Restricted Schur multipliers and their applications
HTML articles powered by AMS MathViewer
- by Timur Oikhberg PDF
- Proc. Amer. Math. Soc. 138 (2010), 1739-1750 Request permission
Abstract:
We compute the norm of the restriction of a Schur multiplier, arising from a multiplication operator, to a coordinate subspace. This result is used to generalize Wielandt’s minimax inequality. Furthermore, we compute various $s$-numbers of an elementary Schur multiplier and determine criteria for membership of such multipliers in certain operator ideals.References
- Ali R. Amir-Moéz, Extreme properties of linear transformations and geometry in unitary spaces, Mathematics Series, Nos. 2 and 3, revised edition, Texas Tech University, Department of Mathematics, Lubbock, Texas, 1971. MR 0347865
- M. Anoussis, V. Felouzis, and I. Todorov, S-numbers of elementary operators on C*-algebras, preprint, J. Operator Theory, to appear.
- Jonathan Arazy, Some remarks on interpolation theorems and the boundness of the triangular projection in unitary matrix spaces, Integral Equations Operator Theory 1 (1978), no. 4, 453–495. MR 516764, DOI 10.1007/BF01682937
- William D. Banks and Asma Harcharras, On the norm of an idempotent Schur multiplier on the Schatten class, Proc. Amer. Math. Soc. 132 (2004), no. 7, 2121–2125. MR 2053985, DOI 10.1090/S0002-9939-04-07340-X
- Rajendra Bhatia, Matrix analysis, Graduate Texts in Mathematics, vol. 169, Springer-Verlag, New York, 1997. MR 1477662, DOI 10.1007/978-1-4612-0653-8
- B. Carl, A. Defant, and M. S. Ramanujan, On tensor stable operator ideals, Michigan Math. J. 36 (1989), no. 1, 63–75. MR 989937, DOI 10.1307/mmj/1029003882
- Kenneth R. Davidson and Allan P. Donsig, Norms of Schur multipliers, Illinois J. Math. 51 (2007), no. 3, 743–766. MR 2379721
- Andreas Defant, Mieczysław Mastyło, and Carsten Michels, Summing norms of identities between unitary ideals, Math. Z. 252 (2006), no. 4, 863–882. MR 2206631, DOI 10.1007/s00209-005-0893-7
- P. G. Dodds, T. K. Dodds, B. de Pagter, and F. A. Sukochev, Lipschitz continuity of the absolute value and Riesz projections in symmetric operator spaces, J. Funct. Anal. 148 (1997), no. 1, 28–69. MR 1461493, DOI 10.1006/jfan.1996.3055
- Ian Doust and T. A. Gillespie, Schur multiplier projections on the von Neumann-Schatten classes, J. Operator Theory 53 (2005), no. 2, 251–272. MR 2153148
- Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
- I. C. Gohberg and M. G. Kreĭn, Introduction to the theory of linear nonselfadjoint operators, Translations of Mathematical Monographs, Vol. 18, American Mathematical Society, Providence, R.I., 1969. Translated from the Russian by A. Feinstein. MR 0246142
- Asma Harcharras, Fourier analysis, Schur multipliers on $S^p$ and non-commutative $\Lambda (p)$-sets, Studia Math. 137 (1999), no. 3, 203–260. MR 1736011, DOI 10.4064/sm-137-3-203-260
- Asma Harcharras, Stefan Neuwirth, and Krzysztof Oleszkiewicz, Lacunary matrices, Indiana Univ. Math. J. 50 (2001), no. 4, 1675–1689. MR 1889075, DOI 10.1512/iumj.2001.50.1952
- Milan Hladnik, Compact Schur multipliers, Proc. Amer. Math. Soc. 128 (2000), no. 9, 2585–2591. MR 1766604, DOI 10.1090/S0002-9939-00-05708-7
- S. Kwapień and A. Pełczyński, The main triangle projection in matrix spaces and its applications, Studia Math. 34 (1970), 43–68. MR 270118, DOI 10.4064/sm-34-1-43-67
- Timur Oikhberg, Direct sums of operator spaces, J. London Math. Soc. (2) 64 (2001), no. 1, 144–160. MR 1840776, DOI 10.1017/S0024610701002174
- Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR 1976867
- Vern I. Paulsen, Stephen C. Power, and Roger R. Smith, Schur products and matrix completions, J. Funct. Anal. 85 (1989), no. 1, 151–178. MR 1005860, DOI 10.1016/0022-1236(89)90050-5
- Albrecht Pietsch, $s$-numbers of operators in Banach spaces, Studia Math. 51 (1974), 201–223. MR 361883, DOI 10.4064/sm-51-3-201-223
- Albrecht Pietsch, Weyl numbers and eigenvalues of operators in Banach spaces, Math. Ann. 247 (1980), no. 2, 149–168. MR 568205, DOI 10.1007/BF01364141
- Albrecht Pietsch, Operator ideals, North-Holland Mathematical Library, vol. 20, North-Holland Publishing Co., Amsterdam-New York, 1980. Translated from German by the author. MR 582655
- Albrecht Pietsch, Eigenvalues and $s$-numbers, Cambridge Studies in Advanced Mathematics, vol. 13, Cambridge University Press, Cambridge, 1987. MR 890520
- Gilles Pisier, Multipliers and lacunary sets in non-amenable groups, Amer. J. Math. 117 (1995), no. 2, 337–376. MR 1323679, DOI 10.2307/2374918
- Gilles Pisier, Similarity problems and completely bounded maps, Second, expanded edition, Lecture Notes in Mathematics, vol. 1618, Springer-Verlag, Berlin, 2001. Includes the solution to “The Halmos problem”. MR 1818047, DOI 10.1007/b55674
- Gilles Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003. MR 2006539, DOI 10.1017/CBO9781107360235
- D. Potapov and F. Sukochev, Operator-Lipschitz functions in Schatten-von Neumann classes, preprint, available at http://xxx.lanl.gov/abs/0904.4095.
- Barry Simon, Trace ideals and their applications, 2nd ed., Mathematical Surveys and Monographs, vol. 120, American Mathematical Society, Providence, RI, 2005. MR 2154153, DOI 10.1090/surv/120
Additional Information
- Timur Oikhberg
- Affiliation: Department of Mathematics, University of California - Irvine, Irvine, California 92697 – and – Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801
- MR Author ID: 361072
- Email: toikhber@math.uci.edu
- Received by editor(s): May 3, 2009
- Received by editor(s) in revised form: September 2, 2009
- Published electronically: January 19, 2010
- Additional Notes: The author is grateful to I. Todorov for many stimulating conversations
- Communicated by: Nigel J. Kalton
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 1739-1750
- MSC (2000): Primary 47B06, 47B10, 47B49, 47L20
- DOI: https://doi.org/10.1090/S0002-9939-10-10203-2
- MathSciNet review: 2587459