# Permutator

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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)

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.