Permutator

An eigen value $ \lambda $ of a stochastic kernel that it is different from one and such that $ | \lambda | = 1 $. A non-negative continuous function $ K( x, y) $ given on a compact set $ \Omega $ in a finite-dimensional space is called a stochastic kernel if

$$ \int\limits _ \Omega K( x, y) dy = 1,\ \ x \in \Omega . $$

The eigen values of such a kernel satisfy the condition $ | \lambda | \leq 1 $. In operator theory, the name permutator is also given to an operator $ A: E \rightarrow E $ if the range of its values, $ A( E) $, is finite dimensional and if there exists a basis $ e _ {1} \dots e _ {n} $ in $ A ( E) $ such that $ Ae _ {j} = e _ {k _ {j} } $, $ j = 1 \dots n $.

References

[1]	P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian)

Comments

Given a kernel $ K( x, t) $, one considers the homogeneous integral equation

$$ u( x) - \lambda \int\limits _ \Omega K( x, t) u( t) dt = 0. $$

A regular point of a kernel $ K( x, t) $ is one for which this equation has only the trivial solution; a characteristic point (value) is one for which there is a non-trivial solution. If $ \lambda $ is characteristic, $ \lambda ^ {-} 1 $ is called an eigen value of the kernel $ K( x, t) $. Note that it is then an eigen value of the integral operator defined by $ K( x, t) $; cf. [1], pp. 27ff.