Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Nonnegative unitary operators

Author(s): K.-H. Förster; B. Nagy
Journal: Proc. Amer. Math. Soc. 132 (2004), 1181-1193.
MSC (2000): Primary 47B15, 47B65
Posted: October 3, 2003
Retrieve article in: PDF

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:

[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: 10.1090/S0002-9939-03-07202-2
PII: S 0002-9939(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
Posted: October 3, 2003
Additional Notes: This work was supported by the Hungarian National Scientific Grant OTKA No. T-030042
Communicated by: Joseph A. Ball
Copyright of article: Copyright 2003, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google