Geometric entropy in Banach spaces

Andrey Dorogovtsev ; Mikhail Popov

Teoriâ slučajnyh processov, Tome 19 (2014) no. 2, pp. 10-30.

Résumé

We introduce and study two notions of entropy in a Banach space $X$ with a normalized Schauder basis $\mathcal B = (e_n)$. The geometric entropy $\mathbf{E}(A)$ of a subset $A$ of $X$ is defined to be the infimum of radii of compact bricks containing $A$, where a brick $K_{\mathcal B, \mathcal E}$ is the set of all sums of convergent series $\sum a_n e_n$ with $|a_n| \leq \varepsilon_n$, $\mathcal E = (\varepsilon_n)$, $\varepsilon_n \geq 0$. The unconditional entropy $\mathbf{E}_0(A)$ is defined similarly, with respect to $1$-unconditional bases of $X$. We obtain several compactness characterizations for bricks (Theorem 3.7) useful for main results. If $X = c_0$ then the two entropies of a set coincide, and equal the radius of a set. However, for $X = \ell_2$ the entropies are distinct. The unconditional entropy of the image $T(B_H)$ of the unit ball of a separable Hilbert space $H$ under an operator $T$ is finite if and only if $T$ is a Hilbert-Schmidt operator, and moreover, $\mathbf{E}_0 \bigl(T(B_H)\bigr) = \|T\|_{HS}$, the Hilbert-Schmidt norm of $T$. We also obtain sufficient conditions on a set in a Hilbert space to have finite unconditional entropy. For Banach spaces without a Schauder basis we offer another entropy, called the Auerbach entropy. Finally, we pose some open problems.

Détail

Keywords: Geometric entropy in Banach spaces, distributions in Banach spaces, precompact sets, compact bricks
Mots-clés : Schauder bases.

Bibliographie

[1] F. Albiac, N. Kalton, Topics in Banach Space Theory, Graduate Texts in Math., 233, Springer, New York, 2006 | MR

[2] A. V. Balakrishnan, Applied Functional Analysis, Springer–Verlag, Berlin–Heidelberg–New York, 1976 | MR | Zbl

[3] S. M. Berman, “Local nondeterminism and local times of Gaussian processes”, Bull. Amer. Math. Soc., 79 (1973), 475–477 | DOI | MR | Zbl

[4] J. Bobok and H. Bruin, “The topological entropy of Banach spaces”, J. Difference Equ. Appl., 18:4 (2012), 569–578 | DOI | MR | Zbl

[5] V. I. Bogachev, Gaussian measures, Math. Surv. and Monographs, 62, 1998 | DOI | MR | Zbl

[6] B. Carl, I. Kyrezi, A. Pajor, “Metric entropy of convex hulls in Banach spaces”, J. London Math. Soc., 60:3 (1999), 871–896 | DOI | MR | Zbl

[7] Sb. Math., 201:5-6, 645–653 | DOI | MR | Zbl

[8] Ukrainian Math. J., 57:9 (2005), 1424–1441 | DOI | MR | Zbl

[9] W. B. Johnson, H. P. Rosenthal, M. Zippin, “On bases, finite-dimensional decompositions and weaker structures in Banach spaces”, Isr. J. Math., 9 (1971), 488–506 | DOI | MR | Zbl

[10] M. I. Kadets, V. M. Kadets, Series in Banach spaces. Conditional and unconditional convergence, Operator Theory Advances and Applications, 94, Birkhäuser, Basel-Boston-Berlin, 1997 | MR | Zbl

[11] V. M. Kadets, A course of Functional Analysis and Measure Theory, Chyslo (Published by I. E. Chyzhykov), Lviv, 2012 (in Ukrainian)

[12] D. Kerr, H. Li, “Dynamical entropy in Banach spaces”, Invent. Math., 162:3 (2005), 649–686 | DOI | MR | Zbl

[13] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces, v. 1, Sequence spaces, Springer–Verlag, Berlin–Heidelberg–New York, 1977 | MR | Zbl

[14] M. B. Marcus, J. Rosen, Markov processes, Gaussian processes and local times, Cambridge Univ. Press, 2006 | MR | Zbl

[15] M. Martín, J. Merí, M. Popov, “On the numerical radius of operators in Lebesgue spaces”, J. Funct. Anal., 261 (2011), 149–168 | DOI | MR | Zbl

[16] B. Simon, The $P(\phi)_2$ Euclidean (quantum) field theory, Princeton Series in Physics, Princeton University Press, Princeton, N.J., 1974 | MR

[17] I. Singer, Bases in Banach Spaces, v. II, Springer-Verlag, Berlin-Heidelberg-New York, 1981 | MR | Zbl

[18] V. N. Sudakov, “A class of compacta of a Hilbert space”, Uspekhi Mat. Nauk, 18:1 (1963), 181-187 (in Russian) | MR | Zbl

[19] N. N. Vakhaniya, V. I. Tarieladze, S. A. Chobanyan, Probability distributions in Banach spaces, Nauka, Moscow, 1985 (in Russian) | MR | Zbl