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Invariant Subspaces for Generalized Differentiation and Volterra Operators

Aleman, Alexandru LU and Bergman, Alex LU orcid (2025)
Abstract
In this paper we provide a far-reaching generalization of the existent results about invariant subspaces of the differentiation operator D=∂t on C^{\infty}(0,1) and the Volterra operator Vf(t)=∫f(s)ds, on L^{2}(0,1). We use an abstract approach to study invariant subspaces of pairs D,V with DV=I, where V is compact and quasi-nilpotent and D is unbounded densely defined and closed on the same Hilbert space. Our results cover many differential operators, like Schrödinger operators and a large class of other canonical systems, as well as the so-called compact self-adjoint operators with removable spectrum recently studied by Baranov and Yakubovich. Our methods are based on a model for such pairs which involves de Branges spaces of entire... (More)
In this paper we provide a far-reaching generalization of the existent results about invariant subspaces of the differentiation operator D=∂t on C^{\infty}(0,1) and the Volterra operator Vf(t)=∫f(s)ds, on L^{2}(0,1). We use an abstract approach to study invariant subspaces of pairs D,V with DV=I, where V is compact and quasi-nilpotent and D is unbounded densely defined and closed on the same Hilbert space. Our results cover many differential operators, like Schrödinger operators and a large class of other canonical systems, as well as the so-called compact self-adjoint operators with removable spectrum recently studied by Baranov and Yakubovich. Our methods are based on a model for such pairs which involves de Branges spaces of entire functions and plays a crucial role in the development. However, a number of difficulties arise from the fact that our abstract operators do not necessarily identify with the usual operators on such spaces, but with rank one perturbations of those, which, in terms of invariant subspaces creates a number of challenging problems. (Less)
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author
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publishing date
type
Working paper/Preprint
publication status
published
subject
publisher
arXiv.org
DOI
10.48550/arXiv.2503.08443
language
English
LU publication?
yes
id
94cdcb2b-64a1-4f69-af9f-da629daff57e
date added to LUP
2025-03-12 08:46:07
date last changed
2025-04-21 09:56:06
@misc{94cdcb2b-64a1-4f69-af9f-da629daff57e,
  abstract     = {{In this paper we provide a far-reaching generalization of the existent results about invariant subspaces of the differentiation operator D=∂t on C^{\infty}(0,1) and the Volterra operator Vf(t)=∫f(s)ds, on L^{2}(0,1). We use an abstract approach to study invariant subspaces of pairs D,V with DV=I, where V is compact and quasi-nilpotent and D is unbounded densely defined and closed on the same Hilbert space. Our results cover many differential operators, like Schrödinger operators and a large class of other canonical systems, as well as the so-called compact self-adjoint operators with removable spectrum recently studied by Baranov and Yakubovich. Our methods are based on a model for such pairs which involves de Branges spaces of entire functions and plays a crucial role in the development. However, a number of difficulties arise from the fact that our abstract operators do not necessarily identify with the usual operators on such spaces, but with rank one perturbations of those, which, in terms of invariant subspaces creates a number of challenging problems.}},
  author       = {{Aleman, Alexandru and Bergman, Alex}},
  language     = {{eng}},
  month        = {{03}},
  note         = {{Preprint}},
  publisher    = {{arXiv.org}},
  title        = {{Invariant Subspaces for Generalized Differentiation and Volterra Operators}},
  url          = {{http://dx.doi.org/10.48550/arXiv.2503.08443}},
  doi          = {{10.48550/arXiv.2503.08443}},
  year         = {{2025}},
}