Cyclic vector
Let be an endomorphism of a finite-dimensional vector space
. A cyclic vector for
is a vector
such that
form a basis for
, i.e. such that the pair
is completely reachable (see also Pole assignment problem; Majorization ordering; System of subvarieties; Frobenius matrix).
A vector in an (infinite-dimensional) Banach space or Hilbert space with an operator
on it is said to be cyclic if the linear combinations of the vectors
,
, form a dense subspace, [a1].
More generally, let be a subalgebra of
, the algebra of bounded operators on a Hilbert space
. Then
is cyclic if
is dense in
, [a2], [a5].
If is a unitary representation of a (locally compact) group
in
, then
is called cyclic if the linear combinations of the
,
, form a dense set, [a3], [a4]. For the connection between positive-definite functions on
and the cyclic representations (i.e., representations that admit a cyclic vector), see Positive-definite function on a group. An irreducible representation is cyclic with respect to every non-zero vector.
References
[a1] | M. Reed, B. Simon, "Methods of mathematical physics: Functional analysis" , 1 , Acad. Press (1972) pp. 226ff |
[a2] | R.V. Kadison, J.R. Ringrose, "Fundamentals of the theory of operator algebras" , 1 , Acad. Press (1983) pp. 276 |
[a3] | S.A. Gaal, "Linear analysis and representation theory" , Springer (1973) pp. 156 |
[a4] | A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) pp. 53 (In Russian) |
[a5] | M.A. Naimark, "Normed rings" , Noordhoff (1964) pp. 239 (In Russian) |
Cyclic vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cyclic_vector&oldid=18561