View source for Invariant subspace problem

{{Short description|Partially unsolved problem in mathematics}}
[[File:Eigenvalue equation.svg|thumb|right|250px|The vector <math>x</math> is an [[eigenvector]] of the [[matrix (mathematics)|matrix]] <math>A</math>. Every operator on a non-trivial complex [[dimension (vector space)|finite-dimensional]] [[vector space]] has an eigenvector, solving the invariant subspace problem for these spaces.]]

In the field of [[mathematics]] known as [[functional analysis]], the '''invariant subspace problem''' is a partially unresolved problem asking whether every [[bounded operator]] on a [[complex number|complex]] [[Banach space]] sends some non-trivial [[closed set|closed]] [[linear subspace|subspace]] to itself. Many variants of the problem have been solved, by restricting the class of bounded operators considered or by specifying a particular class of Banach spaces. The problem is still [[open problem|open]] for [[separable space|separable]] [[Hilbert space]]s (in other words, each example, found so far, of an operator with no non-trivial invariant subspaces is an operator that acts on a Banach space that is not [[isomorphic]] to a separable Hilbert space).

==History==
The problem seems to have been stated in the mid-20th century after work by [[Arne Beurling|Beurling]] and [[John von Neumann|von Neumann]],<ref name="Yadav, page 292">{{Harvp|Yadav|2005|p=292}}.</ref> who found (but never published) a positive solution for the case of [[compact operator]]s. It was then posed by [[Paul Halmos]] for the case of operators <math>T</math> such that <math>T^2</math> is compact. This was resolved affirmatively, for the more general class of polynomially compact operators (operators <math>T</math> such that <math>p(T)</math> is a compact operator for a suitably chosen nonzero [[polynomial]] <math>p</math>), by [[Allen R. Bernstein]] and [[Abraham Robinson]] in 1966 (see {{slink|Non-standard analysis|Invariant subspace problem}} for a summary of the proof).

For [[Banach space]]s, the first example of an operator without an invariant subspace was constructed by [[Per Enflo]]. He proposed a [[counterexample]] to the invariant subspace problem in 1975, publishing an outline in 1976. Enflo submitted the full article in 1981 and the article's complexity and length delayed its publication to 1987.<ref name="Beauzamy 1988; Yadav">{{Harvp|Beauzamy|1988}}; {{Harvp|Yadav|2005}}.</ref> Enflo's long "manuscript had a world-wide circulation among mathematicians"<ref name="Yadav, page 292"/> and some of its ideas were described in publications besides Enflo (1976).<ref>See, for example, {{Harvp|Radjavi|Rosenthal|1982}}.</ref> Enflo's works inspired a similar construction of an operator without an invariant subspace for example by [[Bernard Beauzamy]], who acknowledged Enflo's ideas.<ref name="Beauzamy 1988; Yadav" />

In the 1990s, Enflo developed a "constructive" approach to the invariant subspace problem on Hilbert spaces.<ref>Page 401 in {{cite journal|last1=Foiaş|first1=Ciprian|last2=Jung|first2=Il Bong|last3=Ko|first3=Eungil|last4=Pearcy|first4=Carl | title=On quasinilpotent operators. III | journal=Journal of Operator Theory | volume=54|year=2005|issue=2|pages=401–414}}. Enflo's method of ("forward") "minimal vectors" is also noted in the review of this research article by Gilles Cassier in ''[[Mathematical Reviews]]'': {{MR|2186363}}</ref>

In May 2023, a preprint of Enflo appeared on arXiv,<ref>{{Cite arXiv |last=Enflo |first=Per H. |date=May 26, 2023 |title=On the invariant subspace problem in Hilbert spaces |class=math.FA |eprint=2305.15442 }}</ref> which, if correct, solves the problem for Hilbert spaces and completes the picture.

In July 2023, a second and independent preprint of Neville appeared on arXiv,<ref>{{Cite arXiv |last=Neville |first=Charles W. |date=July 21, 2023 |title=a proof of the invariant subspace conjecture for separable Hilbert spaces |class=math.FA |eprint=2307.08176 }}</ref> claiming the solution of the problem for separable Hilbert spaces.

In September 2024, a peer-reviewed article published in [[Axioms (journal)|Axioms]] by a team of four Jordanian academic researchers announced that they had solved the invariant subspace problem.<ref>{{Cite journal |last1=Khalil |first1=Roshdi |last2=Yousef |first2=Abdelrahman |last3=Alshanti |first3=Waseem Ghazi |last4=Hammad |first4=Ma’mon Abu |date=2024-09-02|title=The Invariant Subspace Problem for Separable Hilbert Spaces |journal=Axioms |language=en |volume=13 |issue=9 |pages=598 |doi=10.3390/axioms13090598 |doi-access=free |issn=2075-1680}}</ref> However, basic mistakes in the proof were pointed out.<ref>{{Cite arXiv |last=Ghatasheh|first=Ahmed|date=Nov 28, 2024|title=Refuting a recent proof of the invariant subspace problem|class=math.FA |eprint=2411.19409}}</ref><ref>See mathoverflow: [https://mathoverflow.net/questions/478156/a-claim-for-a-proof-of-the-invariant-subspace-problem].</ref>

==Precise statement==

Formally,  the '''invariant subspace problem''' for a complex [[Banach space]] <math>H</math> of [[dimension (vector space)|dimension]]&nbsp;>&nbsp;1 is the question whether every [[bounded operator|bounded linear operator]] <math>T: H \to H </math>  has a non-trivial [[closed set|closed]] [[invariant subspace|<math>T</math>-invariant subspace]]: a closed [[linear subspace]] <math>W</math> of <math>H</math>, which is different from <math>\{0\}</math> and from <math>H</math>, such that <math> T(W)\subset W </math>.

A negative answer to the problem is closely related to properties of the [[Orbit_(dynamics)| orbits]] <math>T</math>. If <math>x</math> is an element of the Banach space <math>H</math>, the orbit of <math>x</math> under the action of <math>T</math>, denoted by <math>[x]</math>, is the subspace generated by the sequence <math>\{ T^{n}(x)\,:\, n \ge 0\}</math>. This is also called the [[cyclic subspace|<math>T</math>-cyclic subspace]] generated by <math>x</math>. From the definition it follows that <math>[x]</math> is a <math>T</math>-invariant subspace. Moreover, it is the ''minimal'' <math>T</math>-invariant subspace containing <math>x</math>: if <math>W</math> is another invariant subspace containing <math>x</math>, then necessarily <math>T^n(x) \in W</math> for all <math>n \ge 0</math> (since <math>W</math> is <math>T</math>-invariant), and so <math>[x]\subset W</math>. If <math>x</math> is non-zero, then <math>[x]</math> is not equal to <math>\{0\}</math>, so its closure is either the whole space <math>H</math> (in which case <math>x</math> is said to be a [[cyclic vector]] for <math>T</math>) or it is a non-trivial <math>T</math>-invariant subspace. Therefore, a counterexample to the invariant subspace problem would be a Banach space <math>H</math> and a bounded operator <math>T: H \to H </math> for which every non-zero vector <math>x\in H</math> is a [[cyclic vector]] for <math>T</math>. (Where a "cyclic vector" <math>x</math> for an operator <math>T</math> on a Banach space <math>H</math> means one for which the orbit <math>[x]</math> of <math>x</math> is dense in <math>H</math>.)


===Known special cases===

While the case of the invariant subspace problem for separable Hilbert spaces is still open, several other cases have been settled for [[topological vector space]]s (over the [[field (mathematics)|field]] of complex numbers):
*For finite-dimensional complex vector spaces, every operator admits an eigenvector, so it has a 1-dimensional invariant subspace.
* The conjecture is true if the Hilbert space <math>H</math> is not [[separable space|separable]] (i.e. if it has an [[uncountable]] [[orthonormal basis]]). In fact, if <math>x</math> is a non-zero vector in <math>H</math>, the norm closure of the linear orbit <math>[x]</math> is separable (by construction) and hence a proper subspace and also invariant.
*von Neumann showed<ref>Von Neumann's proof was never published, as relayed in a private communication to the authors of {{harvtxt|Aronszajn|Smith|1954}}. A version of that proof, independently discovered by Aronszajn, is included at the end of that paper.</ref> that any compact operator on a Hilbert space of dimension at least 2 has a non-trivial invariant subspace.
* The [[spectral theorem]] shows that all [[normal operator]]s admit invariant subspaces.
* {{harvtxt|Aronszajn|Smith|1954}} proved that every [[compact operator]] on any Banach space of dimension at least 2 has an invariant subspace.
* {{harvtxt|Bernstein|Robinson|1966}} proved using [[non-standard analysis]] that if  the operator <math>T</math> on a Hilbert space is polynomially compact (in other words <math>p(T)</math> is compact for some nonzero polynomial <math>p</math>) then <math>T</math> has an invariant subspace. Their proof uses the original idea of embedding the infinite-dimensional Hilbert space in a [[hyperfinite set|hyperfinite]]-dimensional Hilbert space (see [[Non-standard analysis#Invariant subspace problem]]).
* {{harvtxt|Halmos|1966}}, after having seen Robinson's preprint, eliminated the non-standard analysis from it and provided a shorter proof in the same issue of the same journal.
* {{harvtxt|Lomonosov|1973}} gave a very short proof using the [[Schauder fixed point theorem]] that if the operator <math>T</math> on a Banach space commutes with a non-zero compact operator then <math>T</math> has a non-trivial invariant subspace.  This includes the case of polynomially compact operators because an operator commutes with any polynomial in itself. More generally, he showed that if <math>S</math> commutes with a non-scalar operator <math>T</math> that commutes with a non-zero compact operator, then <math>S</math> has an invariant subspace.<ref>See {{harvtxt|Pearcy|Shields|1974}} for a review.</ref>
*The first example of an operator on a Banach space with no non-trivial invariant subspaces was found by  {{harvs|txt|authorlink=Per Enflo|first=Per|last=Enflo|year1=1976|year2=1987}}, and his example was simplified by {{harvtxt|Beauzamy|1985}}.
*The first counterexample on a "classical" Banach space was found by {{harvs|txt|authorlink=Charles Read (mathematician)|first=Charles|last= Read|year1=1984|year2=1985}}, who described an operator on the classical Banach space <math>l_1</math> with no invariant subspaces.
*Later {{harvs|txt|authorlink=Charles Read (mathematician)|first=Charles|last=Read|year=1988}} constructed an operator on <math>l_1</math> without even a non-trivial closed invariant ''subset'', that is that for every vector <math>x</math> the ''set'' <math>\{ T^{n}(x)\,:\, n \ge 0\}</math> is dense, in which case the vector is called [[hypercyclic vector|hypercyclic]] (the difference with the case of cyclic vectors is that we are not taking the subspace generated by the points <math>\{ T^{n}(x)\,:\, n \ge 0\}</math> in this case).
*{{harvtxt|Atzmon|1983}} gave an example of an operator without invariant subspaces on a [[nuclear space|nuclear]] [[Fréchet space]].
*{{harvtxt|Śliwa|2008}} proved that any infinite-dimensional Banach space of countable type over a [[non-Archimedean field]] admits a bounded linear operator without a non-trivial closed invariant subspace. This completely solves the non-Archimedean version of this problem, posed by van Rooij and Schikhof in 1992.
*{{harvtxt|Argyros|Haydon|2011}} gave the construction of an infinite-dimensional Banach space such that every continuous operator is the sum of a compact operator and a scalar operator, so in particular every operator has an invariant subspace.

==Notes==
{{Reflist|2}}

==References==

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{{Functional analysis}}

[[Category:Invariant subspaces]] <!-- Specific -->
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[[Category:Unsolved problems in mathematics]]
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