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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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