On geometric entropy in Hilbert spaces (Q2326014)

If \(X\) is a Banach space with a normalized basis \(\mathcal{B}=(e_n)\) and biorthogonal functionals \((e_n^*)\) and \(\mathcal{E}=(\varepsilon_n)\) is a sequence of nonnegative numbers, the set \[ K_{\mathcal{B},\mathcal{E}} =\{x\in X: |e_n^*(x)|\leq \varepsilon_n\text{ for all } n\in \mathbb{N}\} \] is called the brick associated with \(\mathcal{B}\) and \(\mathcal{E}\). The geometric entropy of a set \(A\) in \(X\) is defined to be \[ E(A) = \inf \{ r(K_{\mathcal{B},\mathcal{E}}) : K_{\mathcal{B},\mathcal{E}}\text{ is a compact brick containing } A\}, \] where \(r(K) = \sup \{ \|x\| : x\in K\}\) for a subset \(K\) of \(X\). The unconditional entropy \(E_0(A)\) is similarly defined using only compact bricks associated with a \(1\)-unconditional basis \(\mathcal{B}\). These notions were introduced by \textit{A. Dorogovtsev} and the second author in [Theory Stoch. Process. 19, No. 2, 10--30 (2014; Zbl 1340.46025)]. In the article under review, the authors present results illustrating how sensitive the notion of unconditional entropy can be. The authors construct a subset \(A\) of \(\ell_2\) with finite unconditional entropy and a compact operator \(T:\ell_2\to \ell_2\) such that the image \(T(A)\) has infinite unconditional entropy. A second result shows the existence of two bricks in \(\ell_2\), each with unconditional entropy \(1\), whose union has infinite unconditional entropy. For other results and questions concerning geometric and unconditional entropy, the interested reader can consult the article of Dorogovtsev and the second author cited above.