Induced Operators

Let $(X_1, \mathfrak{B}_1, \mu_1)$ and $(X_2, \mathfrak{B}_2, \mu_2)$ be measure spaces and denote by $L^0(X_1)$ and $L^0(X_2)$ the corresponding spaces of measurable functions (with values in $\mathbb{C}$ ).

Definition - If $T: X_1 \longrightarrow X_2$ is a measure-preserving transformation we can define an operator $U_T:L^0(X_2) \longrightarrow L^0(X_1)$ by

The operator $U_T$ is called the operator induced by $T$ .

Many ideas in ergodic theory can be explored by studying this operator.

Basic Properties

The following properties are clear:

• $U_T$ is linear.
• $U_T$ maps real valued functions to real valued functions.
• If $f \geq 0$ then $U_Tf \geq 0$
• $U_T k = k$ for every constant function $k$ .
• $U_T(fg)=U_T(f)U_T(g)$ .
• $U_T$ maps characteristic functions to characteristic functions. Moreover, $U_T \chi_B = \chi_{T^{-1}B}$ , for every measurable set $B \in \mathfrak{B}_2$ .
• If $T_1:X_1 \longrightarrow X_2$ and $T_2:X_2 \longrightarrow X_3$ are measure preserving maps, then $U_{T_2 \circ T_1} = U_{T_1}U_{T_2}$ .

Preserving Integrals

Theorem 1 - If $f \in L^0(X_2)$ then $\int_{X_1} U_Tf\;d\mu_1 = \int_{X_2} f\;d\mu_2$ , where if one side does not exist or is infinite, then the other side has the same property.

Induced Isometries

Theorem 2 - Let $p \geq 1$ . We have that $U_T(L^p(X_2)) \subseteq L^p(X_1)$ and, moreover,

   for all

$\,$

Thus, when restricted to $L^p$ -spaces, $U_T$ is called the isometry induced by $T$ .