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Cyclic vector

Wikipedia contributorsAdapted for Fathomly · View authors
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Source updated July 8, 2026

In the mathematics of operator theory, an operator A on an (infinite-dimensional) Banach space or Hilbert space H has a cyclic vector f if the vectors f,Af,A^{2}f,\dots span H. Equivalently, f is a cyclic vector for A in case the set of all vectors of the form p(A)f, where p varies over all polynomials, is dense in H.

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This article is adapted from the Wikipedia article “Cyclic vector”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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Related topics

Cyclic and separating vector

In mathematics, the notion of a cyclic and separating vector is important in the theory of von Neumann algebras, and, in particular, in Tomita-Takesaki theory. A related notion is that of a vector that is cyclic for a given operator.