Regular Functions in de Branges-Rovnyak Spaces

Bartosz Malman ()
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Bartosz Malman: Mälardalen University, Division of Mathematics and Physics

Chapter 79 in Operator Theory, 2026, pp 2433-2460 from Springer

Abstract: Abstract In this expository chapter, we discuss the problem of the containment of a non-zero function satisfying various boundary regularity conditions in the de Branges-Rovnyak space ℋ ( b ) $$\mathcal {H}(b)$$ . We review what is known and highlight the connection of this problem to the Korenblum-Roberts cyclicity theory in Bergman-type spaces, the works of Khrushchev on simultaneous approximation by polynomials, and some aspects of the Beurling-Malliavin theory. New proofs of previously known results are given to emphasize these connections. As a new contribution, we fully characterize the symbols b for which the space ℋ ( b ) $$\mathcal {H}(b)$$ contains a non-zero function with C ∞ $$C^\infty $$ extension to the boundary. This extends an earlier result of Dyakonov and Khavinson dealing with the case of inner b. We end the chapter by stating a few open problems.

Keywords: de Branges-Rovnyak spaces; Containment of regular functions; Cyclic singular inner functions; Simultaneous approximation (search for similar items in EconPapers)
Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-16356-1_110

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DOI: 10.1007/978-3-032-16356-1_110

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