Eigenfunction for a probability system

Let $(X,\mathcal{A},\mu ,T)$ be a measure-preserving probability system. Work in complex $L^2(\mu )$ with the pairing $⟨f,g⟩=\int f\bar{g}$, linear in its first variable, as in The complex $L^2$ pairing is well-defined and satisfies Cauchy–Schwarz. Set $U_{T}f=f\circ T$ on almost-everywhere classes and $H_0=\{f\in L^2(\mu ):\int f=0\}$.

An eigenfunction is a nonzero class $f\in L^2(\mu )$ for which $U_{T}f=\lambda f$ for some $\lambda \in \mathbb{C}$. It is nonconstant when $f\mathrm{\notin }\mathbb{C}\mathbf{1}$. Here equality and constancy always mean equality almost everywhere.

We verify the integral interface used here directly. For a nonnegative simple function, augment any finite disjoint display by the measurable complement of its displayed sets, with coefficient $0$. Given two such augmented displays, their pairwise intersections partition $X$, and equality of the functions forces the two coefficients to agree on every nonempty intersection. Finite additivity, with $0\cdot (+\mathrm{\infty })=0$, therefore proves representation independence. On a common augmented refinement the simple integral is monotone and additive; homogeneity is direct when the scalar is $0$ and termwise when it is positive. Now let $0\le g_j\uparrow g$ and put $L={\sup }_j\int g_j$. For a simple $s\le g$ and $0<c<1$, the sets $A_j=\{g_j\ge cs\}$ increase to $X$, including on the zero level of $s$. Since $A\mapsto {\int }_{A}s$ is the finite sum of the measures of the nonzero level sets, continuity from below gives $c\int s={\lim }_{j}c{\int }_{A_j}s\le L$. Letting $c\uparrow 1$ and then taking the supremum over $s$ proves monotone convergence from the definitions in The integral of a nonnegative simple function and The nonnegative Lebesgue integral.

Applying this monotone convergence result to sums of increasing simple approximants gives nonnegative additivity. Positive/negative and real/imaginary decompositions then give finite complex $L^1$ linearity. Applying nonnegative integral invariance to those four parts gives $\int h\circ T=\int h$ for every complex $h\in L^1(\mu )$.

For a measure-preserving $T$, canonical level sets give $\int s\circ T=\int s$ for every nonnegative simple $s$. Choose $s_j\uparrow g$ using Every nonnegative measurable function is the increasing limit of simple measurable functions and apply the preceding monotone-convergence argument to $s_j$ and $s_j\circ T$; hence $\int g\circ T=\int g$ for every nonnegative measurable $g$. Applying this to $g=\mathrm{\mid }f{\mathrm{\mid }}^2$ proves directly that the pullback of The Koopman operator is an $L^2$ isometry. Thus $\mathrm{\parallel }f{\mathrm{\parallel }}_2=\mathrm{\mid }\lambda \mathrm{\mid }\mathrm{\parallel }f{\mathrm{\parallel }}_2$ forces $\mathrm{\mid }\lambda \mathrm{\mid }=1$. For a system invertible modulo null sets in Invertible measure-preserving systems, pullback by the inverse is an inverse isometry, so $U_T$ is unitary. The name does not assume invertibility for every probability system.

Cauchy–Schwarz with $\mathbf{1}$, whose norm is one, gives $\mathrm{\mid }\int f\mathrm{\mid }\le \mathrm{\parallel }f{\mathrm{\parallel }}_2$ and integrability of $f$. Thus $H_0$ is a closed linear subspace. The locally proved integral invariance gives $(\lambda -1)\int f=0$. If $\lambda \mathrm{\ne }1$, the eigenfunction already lies in $H_0$. If $\lambda =1$ and $f$ is nonconstant, $f-(\int f)\mathbf{1}$ is a nonzero eigenfunction in $H_0$.

In this page's spectral criterion, a completed Lebesgue probability space has the usual interval-and-atoms model modulo null sets, with completed measure; no classification theorem for arbitrary probability spaces is used. The Chacon model below is the completed unit interval. These definitions and the displayed finite calculations make no simultaneous choices of representatives.