Metric entropy conditions for an operator to be of trace class
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- by José M. González-Barrios and Richard M. Dudley
- Proc. Amer. Math. Soc. 118 (1993), 175-180
- DOI: https://doi.org/10.1090/S0002-9939-1993-1145418-3
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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
- Bernd Carl, Entropy numbers of diagonal operators with an application to eigenvalue problems, J. Approx. Theory 32 (1981), no. 2, 135–150. MR 633698, DOI 10.1016/0021-9045(81)90110-6
- Bernd Carl and Irmtraud Stephani, Entropy, compactness and the approximation of operators, Cambridge Tracts in Mathematics, vol. 98, Cambridge University Press, Cambridge, 1990. MR 1098497, DOI 10.1017/CBO9780511897467
- 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, DOI 10.1090/surv/015
- R. M. Dudley, The sizes of compact subsets of Hilbert space and continuity of Gaussian processes, J. Functional Analysis 1 (1967), 290–330. MR 0220340, DOI 10.1016/0022-1236(67)90017-1
- R. M. Dudley, Sample functions of the Gaussian process, Ann. Probability 1 (1973), no. 1, 66–103. MR 346884, DOI 10.1214/aop/1176997026
- R. M. Dudley, Universal Donsker classes and metric entropy, Ann. Probab. 15 (1987), no. 4, 1306–1326. MR 905333, DOI 10.1214/aop/1176991978
- X. Fernique, Regularité des trajectoires des fonctions aléatoires gaussiennes, École d’Été de Probabilités de Saint-Flour, IV-1974, Lecture Notes in Math., Vol. 480, Springer, Berlin, 1975, pp. 1–96 (French). MR 0413238 J. M. Gonzalez-Barrios, On von Mises functionals with emphasis on trace class kernels, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 1990.
- Einar Hille and J. D. Tamarkin, On the characteristic values of linear integral equations, Acta Math. 57 (1931), no. 1, 1–76. MR 1555331, DOI 10.1007/BF02403043 A. N. Kolmogorov, A note to the papers of R. A. Minlos and V. Sazonov, Theor. Probab. Appl. 4 (1959), 221-223.
- A. N. Kolmogorov and V. M. Tihomirov, $\varepsilon$-entropy and $\varepsilon$-capacity of sets in functional space, Amer. Math. Soc. Transl. (2) 17 (1961), 277–364. MR 0124720, DOI 10.1090/trans2/017/10
- B. Ja. Levin, Distribution of zeros of entire functions, Revised edition, Translations of Mathematical Monographs, vol. 5, American Mathematical Society, Providence, R.I., 1980. Translated from the Russian by R. P. Boas, J. M. Danskin, F. M. Goodspeed, J. Korevaar, A. L. Shields and H. P. Thielman. MR 589888
- G. G. Lorentz, Metric entropy and approximation, Bull. Amer. Math. Soc. 72 (1966), 903–937. MR 203320, DOI 10.1090/S0002-9904-1966-11586-0
- M. B. Marcus, The $\varepsilon$-entropy of some compact subsets of $l^{p}$, J. Approximation Theory 10 (1974), 304–312. MR 350279, DOI 10.1016/0021-9045(74)90102-6
- R. A. Minlos, Generalized random processes and their extension to a measure, Selected Transl. Math. Statist. and Prob., Vol. 3, Amer. Math. Soc., Providence, R.I., 1963, pp. 291–313. MR 0154317
- B. S. Mitjagin, Approximate dimension and bases in nuclear spaces, Uspehi Mat. Nauk 16 (1961), no. 4 (100), 63–132 (Russian). MR 0152865
- R. Oloff, Entropieeigenschaften von Diagonaloperatoren, Math. Nachr. 86 (1978), 157–165 (German). MR 532781, DOI 10.1002/mana.19780860115
- V. Sazonov, On characteristic functionals, Teor. Veroyatnost. i Primenen. 3 (1958), 201–205 (Russian, with English summary). MR 0098423
- Robert Schatten, Norm ideals of completely continuous operators, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 27, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0119112, DOI 10.1007/978-3-642-87652-3
- Laurent Schwartz, Probabilités cylindriques et applications radonifiantes, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18 (1971), 139–286 (French). MR 320747 F. Smithies, The eigen-values and singular values of integral equations, Proc. London Math. Soc. (2) 43 (1937), 255-279.
- W. Forrest Stinespring, A sufficient condition for an integral operator to have a trace, J. Reine Angew. Math. 200 (1958), 200–207. MR 98986, DOI 10.1515/crll.1958.200.200
- V. N. Sudakov, Gauss and Cauchy measures and $\varepsilon$-entropy, Dokl. Akad. Nauk SSSR 185 (1969), 51–53 (Russian). MR 0247034
Bibliographic Information
- © Copyright 1993 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 118 (1993), 175-180
- MSC: Primary 47B10; Secondary 47G10, 60B11, 60G15, 60G17
- DOI: https://doi.org/10.1090/S0002-9939-1993-1145418-3
- MathSciNet review: 1145418