Hello everybody! Recently in my research, I came across the PerronFrobenius operator .. I would like to intuitive interpretation of this operator, ie, physical interpretations are possible, articles (can be physical). I wonder how this operator originated.
The place to start is with the PerronFrobenius theorem, which (in its most basic form) says that a $d\times d$ matrix $A$ with only positive entries has exactly one positive eigenvector $\vec{v}$, which corresponds to the largest eigenvalue (which is real and positive). Furthermore, all the other eigenvalues are strictly smaller in absolute value, and the iterates of any nonnegative vector under the matrices $A^n$ converge exponentially to the eigendirection spanned by $\vec{v}$. This can be interpreted very intuitively by looking at the action on the unit simplex $\Delta = \{\vec{w}\in \mathbb{R}^d \mid w_i \geq 0, \\vec{w}\_1 = 1\}$ given by $f(\vec{w}) = A\vec{w} / \\vec{w}\_1$. This is discussed at more length in the books by Brin and Stuck and by Katok and Hasselblatt. Once you understand that case, you have at least a general picture for what you're trying to prove in the more general setting, when $\vec{w} \in \mathbb{R}^d$ is replaced by a function $\phi$ in some Banach space $X$. The idea in a wide variety of settings is that you have a dynamical system which defines a natural action on spaces of functions, and you'd like to find a Banach space of functions on which the properties of that action (which is the Ruelle PerronFrobenius operator) mimic those of the positive matrix $A$ above. The two cases are linked by the fact that if $A$ is a $0$$1$ matrix that determines the admissible transitions for a mixing topological Markov chain on $d$ symbols, then the matrix $A$ actually is the PerronFrobenius operator. (Note that the mixing property says that some power of $A$ is positive, so then you can apply the standard PerronFrobenius theorem.) 


I like to explain the Ruelle Perron Frobenius operator this way: Suppose that $T$ is a map from some space $X$ to itself. Suppose also that $X$ has some distinguished ambient measure (\lambda) on it (think of Lebesgue measure). One more assumption: the ambient measure is nonsingular with respect to $T$ (i.e. if $A$ is a set of measure 0, then $T^{1}A$ has measure 0 also). Given a measure $\mu$ on $X$, the "pushforward" of $\mu$ is defined to be $\mu\circ T^{1}$. Seems counterintuitive, but in the case where $\mu$ is concentrated at a single point $a$ you can check that the pushforward is concentrated at $T(a)$. The nonsingularity condition implies that if $\mu$ is a measure absolutely continuous with respect to $\lambda$, then its pushforward is also absolutely continuous with respect to $\lambda$. By the RadonNikodym theorem, measures absolutely continuous with respect to $\lambda$ have essentially unique densities. The RuellePerronFrobenius operator applied to $f$ gives the density of the pushforward of the measure whose density is $f$. In more picturesque language: if $X$ is a random variable with density $f(x)$, then $T(X)$ is a random variable with density $L[f](x)$. 


I'll try to complement Vaughn's answer. In general the (Ruelle) PerronFrobenius or transfer operator $\mathcal{F}$ is given by $\int_X \mathcal{F}f = \int_{T^{1}(X)} f$ for $f$ and $X$ generic and nice, and $T$ some invertible transformation of the ambient space. (The "transfer" nomenclature is related to the "transfer matrix" method in statistical physics.) So let's think about what this means on a finite space $X \equiv \{1,\dots,n\}$ with (probability) measure $p$ corresponding to a row vector, and with functions corresponding to column vectors. Let $\mathcal{P}_X$ be the projection onto $X$: then $\int_X g \equiv p \mathcal{P}_X g$ and $\mathcal{F}$ is defined via $p \mathcal{P}_X \mathcal{F}f = p \mathcal{P}_{T^{1}(X)} f$. W/l/o/g, let $f = e_j$ for some $j$. Then $p \mathcal{P}_X \mathcal{F}e_j = p \mathcal{P}_{T^{1}(X)} e_j$ and the RHS is either an entry of $p$ or zero according to whether or not $j \in T^{1}(X)$, respectively. In the special case where $T$ is a permutation, $\mathcal{F}$ is the corresponding permutation matrix. The general idea is similar: $\mathcal{F}$ is an operator that represents the action of $T$ on a suitable function space. 

