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An overview of some recent developments on the Invariant Subspace Problem

100%
Chalendar I., Partington J. R.
Concrete Operators
|
2013
|
tom 1
1-10
EN
This paper presents an account of some recent approaches to the Invariant Subspace Problem. It contains a brief historical account of the problem, and some more detailed discussions of specific topics, namely, universal operators, the Bishop operators, and Read’s Banach space counter-example involving a finitely strictly singular operator.
2
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L¹ factorizations, moment problems and invariant subspaces

81%
Chalendar I., Partington J., Smith R.
Studia Mathematica
|
2005
|
tom 167
|
nr 2
183-194
EN
For an absolutely continuous contraction T on a Hilbert space 𝓗, it is shown that the factorization of various classes of L¹ functions f by vectors x and y in 𝓗, in the sense that ⟨Tⁿx,y⟩ = f̂(-n) for n ≥ 0, implies the existence of invariant subspaces for T, or in some cases for rational functions of T. One of the main tools employed is the operator-valued Poisson kernel. Finally, a link is established between L¹ factorizations and the moment sequences studied in the Atzmon-Godefroy method, from which further results on invariant subspaces are derived.
3
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Boundary values of analytic semigroups and associated norm estimates

81%
Chalendar I., Esterle J., Partington J.
Banach Center Publications
|
2010
|
tom 91
|
nr 1
87-103
EN
The theory of quasimultipliers in Banach algebras is developed in order to provide a mechanism for defining the boundary values of analytic semigroups on a sector in the complex plane. Then, some methods are presented for deriving lower estimates for operators defined in terms of quasinilpotent semigroups using techniques from the theory of complex analysis.
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