In the field of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 known as functional analysis

Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...

, the invariant subspace problem for a complex Banach space

Banach space

In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...

 H of dimension

Dimension (vector space)

In mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...

 > 1 is the question whether
every bounded linear operator

Bounded operator

In functional analysis, a branch of mathematics, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L to that of v is bounded by the same number, over all non-zero vectors v in X...

 T : H → H has a non-trivial closed

Closed set

In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...

 T-invariant subspace

Invariant subspace

In mathematics, an invariant subspace of a linear mappingfrom some vector space V to itself is a subspace W of V such that T is contained in W...

 (a closed linear subspace

Linear subspace

The concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....

 W of H which is different from {0} and H such that T(W) ⊆ W).

To find a "counterexample" to the invariant subspace problem, means to answer affirmatively the following equivalent question: does there exist a bounded linear operator T : H → H such that for every non-zero vector x, the vector space generated by the sequence {T ⁿ(x) : n ≥ 0} is norm dense

Dense set

In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...

 in H? Such operators are called cyclic.

For the most important case of Hilbert spaces H this is still open (as of 2010), though Per Enflo

Per Enflo

Per H. Enflo is a mathematician who has solved fundamental problems in functional analysis. Three of these problems had been open for more than forty years:* The basis problem and the approximation problem and later...

 showed that the invariant subspace problem is false for some Banach spaces.

Known special cases

While the general case of the invariant subspace problem is still open, several special cases have been settled for topological vector spaces (over the field of complex numbers):

• For non-zero finite-dimensional vector spaces every operator has an eigenvector, so has a 1-dimensional invariant subspace.
• The conjecture is true if the Hilbert space H is not separable (i.e. if it has an uncountable orthonormal basis
  Orthonormal basis
  In mathematics, particularly linear algebra, an orthonormal basis for inner product space V with finite dimension is a basis for V whose vectors are orthonormal. For example, the standard basis for a Euclidean space Rn is an orthonormal basis, where the relevant inner product is the dot product of...
  
  ). In fact, if x is a non-zero vector in H, the norm closure of the vector space generated by the infinite sequence {T ⁿ(x) : n ≥ 0} is separable and hence a proper subspace and also invariant.
• von Neumann showed that any compact operator on a Hilbert space of dimension at least 2 has a non-trivial invariant subspace.
• The spectral theorem
  Spectral theorem
  In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
  
   shows that all normal operator
  Normal operator
  In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operatorN:H\to Hthat commutes with its hermitian adjoint N*: N\,N^*=N^*N....
  
  s admit invariant subspaces. proved that every compact operator
  Compact operator
  In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...
  
   on any Banach space of dimension at least 2 has an invariant subspace. proved using non-standard analysis
  Non-standard analysis
  Non-standard analysis is a branch of mathematics that formulates analysis using a rigorous notion of an infinitesimal number.Non-standard analysis was introduced in the early 1960s by the mathematician Abraham Robinson. He wrote:...
  
   that if the operator T on a Hilbert space is polynomially compact (in other words P(T) is compact for some non-zero polynomial P) then T has an invariant subspace. Their proof uses the original idea of embedding the infinite-dimensional Hilbert space in a hyperfinite-dimensional Hilbert space., 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. gave a very short proof using the Schauder fixed point theorem
  Schauder fixed point theorem
  The Schauder fixed point theorem is an extension of the Brouwer fixed point theorem to topological vector spaces, which may be of infinite dimension...
  
   that if the operator T on a Banach space commutes with a non-zero compact operator then T 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 S commutes with a non-scalar operator T that commutes with a non-zero compact operator, then S has an invariant subspace.
• The first example of an operator on a Banach space with no invariant subspaces was found by , and his example was simplified by .
• The first counterexample on a "classical" Banach space was found by , who described an operator on the classical Banach space l¹ with no invariant subspaces.
• Later constructed an operator on l₁ even without a non-trivial closed invariant subset, that is, with every vector hypercyclic, solving in the negative the invariant subset problem for the class of Banach spaces. gave an example of an operator without invariant subspaces on a nuclear
  Nuclear space
  In mathematics, a nuclear space is a topological vector space with many of the good properties of finite-dimensional vector spaces. The topology on them can be defined by a family of seminorms whose unit balls decrease rapidly in size...
  
   Fréchet space
  Fréchet space
  In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces...
  
  . 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 Shikhof in 1992. announced 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.