Nonnegative unitary operators

Förster, K.-H.; Nagy, B.

doi:10.1090/S0002-9939-03-07202-2

Nonnegative unitary operators
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by K.-H. Förster and B. Nagy

Proc. Amer. Math. Soc. 132 (2004), 1181-1193

DOI: https://doi.org/10.1090/S0002-9939-03-07202-2

Published electronically: October 3, 2003

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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

Arlen Brown, A version of multiplicity theory, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 129–160. MR 0420322
R. Bru, C. Coll, S. Romero, and E. Sanchez, Positively similar linear systems, Abstract of 10th ILAS Conference, Auburn, Alabama, 2002.
Stanley S. Chang and Chi-Kwong Li, Certain isometries on $\textbf {R}^n$, Linear Algebra Appl. 165 (1992), 251–265. MR 1149758, DOI 10.1016/0024-3795(92)90241-2
Zhuorong Chen and Wen Li, A note on nonnegative normal matrices, Linear Algebra Appl. 279 (1998), no. 1-3, 281–283. MR 1637917, DOI 10.1016/S0024-3795(98)10011-3
John B. Conway, A course in functional analysis, Graduate Texts in Mathematics, vol. 96, Springer-Verlag, New York, 1985. MR 768926, DOI 10.1007/978-1-4757-3828-5
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part III: Spectral operators, Pure and Applied Mathematics, Vol. VII, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1971. With the assistance of William G. Bade and Robert G. Bartle. MR 0412888
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952, DOI 10.1007/978-1-4684-9330-6
S. K. Jain and L. E. Snyder, Nonnegative normal matrices, Linear Algebra Appl. 182 (1993), 147–155. MR 1207080, DOI 10.1016/0024-3795(93)90497-C
D. König, Theorie der endlichen und unendlichen Graphen, Akademische Verlagsgesellschaft, Leipzig, 1936.
K. Kuratowski and A. Mostowski, Set theory, PWN—Polish Scientific Publishers, Warsaw; North-Holland Publishing Co., Amsterdam, 1968. Translated from the Polish by M. Maczyński. MR 0229526
Zhongshan Li, Frank Hall, and Fuzhen Zhang, Sign patterns of nonnegative normal matrices, Proceedings of the Fifth Conference of the International Linear Algebra Society (Atlanta, GA, 1995), 1997, pp. 335–354. MR 1436685, DOI 10.1016/S0024-3795(96)00469-7
C. K. Li and W. So, Isometries of $l_p$-norm, Amer. Math. Monthly 101 (1994), 452-453.
Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 92, Springer-Verlag, Berlin-New York, 1977. Sequence spaces. MR 0500056, DOI 10.1007/978-3-642-66557-8
Oystein Ore, Theory of graphs, American Mathematical Society Colloquium Publications, Vol. XXXVIII, American Mathematical Society, Providence, R.I., 1962. MR 0150753
Abram I. Plesner, Spectral theory of linear operators. Vols. I, II, Frederick Ungar Publishing Co., New York, 1969. Translated from the Russian by Merlynd K. Nestell and Alan G. Gibbs. MR 0244792
P. Hebroni, Sur les inverses des éléments dérivables dans un anneau abstrait, C. R. Acad. Sci. Paris 209 (1939), 285–287 (French). MR 14
Richard Sinkhorn, Power symmetric stochastic matrices, Linear Algebra Appl. 40 (1981), 225–228. MR 629619, DOI 10.1016/0024-3795(81)90152-X
Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
Cheng Wang, Dao Sheng Zheng, Guo-Liang Chen, and Shu Qin Zhao, Structures of $p$-isometric matrices and rectangular matrices with minimum $p$-norm condition number, Linear Algebra Appl. 184 (1993), 261–278. MR 1209396, DOI 10.1016/0024-3795(93)90383-Y
Bo-Ying Wang and Fuzhen Zhang, On normal matrices of zeros and ones with fixed row sum, Proceedings of the Sixth Conference of the International Linear Algebra Society (Chemnitz, 1996), 1998, pp. 617–626. MR 1628414, DOI 10.1016/S0024-3795(97)10040-4