Invariant subspace

2024-09-20T13:52:12Z

2A01:4B00:B601:2100:6DE4:F5A8:BFA6:CCED: /* For a single operator */

{{Short description|Subspace preserved by a linear mapping}}
In [[mathematics]], an '''invariant subspace''' of a [[linear mapping]] ''T'' : ''V'' → ''V '' i.e. from some [[vector space]] ''V'' to itself, is a [[linear subspace|subspace]] ''W'' of ''V'' that is preserved by ''T''. More generally, an invariant subspace for a collection of linear mappings is a subspace preserved by each mapping individually.

== For a single operator ==
Consider a vector space <math>V</math> and a linear map <math>T: V \to V.</math> A subspace <math>W \subseteq V</math> is called an '''invariant subspace for''' <math>T</math>, or equivalently, {{Mvar|T}}-invariant, if {{Mvar|T}} transforms any vector <math>\mathbf{v} \in W</math> back into {{Mvar|W}}. In formulas, this can be written<math display="block">\mathbf{v} \in W \implies T(\mathbf{v}) \in W</math>or<ref>{{harvnb|Roman|2008|loc=p. 73 §2}}</ref> <math display="block">TW\subseteq W\text{.}</math>

In this case, {{Mvar|T}} [[restriction (mathematics)|restricts]] to an [[endomorphism]] of {{Mvar|W}}:<ref>{{harvnb|Roman|2008|loc=p. 73 §2}}</ref><math display="block">T|_W : W \to W\text{;}\quad T|_W(\mathbf{w}) = T(\mathbf{w})\text{.}</math>

The existence of an invariant subspace also has a [[Matrix representation|matrix formulation]]. Pick a [[basis (linear algebra)|basis]] ''C'' for ''W'' and complete it to a basis ''B'' of ''V''. With respect to {{Mvar|B}}, the operator {{Mvar|T}} has form <math display=block> T = \begin{bmatrix} T|_W & T_{12} \\ 0 & T_{22} \end{bmatrix} </math> for some {{Math|''T''<sub>12</sub>}} and {{Math|''T''<sub>22</sub>}}, where <math>T|_W</math> here denotes the matrix of <math>T|_W</math> with respect to the basis ''C''.

== Examples ==
Any linear map <math>T : V \to V</math> admits the following invariant subspaces:
* The vector space <math>V</math>, because <math>T</math> maps every vector in <math>V</math> into <math>V.</math>
* The set <math>\{0\}</math>, because <math>T(0) = 0</math>.
These are the improper and trivial invariant subspaces, respectively. Certain linear operators have no proper non-trivial invariant subspace: for instance, [[rotation (mathematics)|rotation]] of a two-dimensional [[real number|real]] vector space. However, the [[Rotation axis|axis]] of a rotation in three dimensions is always an invariant subspace.

===1-dimensional subspaces===
If {{Mvar|U}} is a 1-dimensional invariant subspace for operator {{Mvar|T}} with vector {{Math|'''v''' ∈ ''U''}}, then the vectors {{Math|'''v'''}} and {{Math|''T'''''v'''}} must be [[linearly dependent]]. Thus <math display="block"> \forall\mathbf{v}\in U\;\exists\alpha\in\mathbb{R}: T\mathbf{v}=\alpha\mathbf{v}\text{.}</math>In fact, the scalar {{Mvar|α}} does not depend on {{Math|'''v'''}}.

The equation above formulates an [[eigenvalue]] problem. Any [[eigenvector]] for {{Mvar|T}} spans a 1-dimensional invariant subspace, and vice-versa. In particular, a nonzero '''invariant vector''' (i.e. a [[Fixed point (mathematics)|fixed point]] of ''T'') spans an invariant subspace of dimension 1.

As a consequence of the [[fundamental theorem of algebra]], every linear operator on a nonzero [[dimension (vector space)|finite-dimensional]] [[complex number|complex]] vector space has an eigenvector. Therefore, every such linear operator in at least two dimensions has a proper non-trivial invariant subspace.

== Diagonalization via projections ==
Determining whether a given subspace ''W'' is invariant under ''T'' is ostensibly a problem of geometric nature. Matrix representation allows one to phrase this problem algebraically.

Write {{Mvar|V}} as the [[direct sum]] {{Math|''W'' &oplus; ''W''′}}; a suitable {{Math|''W''′}} can always be chosen by extending a basis of {{mvar|W}}. The associated [[projection operator]] ''P'' onto ''W'' has matrix representation
:<math>
P = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} : \begin{matrix}W \\ \oplus \\ W' \end{matrix} \rightarrow \begin{matrix}W \\ \oplus \\ W' \end{matrix}.
</math>

A straightforward calculation shows that ''W'' is {{Mvar|T}}-invariant if and only if ''PTP'' = ''TP''.

If 1 is the [[identity operator]], then {{Math|1-''P''}} is projection onto {{Math|''W''′}}. The equation {{math|''TP'' {{=}} ''PT''}} holds if and only if both im(''P'') and im(1 − ''P'') are invariant under ''T''. In that case, ''T'' has matrix representation <math display=block>
T = \begin{bmatrix} T_{11} & 0 \\ 0 & T_{22} \end{bmatrix} : \begin{matrix} \operatorname{im}(P) \\ \oplus \\ \operatorname{im}(1-P) \end{matrix} \rightarrow \begin{matrix} \operatorname{im}(P) \\ \oplus \\ \operatorname{im}(1-P) \end{matrix} \;.
</math>

Colloquially, a projection that commutes with ''T'' "diagonalizes" ''T''.

== Lattice of subspaces ==
As the above examples indicate, the invariant subspaces of a given linear transformation ''T'' shed light on the structure of ''T''. When ''V'' is a finite-dimensional vector space over an [[algebraically closed field]], linear transformations acting on ''V'' are characterized (up to similarity) by the [[Jordan canonical form]], which decomposes ''V'' into invariant subspaces of ''T''. Many fundamental questions regarding ''T'' can be translated to questions about invariant subspaces of ''T''.

The set of {{Mvar|T}}-invariant subspaces of {{Mvar|V}} is sometimes called the '''invariant-subspace lattice''' of {{Mvar|T}} and written {{Math|Lat(''T'')}}. As the name suggests, it is a ([[Modular lattice|modular]]) [[lattice (order)|lattice]], with [[Join and meet|meets and joins]] given by (respectively) [[set intersection]] and [[linear span]]. A [[minimal element]] in {{Math|Lat(''T'')}} in said to be a '''minimal invariant subspace'''.

In the study of infinite-dimensional operators, {{Math|Lat(''T'')}} is sometimes restricted to only the [[Closed (mathematics)|closed]] invariant subspaces.

== For multiple operators ==
Given a collection {{Math|{{mathcal|T}}}} of operators, a subspace is called {{Math|{{mathcal|T}}}}-invariant if it is invariant under each {{Math|''T'' ∈ {{mathcal|T}}}}.

As in the single-operator case, the invariant-subspace lattice of {{Math|{{mathcal|T}}}}, written {{Math|Lat({{mathcal|T}})}}, is the set of all {{Math|{{mathcal|T}}}}-invariant subspaces, and bears the same meet and join operations. Set-theoretically, it is the intersection <math display="block">\mathrm{Lat}(\mathcal{T})=\bigcap_{T\in\mathcal{T}}{\mathrm{Lat}(T)}\text{.}</math>

=== Examples ===
Let {{Math|End(''V'')}} be the set of all linear operators on {{Mvar|V}}. Then {{Math|1=Lat(End(''V''))={0,''V''}<nowiki />}}.

Given a [[Group representation|representation]] of a [[group (mathematics)|group]] ''G'' on a vector space ''V'', we have a linear transformation ''T''(''g'') : ''V'' → ''V'' for every element ''g'' of ''G''. If a subspace ''W'' of ''V'' is invariant with respect to all these transformations, then it is a [[subrepresentation]] and the group ''G'' acts on ''W'' in a natural way. The same construction applies to [[Algebra representation|representations of an algebra]].

As another example, let {{Math|''T'' ∈ End(''V'')}} and {{Mvar|Σ}} be the algebra generated by {1, ''T'' }, where 1 is the identity operator. Then Lat(''T'') = Lat(Σ).

=== Fundamental theorem of noncommutative algebra ===
Just as the fundamental theorem of algebra ensures that every linear transformation acting on a finite-dimensional complex vector space has a non-trivial invariant subspace, the ''fundamental theorem of noncommutative algebra'' asserts that Lat(Σ) contains non-trivial elements for certain Σ.
{{Math theorem
| math_statement = Assume {{mvar|V}} is a complex vector space of finite dimension. For every proper subalgebra {{mvar|Σ}} of {{math|End(''V'')}}, {{math|Lat(''Σ'')}} contains a non-trivial element.
| note = Burnside
}}
One consequence is that every commuting family in ''L''(''V'') can be simultaneously [[upper-triangular|upper-triangularized]]. To see this, note that an upper-triangular matrix representation corresponds to a [[Flag (linear algebra)|flag]] of invariant subspaces, that a commuting family generates a commuting algebra, and that {{Math|End(''V'')}} is not commutative when {{Math|dim(''V'') &geq; 2}}.

== Left ideals ==

If ''A'' is an [[algebra over a field|algebra]], one can define a [[regular representation|''left regular representation'']] Φ on ''A'': Φ(''a'')''b'' = ''ab'' is a [[algebra homomorphism|homomorphism]] from ''A'' to ''L''(''A''), the algebra of linear transformations on ''A''

The invariant subspaces of Φ are precisely the left ideals of ''A''. A left ideal ''M'' of ''A'' gives a subrepresentation of ''A'' on ''M''.

If ''M'' is a left [[Algebra_over_a_field#Subalgebras_and_ideals|ideal]] of ''A'' then the left regular representation Φ on ''M'' now descends to a representation Φ' on the [[quotient vector space]] ''A''/''M''. If [''b''] denotes an [[equivalence class]] in ''A''/''M'', Φ'(''a'')[''b''] = [''ab'']. The kernel of the representation Φ' is the set {''a'' ∈ ''A'' | ''ab'' ∈ ''M'' for all ''b''}.

The representation Φ' is [[irreducible representation|irreducible]] if and only if ''M'' is a [[maximal ideal|maximal]] left ideal, since a subspace ''V'' ⊂ ''A''/''M'' is an invariant under {Φ'(''a'') | ''a'' ∈ ''A''} if and only if its [[preimage]] under the [[quotient map]], ''V'' + ''M'', is a left ideal in ''A''.

== Invariant subspace problem ==
:{{main|Invariant subspace problem}}

The invariant subspace problem concerns the case where ''V'' is a separable [[Hilbert space]] over the [[complex number]]s, of dimension > 1, and ''T'' is a [[bounded operator]]. The problem is to decide whether every such ''T'' has a non-trivial, closed, invariant subspace. It is unsolved.

In the more general case where ''V'' is assumed to be a [[Banach space]], [[Per Enflo]] (1976) found an example of an operator without an invariant subspace. A concrete example of an operator without an invariant subspace was produced in 1985 by [[Charles Read (mathematician)|Charles Read]].

==Almost-invariant halfspaces==

Related to invariant subspaces are so-called almost-invariant-halfspaces ('''AIHS's'''). A closed subspace <math>Y</math> of a Banach space <math>X</math> is said to be '''almost-invariant''' under an operator <math>T \in \mathcal{B}(X)</math> if <math>TY \subseteq Y+E</math> for some finite-dimensional subspace <math>E</math>; equivalently, <math>Y</math> is almost-invariant under <math>T</math> if there is a [[finite-rank operator]] <math>F \in \mathcal{B}(X)</math> such that <math>(T+F)Y \subseteq Y</math>, i.e. if <math>Y</math> is invariant (in the usual sense) under <math>T+F</math>. In this case, the minimum possible dimension of <math>E</math> (or rank of <math>F</math>) is called the '''defect'''.

Clearly, every finite-dimensional and finite-codimensional subspace is almost-invariant under every operator. Thus, to make things non-trivial, we say that <math>Y</math> is a halfspace whenever it is a closed subspace with infinite dimension and infinite codimension.

The AIHS problem asks whether every operator admits an AIHS. In the complex setting it has already been solved; that is, if <math>X</math> is a complex infinite-dimensional Banach space and <math>T \in \mathcal{B}(X)</math> then <math>T</math> admits an AIHS of defect at most 1. It is not currently known whether the same holds if <math>X</math> is a real Banach space. However, some partial results have been established: for instance, any [[self-adjoint operator]] on an infinite-dimensional real Hilbert space admits an AIHS, as does any strictly singular (or compact) operator acting on a real infinite-dimensional reflexive space.

==See also==
* [[Invariant manifold]]
* [[Lomonosov's invariant subspace theorem]]

==References==
{{reflist}}

==Sources==
* {{cite book
|first1=Yuri A.
|last1= Abramovich
|first2= Charalambos D.
|last2= Aliprantis
|author2-link=Charalambos D. Aliprantis
|title=An Invitation to Operator Theory
|publisher=American Mathematical Society
|year=2002
|isbn=978-0-8218-2146-6}}

*{{cite book
|last=Beauzamy
|first= Bernard
|title=Introduction to Operator Theory and Invariant Subspaces
|year=1988
|publisher=North Holland
}}

* {{cite book
|authorlink1=Per Enflo
|last1=Enflo
|first1= Per
|authorlink2=Victor Lomonosov
|last2= Lomonosov
|first2= Victor
|chapter=Some aspects of the invariant subspace problem
|title=Handbook of the geometry of Banach spaces
|volume=I
|pages=533–559
|publisher=North-Holland
|location=Amsterdam|year=2001
}}

*{{cite book
|title=Invariant Subspaces of Matrices with Applications
|first1=Israel
|last1= Gohberg
|first2=Peter
|last2=Lancaster
|first3=Leiba
|last3= Rodman
|edition=Reprint, with list of [[errata]] and new preface, of the 1986 Wiley
|series=Classics in Applied Mathematics
|volume=51
|publisher=Society for Industrial and Applied Mathematics (SIAM)
|year=2006
|pages=xxii+692
|isbn=978-0-89871-608-5
}}

* {{cite book
|first=Yurii I.
|last=Lyubich
|title=Introduction to the Theory of Banach Representations of Groups
|edition=Translated from the 1985 Russian-language
|location=Kharkov, Ukraine
|publisher=Birkhäuser Verlag
|date= 1988
}}

* {{cite book
|first1=Heydar
|last1=Radjavi
|first2=Peter
|last2=Rosenthal
|title=Invariant Subspaces
|year=2003
|edition=Update of 1973 Springer-Verlag
|isbn=0-486-42822-2
|publisher=Dover Publications
}}

*{{cite book
| last=Roman
| first=Stephen
| title=Advanced Linear Algebra
| edition=Third
| series=[[Graduate Texts in Mathematics]] | publisher = Springer
| date=2008
| pages=
| isbn=978-0-387-72828-5
|author-link=Steven Roman}}

[[Category:Linear algebra]]
[[Category:Operator theory]]
[[Category:Representation theory]]

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