More generally, invariant subspaces are defined for sets of operators as subspaces invariant for each operator in the set.
12.
The Wold decomposition and the related Wold's theorem inspired harmonic analysis and related work on invariant subspaces of linear operators.
13.
The degree of an elementary divisor is the size of the corresponding Jordan block, therefore the dimension of the corresponding invariant subspace.
14.
Many fundamental questions regarding " T " can be translated to questions about invariant subspaces of " T ".
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The problem is to decide whether every such " T " has a non-trivial, closed, invariant subspace.
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Correspondingly, the nonlinear system has invariant manifolds, made of trajectories of the nonlinear system, corresponding to each of these invariant subspaces.
17.
In operator theory, it is unknown whether all linear operators on separable complex Hilbert spaces have ( non-trivial ) invariant subspaces.
18.
The minimal invariant subspaces correspond to the first diagonal block in that case . talk ) 20 : 21, 6 August 2014 ( UTC)
19.
It implies that, for irreducible representations of C *-algebras, the only non-zero linear invariant subspace is the whole space.
20.
A unitary representation is completely reducible, in the sense that for any closed invariant subspace, the orthogonal complement is again a closed invariant subspace.
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