# invariant subspace

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## invariant subspace

[in‚ver·ē·ənt ′səb‚spās]
(mathematics)
For a bounded operator on a Banach space, a closed linear subspace of the Banach space such that the operator takes any point in the subspace to another point in the subspace.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Now let U be an invariant subspace with basis U, and let U = U + [DELTA]U with a small perturbation [DELTA]U.
Hasan, "Rational invariant subspace approximations with applications," IEEE Transactions on Signal Processing, vol.
Furthermore, authors have shown that CLBS is closely related to the invariant subspace; namely, exact solutions defined on invariant subspaces for equations or their variant forms can be obtained by using the CLBS method [12-24].
The subspace [K.sup.2.sub.u] = [H.sup.2] [member of] u[H.sup.2] is a proper nontrivial invariant subspace of [S.sup.*], the most general one by the well-known theorem of A.
Furthermore, for the nonlinear problem, the multiple exp-function method [18, 19], the transformed rational function method [20-22], and invariant subspace method [23, 24] are three systematical approaches to handle the nonlinear terms.
The operator A is polaroid if it is polar at every [lambda] [member of] iso[sigma](A), and it is hereditarily polaroid if every restriction [A|.sub.M] of A to an (always closed) invariant subspace M of A is polaroid.
According to (28) and (32), the three-dimensional invariant subspace is obtained as
In fact, the [S.sub.[infinity]]- invariant subspace C{[[summation].sub.i[member of]N] [[upsilon].sub.i]} does not have a closed [CS.sub.[infinity]]-invariant complement, so that all projection operators with range C{[[summation].sub.i[member of]N] [[upsilon].sub.i]} are unbounded.
The invariant subspace method is refined to present more unity and more diversity of exact solutions by taking subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit .
(v) If M is an invariant subspace of T, then the restriction has [T|.sub.M] has property (i).
A subspace M is invariant for T if T(M) [subset not equal to] M and a part of an operator is a restriction of it to an invariant subspace.
To be more precise, we consider two coupled map families such that the family maps all have the same fixed point which is nested within the same topologically transitive invariant set which is nested in turn within the same invariant subspace. We prove in such a case that these point bifurcations which are transversal to the invariant subspace generate two periodic of period 2 points in a neighbourhood of the given point and besides can simultaneously give rise to orbits that are homoclinic to the periodic points.

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