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Interior compact subspaces and differentiation in model subspaces. (Russian, English) Zbl 1096.47506
Zap. Nauchn. Semin. POMI 327, 17-24 (2005); translation in J. Math. Sci., New York 139, No. 2, 6369-6373 (2006).
Let \(\lambda_n\) be a sequence of complex numbers with positive imaginary parts, and assign to them some multiplicities \(m_n \in \mathbb N\). Denote by \(E\) the subspace of \(L_2(0, +\infty)\) spanned by the functions \(t^{k-1}\exp(-i\bar{\lambda}_nt)\), \(n \in \mathbb N\), \(1\leq k \leq m_n\). By \(K\) denote the subspace of \(L_2(-\infty, +\infty)\) spanned by the functions \((t-\bar{\lambda}_n)^{-k}\), \(n \in \mathbb N\), \(1\leq k \leq m_n\). The main result of the paper under review says that the compactness of the differentiation operator \(D:K \to L_2(-\infty, +\infty)\) is equivalent to the compactness of the shift operators in \(E\).
MSC:
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
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