An infinite dimensional Schur-Horn theorem and majorization theory. (English) Zbl 1202.15035
Summary: The main result of this paper is the extension of the Schur-Horn Theorem to infinite sequences: For two nonincreasing nonsummable sequences \(\xi \) and \(\eta \) that converge to 0, there exists a positive compact operator \(A\) with eigenvalue list \(\eta \) and diagonal sequence \(\xi \) if and only if \(\sum ^n_{j=1}\xi _j \leqslant \sum ^n_{j=1}\eta _j\) for every \(n\) if and only if \(\xi =Q\eta \) for some orthostochastic matrix \(Q\). When \(\xi \) and \(\eta \) are summable, requiring additionally equality of their infinite series obtains the same conclusion, extending a theorem by Arveson and Kadison. Our proof depends on the construction and analysis of an infinite product of T-transform matrices.
MSC:
| 15B51 | Stochastic matrices |
References:
| [1] | Antezana, J.; Massey, P.; Ruiz, M.; Stojanoff, D., The Schur-Horn Theorem for operators and frames with prescribed norms and frame operator, Illinois J. Math., 51, 2, 537-560 (2007) · Zbl 1137.42008 |
| [2] | Arveson, W.; Kadison, R. V., Diagonals of self-adjoint operators, (Operator Theory, Operator Algebras, and Applications. Operator Theory, Operator Algebras, and Applications, Contemp. Math., vol. 414 (2006), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 247-263 · Zbl 1113.46064 |
| [3] | Birkhoff, G., Tres observaciones sobre el algebra lineal, Univ. Nac. Tucuman Rev. Ser. A, 5, 147-151 (1946) · Zbl 0060.07906 |
| [4] | Calkin, J. W., Two-sided ideals and congruences in the ring of bounded operators in Hilbert space, Ann. of Math., 42, 2, 839-873 (1941) · Zbl 0063.00692 |
| [5] | P. Casazza, M. Leon, Frames with a given frame operator, preprint, 2002.; P. Casazza, M. Leon, Frames with a given frame operator, preprint, 2002. |
| [6] | Dykema, K.; Figiel, T.; Weiss, G.; Wodzicki, M., The commutator structure of operator ideals, Adv. Math., 185, 1, 1-79 (2004) · Zbl 1103.47054 |
| [7] | Fan, K., On a theorem of Weyl concerning eigenvalues of linear transformations, Proc. Natl. Acad. Sci. USA, 35, 652-655 (1949) · Zbl 0041.00602 |
| [8] | Fan, K., Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA, 37, 760-766 (1951) · Zbl 0044.11502 |
| [9] | Gohberg, I. C.; Markus, A. S., Some relations between eigenvalues and matrix elements of linear operators, Mat. Sb.. Mat. Sb., Amer. Math. Soc. Transl. Ser. 2, vol. 52, 106, 481-496 (1966), pp. 201-216 (in English) · Zbl 0198.17003 |
| [10] | Halmos, P. R., A Hilbert Space Problem Book, Grad. Texts in Math., vol. 19 (1967), Springer-Verlag: D. Van Nostrand Co., Inc.: Springer-Verlag: D. Van Nostrand Co., Inc. Princeton, NJ/Toronto, ON/London, from · Zbl 0144.38704 |
| [11] | Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities (1952), Cambridge University Press · Zbl 0047.05302 |
| [12] | Horn, A., Doubly stochastic matrices and the diagonal of a rotation matrix, Amer. J. Math., 76, 620-630 (1954) · Zbl 0055.24601 |
| [13] | Horn, A.; Johnson, C. R., Topics in Matrix Analysis (1995), Cambridge University Press: Cambridge University Press Cambridge, corrected reprint of the 1991 original |
| [14] | Horn, A.; Johnson, C. R., Matrix Analysis (1996), Cambridge University Press: Cambridge University Press Cambridge, corrected reprint of the 1985 original |
| [15] | Kadison, R., The Pythagorean Theorem I: the finite case, Proc. Natl. Acad. Sci. USA, 99, 7, 4178-4184 (2002) · Zbl 1013.46049 |
| [16] | Kadison, R., The Pythagorean Theorem II: the infinite discrete case, Proc. Natl. Acad. Sci. USA, 99, 8, 5217-5222 (2002) · Zbl 1013.46050 |
| [17] | Kaftal, V.; Weiss, G., Traces, ideals, and arithmetic means, Proc. Natl. Acad. Sci. USA, 99, 11, 7356-7360 (2002) · Zbl 1015.47016 |
| [18] | Kaftal, V.; Weiss, G., Soft ideals and arithmetic mean ideals, Integral Equations Operator Theory, 58, 363-405 (2007) · Zbl 1144.47031 |
| [19] | Kaftal, V.; Weiss, G., Second order arithmetic means in operator ideals, Oper. Matrices, 1, 2, 235-256 (2007) · Zbl 1163.47016 |
| [20] | Kaftal, V.; Weiss, G., A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur-Horn majorization theorem, (Hot Topics in Operator Theory (2008), Theta), 101-135 · Zbl 1199.47166 |
| [21] | Kaftal, V.; Weiss, G., Traces on operator ideals and arithmetic means, J. Operator Theory, 63, 1, 3-46 (2010) · Zbl 1213.47039 |
| [22] | V. Kaftal, G. Weiss, The \(B(H)\) lattices, density and arithmetic mean ideals, Houston J. Math. 37 (1) (2011), in press.; V. Kaftal, G. Weiss, The \(B(H)\) lattices, density and arithmetic mean ideals, Houston J. Math. 37 (1) (2011), in press. · Zbl 1226.47033 |
| [23] | V. Kaftal, G. Weiss, Majorization and arithmetic mean ideals, preprint.; V. Kaftal, G. Weiss, Majorization and arithmetic mean ideals, preprint. · Zbl 1258.15016 |
| [24] | Kalton, N. J., Trace-class operators and commutators, J. Funct. Anal., 86, 41-74 (1989) · Zbl 0684.47017 |
| [25] | Kornelson, K.; Larson, D., Rank-one decompositions of operators and construction of frames, (Wavelets, Frames and Operator Theory. Wavelets, Frames and Operator Theory, Contemp. Math., vol. 345 (2004), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 203-214 · Zbl 1058.42023 |
| [26] | Lorenz, M. O., Methods of measuring concentration of wealth, J. Amer. Statist. Assoc., 9, 209-219 (1905) |
| [27] | Markus, A. S., The eigen- and singular values of the sum and product of linear operators, Uspekhi Mat. Nauk, 4, 118, 93-123 (1964) · Zbl 0133.07205 |
| [28] | Marshall, A. W.; Olkin, I., Inequalities: Theory of Majorization and Its Applications, Math. Sci. Eng., vol. 143 (1979), Academic Press Inc. (Harcourt Brace Jovanovich Publishers) · Zbl 0437.26007 |
| [29] | Mirsky, L., Matrices with prescribed characteristic roots and diagonal elements, J. London Math. Soc., 33, 14-24 (1958) · Zbl 0101.25303 |
| [30] | Muirhead, R. F., Some methods applicable to identities and inequalities of symmetric algebraic functions of \(n\) letters, Proc. Edinb. Math. Soc., 21, 144-157 (1903) · JFM 34.0202.01 |
| [31] | Neumann, A., An infinite-dimensional generalization of the Schur-Horn convexity theorem, J. Funct. Anal., 161, 2, 418-451 (1999) · Zbl 0926.52001 |
| [32] | Ostrowski, A. M., Sur quelques applications des fonctions convexes et concaves au sens de I. Schur, J. Math. Pures Appl. (9), 31, 253-292 (1952) · Zbl 0047.29602 |
| [33] | Schur, I., Über eine Klasse von Mittelbildungen mit Anwendungen auf der Determinantentheorie, Sitzungsber. Berliner Mat. Ges., 22, 9-29 (1923) · JFM 49.0054.01 |
| [34] | G. Weiss, Commutators and operators ideals, dissertation, University of Michigan Microfilm, 1975.; G. Weiss, Commutators and operators ideals, dissertation, University of Michigan Microfilm, 1975. |
| [35] | Weiss, G., Commutators of Hilbert-Schmidt operators. I, Integral Equations Operator Theory, 3, 4, 574-600 (1980) · Zbl 0455.47025 |
| [36] | Weiss, G., Commutators of Hilbert-Schmidt operators. II, Integral Equations Operator Theory, 9, 6, 877-892 (1986) · Zbl 0618.47030 |
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