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Locally finite operator

From Wikipedia, the free encyclopedia

In mathematics, a linear operator is called locally finite if the space is the union of a family of finite-dimensional -invariant subspaces.[1][2]: 40 

In other words, there exists a family of linear subspaces of , such that we have the following:

  • Each is finite-dimensional.

An equivalent condition only requires to be the spanned by finite-dimensional -invariant subspaces.[3][4] If is also a Hilbert space, sometimes an operator is called locally finite when the sum of the is only dense in .[2]: 78–79 

Examples

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  • Every linear operator on a finite-dimensional space is trivially locally finite.
  • Every diagonalizable (i.e. there exists a basis of whose elements are all eigenvectors of ) linear operator is locally finite, because it is the union of subspaces spanned by finitely many eigenvectors of .
  • The operator on , the space of polynomials with complex coefficients, defined by , is not locally finite; any -invariant subspace is of the form for some , and so has infinite dimension.
  • The operator on defined by is locally finite; for any , the polynomials of degree at most form a -invariant subspace.[5]

References

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  1. ^ Yucai Su; Xiaoping Xu (2000). "Central Simple Poisson Algebras". arXiv:math/0011086v1.
  2. ^ a b DeWilde, Patrick; van der Veen, Alle-Jan (1998). Time-Varying Systems and Computations. Dordrecht: Springer Science+Business Media, B.V. doi:10.1007/978-1-4757-2817-0. ISBN 978-1-4757-2817-0.
  3. ^ Radford, David E. (Feb 1977). "Operators on Hopf Algebras". American Journal of Mathematics. 99 (1). Johns Hopkins University Press: 139–158. doi:10.2307/2374012. JSTOR 2374012.
  4. ^ Scherpen, Jacquelien; Verhaegen, Michel (September 1995). On the Riccati Equations of the H Control Problem for Discrete Time-Varying Systems. 3rd European Control Conference (Rome, Italy). CiteSeerX 10.1.1.867.5629.
  5. ^ Joppy (Apr 28, 2018), answer to "Locally Finite Operator". Mathematics StackExchange. StackOverflow.