Invariant subspaces for Bishop operators and beyond☆
Introduction
Perhaps, one of the best-known unsolved problems in Functional Analysis is the Invariant Subspace Problem:
Does every bounded linear operator on a (separable, infinite-dimensional, complex) Hilbert space have a non-trivial closed invariant subspace?
In this regard, one of the earliest and most elegant invariant subspace theorems is the result of von Neumann in the Hilbert space setting (unpublished) and Aronszajn and Smith [3] in the context of Banach spaces which states, in particular, that compact operators have non-trivial closed invariant subspaces. In 1973 operator theorists were stunned by the generalization achieved by Lomonosov [20], who proved one of the most general positive results to provide invariant subspaces, namely: any linear bounded operator T acting on a Banach space commuting with a non-zero compact operator has a non-trivial closed invariant subspace. Moreover, T has a non-trivial hyperinvariant closed subspace, that is, a closed subspace which is invariant under every operator in the commutant of T. Accordingly, any linear bounded operator T has a non-trivial invariant closed subspace if it commutes with a non-scalar operator that commutes with a nonzero compact operator. But, it was not until 1980 that Hadwin, Nordgren, Radjavi, Rosenthal [16] showed the existence of an operator in the Hilbert space setting having non-trivial invariant subspaces to which Lomonosov's Theorem does not apply.
In the meantime, two remarkable counterexamples came into scene. Firstly, in 1975 Enflo announced in the Séminaire Maurey-Schwarz at the École Polytechnique in Paris the existence of a separable Banach space and a linear bounded operator T without non-trivial closed invariant subspaces; though its publication was delayed for more than ten years [13]. Then, in 1985, Read [27] constructed a bounded linear operator without non-trivial closed invariant subspaces in the well-known sequence space (see also [26] for a previous construction). Indeed, the construction carried over in [27] is the first known example of such an operator on any of the classical Banach spaces.
For decades a number of authors worked on extending these results to more general classes of operators, and significant progress has been made by developing deep tools in allied areas like Harmonic Analysis, Function Theory or finite dimension Linear Algebra in the framework of Operator Theory. Among different approaches, two have been specially fruitful in order to provide invariant subspaces for a given operator: one coming from the behavior of such operator acting on finite dimensional subspaces leading to the concept of quasitriangular operators. The other one, mostly based on function theory techniques, consist in developing an “appropriate” functional calculus which allows to produce hyperinvariant subspaces from the fact that two non-zero functions may have pointwise zero product.
Regarding the first approach, recall that a linear bounded operator T in a separable infinite dimensional Hilbert space H is said to be quasitriangular if there exists an increasing sequence of finite rank projections converging to the identity I strongly as such that Based on Aronszajn and Smith's Theorem, Halmos [17] introduced the concept of quasitriangular operators in the sixties to prove the existence of invariant subspaces. It is completely apparent that given a triangular operator in H, that is, a linear bounded operator which admits a representation as an upper triangular matrix with respect to a suitable orthonormal basis, there exists an increasing sequence of finite rank projections converging to the identity I strongly as such that Hence, the definition of quasitriangularity says, roughly speaking, that T has a sequence of “approximately invariant” finite-dimensional subspaces. Compact operators, operators with finite spectrum, decomposable operators or compact perturbations of normal operators are examples of quasitriangular operators. On the other hand, the shift operator of index one is not quasitriangular; and remarkable results due to Douglas and Pearcy [12] and Apostol, Foiaş and Voiculescu [2] yield that the Invariant Subspace Problem is reduced to be proved for quasitriangular operators (see Herrero's book [18] for more on the subject).
In what the second approach refers, Beurling algebras have played an important role in this context. The starting point was a theorem of Wermer [29] in 1952 which states that an invertible linear bounded operator T on H such that the series converges and its spectrum is not a singleton is either a multiple of the identity or has a non-trivial hyperinvariant closed subspace. A stronger variant was proved by Atzmon (see [4] and [5], for instance). The common feature is the definition of a functional calculus, particularly in [5] mapping an algebra of functions defined on the unit circle into , the Banach algebra of linear bounded operators acting on H. For more on the subject we refer to the classical monograph by Radjavi and Rosenthal [25] and the recent one by Chalendar and Partington [9].
The main goal of this work is addressing both approaches in the context of Bishop operators. Given an irrational number , recall that the Bishop operator is defined on , , by where denotes the fractional part. As explained by Davie [11], these examples were suggested by Bishop as candidates for operators without non-trivial closed invariant subspaces. By means of a functional calculus approach, Davie proved the existence of non-trivial closed hyperinvariant subspaces in for whenever α is a non-Liouville irrational number in . Later, subsequent extensions strengthening it due to Blecher and Davie [7], MacDonald [21], [22] and Flattot [14] provided a large class of irrationals including some Liouville numbers.
Our main results in this context will be showing, on one hand, that every Bishop operator as well as its adjoint are quasitriangular operators in , having therefore a good approximation by approximately invariant finite-dimensional subspaces. On the other hand, in Theorem 3.7 we will extend the class of irrationals such that has non-trivial closed hyperinvariant subspaces in by considering arithmetical techniques which allow to strengthen the analysis of the behavior of certain functions associated to the functional calculus model. Indeed, those Liouville irrationals α escaping the condition set up in Theorem 3.7 are so extreme that Theorem 4.1 will show that, essentially, Atzmon's Theorem cannot be applied for such irrationals. Roughly speaking, we prove that when our approach fails to produce invariant subspaces it is actually because Atzmon's Theorem cannot be applied, what establishes, somehow, the threshold limit in the growth of the denominators of the convergents of those α. In some sense, this corroborates an approach to look for invariant subspaces for every based on different functional analytic tools; which will be the goal in the final section.
On the other hand, observe that by Jarník-Besicovitch Theorem (see [8, Section 5.5], for instance), Liouville irrationals form a set of vanishing Hausdorff dimension. Nevertheless, it is possible to measure the difference between those cases covered by Davie and Flattot Theorems and Theorem 3.7, by considering the logarithmic Hausdorff dimension through the use of the family of functions (instead of the usual ). With such a dimension, by means of [8, Theorem 6.8], one can easily deduce that the set of exceptions in Davie, Flattot and our case have dimension and 2, respectively.
The rest of the manuscript is organized as follows. In Section 2 we introduce some preliminaries and prove that every Bishop operator is biquasitriangular in . In Section 3, we recall the functional calculus provided by Davie and its extension by Atzmon (a good reference for that is [9, Chapter 5]); and construct explicit functions in which allow to extend the class of Liouville numbers such that has non-trivial closed hyperinvariant subspaces. In Section 4, we show the limits of Atzmon's Theorem approach in the context of Bishop operators. Finally, in Section 5 we discuss some consequences regarding spectral subspaces, which constitute a class of invariant linear manifolds to look for non-trivial closed hyperinvariant subspaces. We will show, in particular, that does not satisfy Dunford's property (C) in by exhibiting that some spectral subspaces are not closed.
A word about notation. In this paper we employ a form of Vinogradov's notation. We write meaning for some absolute constant . Note that in particular we have .
Section snippets
Quasitriangular Bishop operators
As mentioned in the introduction, a linear bounded operator T in a separable infinite dimensional Hilbert space H is quasitriangular if there exists an increasing sequence of finite rank projections converging to the identity I strongly as such that In this Section, we show that every Bishop operator is indeed biquasitriangular in , that is, both and its adjoint are quasitriangular operators. We will derive some consequences regarding the
Bishop operators with non-trivial invariant subspaces: enlarging the class of irrationals α
In this section, we extend the set of known values of α for which the Bishop operator acting on , , has non-trivial closed invariant subspaces (observe that for , the existence follows since is not separable).
The main goal of this section will be providing a careful approach to those irrationals in order to apply Atzmon's Theorem [5], by means of a functional calculus based on Beurling algebras, that is, algebras of continuous functions on the unit circle with a
The limits of Atzmon's Theorem
In this section we shall show that it is not possible to improve much on Theorem 3.7 by applying Atzmon's Theorem (Theorem 3.4) to . Before stating the main result of the section, observe that if denotes the space of (classes of) measurable functions defined almost everywhere on [0,1), is a bijection in with inverse: Nevertheless, in , , the operator is an injective, dense range operator. Hence, there exists a dense set
Spectral subspaces of Bishop operators
In this section, we deal with local spectral subspaces of Bishop operators, which are hyperinvariant subspaces (not necessarily closed) associated to closed subsets of the spectrum. While local spectral subspaces are closed for a large class of operators, those satisfying the so-called Dunford's property (C), as a consequence of the estimates obtained in the previous section, our main result in this section is that all Bishop operators do not belong to such a class; and therefore there exist
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A study of Bishop operators from the point of view of linear dynamics
2023, Journal of Mathematical Analysis and ApplicationsOperator Theory by Example
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2022, Proceedings of the American Mathematical SocietySpectral Decompositions Arising from Atzmon’s Hyperinvariant Subspace Theorem
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F. Chamizo and A. Ubis are partially supported by Ministerio de Ciencia e Innovación (Plan Nacional I+D) grant no. MTM2017-83496-P (Spain); E.A. Gallardo-Gutiérrez and M. Monsalve-López are partially supported by Ministerio de Ciencia e Innovación (Plan Nacional I+D) grants no. MTM2016-77710-P and PID2019-105979GB-I00 (Spain). The first three authors are also partially supported by “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554) and from the Spanish National Research Council, through the “Ayuda extraordinaria a Centros de Excelencia Severo Ochoa” (20205CEX001). Finally, M. Monsalve-López also acknowledges support of the grant Ayudas de la Universidad Complutense de Madrid para contratos predoctorales de personal investigador en formación, ref. no. CT27/16.