Geometric entropy in Banach spaces

Andrey Dorogovtsev, Mikhail Popov
Theory of Stochastic Processes
Vol.19 (35), no.2, 2014, pp.10-30
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.