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 $\mathrm{\Omega }$ in a finite-dimensional space is called a stochastic kernel if

$$\underset{\mathrm{\Omega }}{\int }K(x,y)dy=1,x\in \mathrm{\Omega }.$$

The eigen values of such a kernel satisfy the condition $|\lambda |\le 1$. In operator theory, the name permutator is also given to an operator $A:E\to E$ 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 $A{e}_j=e_{k_j}$, $j=1\dots n$.

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

$$u(x)-\lambda \underset{\mathrm{\Omega }}{\int }K(x,t)u(t)dt=0.$$

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.