Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Metric entropy conditions for an operator to be of trace class

Authors: José M. González-Barrios and Richard M. Dudley
Journal: Proc. Amer. Math. Soc. 118 (1993), 175-180
MSC: Primary 47B10; Secondary 47G10, 60B11, 60G15, 60G17
MathSciNet review: 1145418
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ A$ be an operator from one Hilbert space $ H$ into another. It was known that $ A$ is of trace class if and only if the metric entropy of $ A(B)$ is integrable where $ B$ is the unit ball in $ H$. We give a new, general sufficient condition for an integral operator to be of trace class, and examples showing it is sharp but not necessary.

References [Enhancements On Off] (What's this?)

  • [C] B. Carl, Entropy numbers of diagonal operators with an application to eigenvalue problems, J. Approx. Theory 32 (1981), 135-150. MR 633698 (83a:47024)
  • [CS] B. Carl and I. Stephani, Entropy, compactness and the approximation of operators, Cambridge Univ. Press, London and New York, 1990. MR 1098497 (92e:47002)
  • [DiU] J. Diestel and J. J. Uhl Jr., Vector measures, Math. Surveys Monographs, vol. 15, Amer. Math. Soc., Providence, RI, 1977. MR 0453964 (56:12216)
  • [Du1] R. M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Funct. Anal. 1 (1967), 290-330. MR 0220340 (36:3405)
  • [Du2] -, Sample functions of the Gaussian process, Ann. Probab. 1 (1973), 66-103. MR 0346884 (49:11605)
  • [Du3] -, Universal Donsker classes and metric entropy, Ann. Probab. 15 (1987), 1306-1326. MR 905333 (88g:60081)
  • [F] X. Fernique, Régularité des trajectoires des fonctions aléatoires gaussiennes, Ecole d'été de Probabilités de St.-Flour IV-1974, Lecture Notes in Math., vol. 480, Springer, Berlin and New York, 1975, pp. 1-96. MR 0413238 (54:1355)
  • [G] J. M. Gonzalez-Barrios, On von Mises functionals with emphasis on trace class kernels, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 1990.
  • [HT] E. Hille and J. D. Tamarkin, On the characteristic values of linear integral equations, Acta Math. 57 (1931), 1-76. MR 1555331
  • [K] A. N. Kolmogorov, A note to the papers of R. A. Minlos and V. Sazonov, Theor. Probab. Appl. 4 (1959), 221-223.
  • [KT] A. N. Kolmogorov and V. M. Tikhomirov, $ \varepsilon $-entropy and $ \varepsilon $-capacity of sets in function spaces, Uspekhi Mat. Nauk 14 (1959), no. 2(86), 1-86; English transl. in Amer. Math. Soc. Transl. 17 (1961), 277-364. MR 0124720 (23:A2031)
  • [Le] B. Ya. Levin, Distribution of zeros of entire functions, rev. ed., Transl. Math. Monographs, vol. 5, Amer. Math. Soc., Providence, RI, 1980. MR 589888 (81k:30011)
  • [Lo] G. G. Lorentz, Metric entropy and approximation, Bull. Amer. Math. Soc. 72 (1966), 903-937. MR 0203320 (34:3173)
  • [Ma] M. B. Marcus, The $ \varepsilon $-entropy of some compact subsets of $ {l^p}$, J. Approx. Theory 10 (1974), 304-312. MR 0350279 (50:2772)
  • [Min] R. A. Minlos, Generalized random processes and their extension to a measure, Trudy Mosk. Mat. Obshch. 8 (1959), 497-518; English transl., Selected Transl. Math. Statist. and Probab., vol. 3, Amer. Math. Soc., Providence, RI, 1962, pp. 291-313. MR 0154317 (27:4266)
  • [Mit] B. S. Mityagin, Approximate dimension and bases in nuclear spaces, Russian Math. Surveys 16 (1961), no. 4, 59-127. MR 0152865 (27:2837)
  • [Ol] R. Oloff, Entropieeigenschaften von Diagonaloperatoren, Math. Nachr. 86 (1978), 157-165. MR 532781 (80i:47035)
  • [Sa] V. V. Sazonov, A remark on characteristic functionals, Theor. Probab. Appl. 3 (1958), 188-192. MR 0098423 (20:4882)
  • [Scha] R. Schatten, Norm ideals of completely continuous operators, Springer-Verlag, Berlin, 1960. MR 0119112 (22:9878)
  • [Schw] Laurent Schwartz, Probabilités cylindriques et applications radonifiantes, J. Fac Sci. Univ. Tokyo Sect. 1A Math. 18 (1971-72), 139-286. MR 0320747 (47:9281)
  • [Sm] F. Smithies, The eigen-values and singular values of integral equations, Proc. London Math. Soc. (2) 43 (1937), 255-279.
  • [St] W. F. Stinespring, A sufficient condition for an integral operator to have a trace, J. Reine Angew. Math. 200 (1958), 200-207. MR 0098986 (20:5431)
  • [Su] V. N. Sudakov, Gaussian measures, Cauchy measures and $ \varepsilon $-entropy, Dokl. Akad. Nauk SSSR 185 (1969), 51-53; English transl. in Soviet Math. Dokl. 10 (1969), 310-313. MR 0247034 (40:303)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47B10, 47G10, 60B11, 60G15, 60G17

Retrieve articles in all journals with MSC: 47B10, 47G10, 60B11, 60G15, 60G17

Additional Information

Keywords: Metric entropy, ellipsoids, trace class, Hilbert-Schmidt operators
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society