Locally finite operator explained

f: V\to V is called locally finite if the space V is the union of a family of finite-dimensional f -invariant subspaces.^{[1] [2]}

In other words, there exists a family

\{ V _{i\vert} i\in I\} of linear subspaces of V , such that we have the following:c up _{i\in} V _{i=V} (\forall i\in I) f[V _{i]\subseteq} V _{i}

• Each

V _{i} is finite-dimensional.

An equivalent condition only requires

V to be the spanned by finite-dimensional f -invariant subspaces.^{[3] [4]} If V is also a Hilbert space, sometimes an operator is called locally finite when the sum of the \{ V _{i\vert} i\in I\} is only dense in V .

Examples

• Every linear operator on a finite-dimensional space is trivially locally finite.
• Every diagonalizable (i.e. there exists a basis of

V whose elements are all eigenvectors of f ) linear operator is locally finite, because it is the union of subspaces spanned by finitely many eigenvectors of f .

• The operator on

C [x] , the space of polynomials with complex coefficients, defined by T(f(x)) = xf(x) , is not locally finite; any T -invariant subspace is of the form C [x]f _{0(x)} for some f _{0(x)\inC[x]} , and so has infinite dimension.

• The operator on

C [x] defined by T(f(x)) = (f(x) - f( 0 ))/(x) is locally finite; for any n , the polynomials of degree at most n form a T -invariant subspace.^[5]

Notes and References

1. Central Simple Poisson Algebras. Yucai Su. Xiaoping Xu. math/0011086v1. 2000.
2. Book: Time-Varying Systems and Computations. Patrick. DeWilde. Alle-Jan. van der Veen. Springer Science+Business Media, B.V.. 978-1-4757-2817-0. 10.1007/978-1-4757-2817-0. 1998. Dordrecht.
3. Operators on Hopf Algebras. David E.. Radford. American Journal of Mathematics. Feb 1977. 99. 1. 139–158. Johns Hopkins University Press. 10.2307/2374012 . 2374012.
4. On the Riccati Equations of the H^∞ Control Problem for Discrete Time-Varying Systems. Jacquelien. Scherpen. Michel . Verhaegen. 3rd European Control Conference (Rome, Italy). September 1995. 10.1.1.867.5629 .
5. https://math.stackexchange.com/users/431940/joppy Joppy