Collectively compact Hankel operator sequences on Hardy spaces (Q1378741)

The authors discuss two different definitions of the collectively compact operator sequences. Let \(H_1\), \(H_2\) be Hilbert spaces, and \(\{T_n\}\) be a sequence of uniformly bounded operators from \(H_1\) to \(H_2\). Definition 1.1. Set \( \varepsilon_{\infty}(\{T_n\})= \lim_{N \rightarrow \infty} \inf \{\varepsilon > 0 \mid \bigcup_{n=N}^{\infty} T_nB\) has a finite \(\varepsilon\)-net\}, where \(B\) is the unit ball of \(H_1\). Then \(\{T_n\}\) is said to be collectively compact if \(\varepsilon_{\infty}(\{T_n\})=0\). Definition 1.2. Set \( \eta_{\infty}(\{T_n\})= \lim_{N \rightarrow \infty} \inf \{\varepsilon > 0 \mid\) for every sequence \(\{x_k\} \in B\) which converges weakly to \(0\), there is \(K\) such that \(\|T_nx_k\|_{H_2} < \varepsilon\) when \(n>N\) and \(k>K \}\). Then \(\{T_n\}\) is said to be collectively compact if \(\eta_{\infty}(\{T_n\})=0\). It is shown that in general these definitions are not equivalent. But for the sequences of Hankel operators these definitions are equivalent, and the authors give the criterium for a sequence of Hankel operators to be collectively compact. This result extends the Hartman compactness characterization of a single Hankel operator.