Abstract
It is shown that thep-summing norm of any operator withn-dimensional domain can be well-aproximated using only “few” vectors in the definition of thep-summing norm. Except for constants independent ofn and logn factors, “few” meansn if 1<p<2 andn p/2 if 2<p<∞.
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References
J. Bourgain,A remark on the behaviour of L p -multipliers and the range of operators acting on L p-spaces, Israel J. Math.79 (1992), 1–11.
J. Bourgain, J. Lindenstrauss and V. D. Milman,Approximation of zonoids by zonotopes, Acta Math.162 (1989), 73–141.
M. Defant and M. Junge,Absolutely summing norms with n vectors, preprint.
M. Defant and M. Junge,On absolutely summing operators with application to the (p, q)-summing norm with few vectors, J. Functional Analysis103 (1992), 62–73.
M. Defant and M. Junge,How many vectors are needed to compute (p, q)-summing norms?, preprint.
R. M. Dudley,The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis1 (1967), 290–330.
G. J. O. Jameson,The number of elements required to determine π 2,1, Illinois J. Math., to appear.
D. R. Lewis,Ellipsoids defined by Banach ideal norms, Mathematika26 (1979), 18–29.
M. B. Marcus and G. Pisier,Random Fourier series with applications to harmonic analysis, Annals of Math. Studies, Vol. 101, Princeton University Press, New Jersey, 1981.
A. Pajor and N. Tomczak-Jaegermann,Subspaces of small codimension of finite-dimensional Banach spaces, Proc. Amer. Math. Soc.97 (1986), 637–642.
G. Schechtman,More on embedding subspaces of L p into lr/n, Comp. Math.61 (1987), 159–170.
V. N. Sudakov,Gaussian random processes and measures of solid angles in Hilbert spaces, Soviet Math. Dokl.12 (1971), 412–415.
S. J. Szarek,Computing summing norms and type constants on few vectors, Studia Math.98 (1990), 147–156.
M. Talagrand,Embedding subspaces of L 1 into lN, Proc. Amer. Math. Soc.108 (1990), 363–369.
N. Tomczak-Jaegermann,Banach-Mazur distances and finite-dimensional operator ideals, Pitman Monographs and Surveys in Pure and Applied Mathematics38, Longman, London, 1989.
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Supported in part by NSF #DMS90-03550 and the U.S.-Israel Binational Science Foundation.
Supported in part by the U.S.-Israel Binational Science Foundation.
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Johnson, W.B., Schechtman, G. Computingp-summing norms with few vectors. Israel J. Math. 87, 19–31 (1994). https://doi.org/10.1007/BF02772980
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DOI: https://doi.org/10.1007/BF02772980