunford and J. T. Schwartz, Linear operators, Part III: Spectral operators, Wiley-Interscience, New York, 1971. MR 54:1009
  • [Hal1] P. R. Halmos, Introduction to Hilbert space and the theory of spectral multiplicity, Chelsea, New York, 1951. MR 13:563a
  • [Hal2] P. R. Halmos, A Hilbert space problem book, 2nd ed., Springer-Verlag, New York, 1982. MR 84e:47001
  • [JS] S. K. Jain and L. E. Snyder, Nonnegative normal matrices, Linear Algebra Appl., 182 (1993), 147-155. MR 94c:15026
  • [Kon] D. König, Theorie der endlichen und unendlichen Graphen, Akademische Verlagsgesellschaft, Leipzig, 1936.
  • [K-M] K. Kuratowski and A. Mostowski, Set theory, North-Holland, Amsterdam, 1968. MR 37:5100
  • [LHZ] Z. Li, F. Hall and F. Zhang, Sign patterns of nonnegative normal matrices, Linear Algebra Appl., 254 (1997), 335-354. MR 97m:15041
  • [LS] C. K. Li and W. So, Isometries of $l_p$-norm, Amer. Math. Monthly 101 (1994), 452-453.
  • [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I: Sequence spaces, Springer-Verlag, Berlin, 1977. MR 58:17766
  • [O] O. Ore, Theory of graphs, Amer. Math. Soc. Colloquium Publications, Vol. 38, Providence, RI, 1962. MR 27:740
  • [Ples] A. I. Plesner, Spectral theory of linear operators, Frederick Ungar, New York, 1969. MR 39:6106
  • [R:Sz-N] F. Riesz and B. Szökefalvi-Nagy, Leçons d'analyse fonctionnelle, Akad. Kiadó, Budapest, 1952. MR 14:286d
  • [S] R. Sinkhorn, Power symmetric stochastic matrices, Linear Algebra Appl., 40 (1981), 225-228. MR 82j:15017
  • [Sz-N:F] B. Szökefalvi-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, Akad. Kiadó, Budapest, 1970. MR 43:947
  • [W] C. Wang, D. Zheng, G.-L. Chen, and S. Zhao, Structures of $p$-isometric matrices and rectangular matrices with minimum $p$-norm condition number, Linear Algebra Appl., 184 (1993), 261-278. MR 94f:65049
  • [WZ] B.-Y. Wang and F. Zhang, On normal matrices of zeros and ones with fixed row sum, Linear Algebra Appl. 275/276 (1998), 617-626. MR 99e:05028

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B15, 47B65

Retrieve articles in all journals with MSC (2000): 47B15, 47B65


Additional Information

K.-H. Förster
Affiliation: Department of Mathematics, Technical University Berlin, Sekr. MA 6-4, Straße des 17. Juni 136, D-10623 Berlin, Germany
Email: foerster@math.tu-berlin.de

B. Nagy
Affiliation: Department of Analysis, Institute of Mathematics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary
Email: bnagy@math.bme.hu

DOI: https://doi.org/10.1090/S0002-9939-03-07202-2
Keywords: Unitary operator, infinite matrix with nonnegative entries, complete set of unitary invariants, multiplicity
Received by editor(s): July 10, 2002
Received by editor(s) in revised form: December 30, 2002
Published electronically: October 3, 2003
Additional Notes: This work was supported by the Hungarian National Scientific Grant OTKA No. T-030042
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society