Which de Branges–Rovnyak spaces are Dirichlet spaces (and vice versa)?

Costara, Constantin; Ransford, Thomas

doi:10.1016/j.jfa.2013.08.015

Cited by 41 publications

(38 citation statements)

References 9 publications

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“…(One can also study a weaker notion of 'identified', where the spaces D ω and H(b) are allowed to carry different norms. This appears to be more subtle: see [3]. )…”

Section: Introductionmentioning

confidence: 98%

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Dirichlet Spaces with Superharmonic Weights and de Branges–Rovnyak Spaces

El-Fallah¹,

Kellay

Klaja

et al. 2015

Complex Anal. Oper. Theory

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We consider Dirichlet spaces with superharmonic weights. This class contains both the harmonic weights and the power weights. Our main result is a characterization of the Dirichlet spaces with superharmonic weights that can be identified as de Branges-Rovnyak spaces. As an application, we obtain the dilation inequalityCommunicated by Isabelle Chalendar. B T. Ransford O. El-Fallah et al.where D ω denotes the Dirichlet integral with superharmonic weight ω, and f r (z) := f (r z) is the r -dilation of the holomorphic function f .

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“…(One can also study a weaker notion of 'identified', where the spaces D ω and H(b) are allowed to carry different norms. This appears to be more subtle: see [3]. )…”

Section: Introductionmentioning

confidence: 98%

“…If there exists a weight ω with0 < ω L 1 (D) < ∞ such that H(b) = D ω ,with equality of norms, then b is nonextreme and its inner factor is exactly z. Consequently the inner factor of the associated function φ is also z.Remark This result is in stark contrast with what happens if we do not insist on equality of norms. See[3, Theorem 4.5].…”

mentioning

confidence: 99%

Dirichlet Spaces with Superharmonic Weights and de Branges–Rovnyak Spaces

El-Fallah¹,

Kellay

Klaja

et al. 2015

Complex Anal. Oper. Theory

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show abstract

“…The connection between de Branges-Rovnyak and Dirichlet spaces is exploited in [8,9]; that between de Branges-Rovnyak spaces and composition operators in [15].…”

Section: Further Readingmentioning

confidence: 99%

A short introduction to de Branges–Rovnyak spaces

Timotin¹

2015

Invariant Subspaces of the Shift Operator

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“…It is worth noting here that if ϕ is rational, then the functions a and b in the representation of ϕ are also rational (see [11]) and in such a case (b, a) is called a rational pair. Recently spaces H(b) for rational pairs have been studied in [1], [2] and [6]. In [2] the authors described also the spaces H(b r ), where b is a rational outer funtion in the closed unit ball of H ∞ and r is a positive number.…”

Section: Introductionmentioning

confidence: 99%