Operator ideal norms on 𝐿^{𝑝}

Rodríguez-Piazza, L.; Romero-Moreno, M.

doi:10.1090/S0002-9947-99-02196-0

Operator ideal norms on $L^p$
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by L. Rodríguez-Piazza and M. C. Romero-Moreno PDF

Trans. Amer. Math. Soc. 352 (2000), 379-395 Request permission

Abstract:

Let $p$ be a real number such that $p \in (1,+\infty )$ and its conjugate exponent $q\not =4,6,8\ldots$. We prove that for an operator $T$ defined on $L^{p}(\lambda )$ with values in a Banach space, the image of the unit ball determines whether $T$ belongs to any operator ideal and its operator ideal norm. We also show that this result fails to be true in the remaining cases of $p$. Finally we prove that when the result holds in finite dimension, the map which associates to the image of the unit ball the operator ideal norm is continuous with respect to the Hausdorff metric.

References

R. Anantharaman and J. Diestel, Sequences in the range of a vector measure, Comment. Math. Prace Mat. 30 (1991), no. 2, 221–235. MR 1122692
Ethan D. Bolker, A class of convex bodies, Trans. Amer. Math. Soc. 145 (1969), 323–345. MR 256265, DOI 10.1090/S0002-9947-1969-0256265-X
Joe Diestel, Hans Jarchow, and Andrew Tonge, Absolutely summing operators, Cambridge Studies in Advanced Mathematics, vol. 43, Cambridge University Press, Cambridge, 1995. MR 1342297, DOI 10.1017/CBO9780511526138
J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Marek Kanter, The $L^{p}$ norm of sums of translates of a function, Trans. Amer. Math. Soc. 79 (1973), 35–47. MR 361617, DOI 10.1090/S0002-9947-1973-0361617-4
Werner Linde, Moments of measures on Banach spaces, Math. Ann. 258 (1981/82), no. 3, 277–287. MR 649200, DOI 10.1007/BF01450683
Wolfgang Lusky, Some consequences of W. Rudin’s paper: “$L_{p}$-isometries and equimeasurability” [Indiana Univ. Math. J. 25 (1976), no. 3, 215–228; MR 53 #14105], Indiana Univ. Math. J. 27 (1978), no. 5, 859–866. MR 503719, DOI 10.1512/iumj.1978.27.27057
Abraham Neyman, Representation of $L_p$-norms and isometric embedding in $L_p$-spaces, Israel J. Math. 48 (1984), no. 2-3, 129–138. MR 770695, DOI 10.1007/BF02761158
A. Pełczyński, $p$-integral operators commuting with group representations and examples of quasi-$p$-integral operators which are not $p$-integral, Studia Math. 33 (1969), 63–70. MR 244811, DOI 10.4064/sm-33-1-63-70
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
Luis Rodríguez-Piazza, The range of a vector measure determines its total variation, Proc. Amer. Math. Soc. 111 (1991), no. 1, 205–214. MR 1025281, DOI 10.1090/S0002-9939-1991-1025281-X
L. Rodríguez-Piazza, Derivability, variation and range of a vector measure, Studia Math. 112 (1995), no. 2, 165–187. MR 1311694, DOI 10.4064/sm-112-2-165-187
L. Rodríguez-Piazza and M. C. Romero-Moreno, Conical measures and properties of a vector measure determined by its range, Studia Math. 125 (1997), no. 3, 255–270. MR 1463145
Walter Rudin, Projections on invariant subspaces, Proc. Amer. Math. Soc. 13 (1962), 429–432. MR 138012, DOI 10.1090/S0002-9939-1962-0138012-4
Walter Rudin, $L^{p}$-isometries and equimeasurability, Indiana Univ. Math. J. 25 (1976), no. 3, 215–228. MR 410355, DOI 10.1512/iumj.1976.25.25018