A new approach to Frobenius-Perron operators
Let \((X,\Sigma, \mu)\) be a complete \(\sigma\)-finite measure space, \(ba(X,\Sigma,\mu)= ba(\mu)\) the set of all finitely additive complex valued measures which vanish on all sets of \(\mu\)-measure 0, \(L^ 1(\mu)= L^ 1(X,\Sigma, \mu)\) and \(L^ \infty(\mu)= L^ \infty(X,\Sigma,\mu)\) denote the usual Lebesgue spaces of complex valued functions on \(X\). Let \(S: X\to X\) be a non-singular transformation: \((S^{-1}(A))= 0\) for each \(A\in \Sigma\) with \(\mu(A)= 0\). The authors define the Koopman operator \(U_ S: L^ \infty(\mu)\to L^ \infty(\mu)\) by the equality \(U_ S\varphi= \varphi\circ S\). They call the conjugate operator \(U^*_ S:= {\mathcal P}_ S: ba(\mu)\to ba(\mu)\) a generalizing Frobenius-Perron operator corresponding to \(S\). Then the classical Frobenius-Perron operator \(P_ S\) is a restriction of \({\mathcal P}_ S\) onto \(L^ 1(\mu)\) (the latter is (up to an isometry) a subspace of \(ba(\mu)\)) and acts by the formula \(P_ S f= h(E(f| S^{- 1}(\Sigma))\circ S^{-1})\), \(f\in L^ 1(\mu)\), where \(E(\cdot| S^{-1}(\Sigma))\) is the conditional expectation operator with respect to the \(\sigma\)-algebra \(S^{-1}(\Sigma)\), \(h= d(\mu\circ S^{- 1})/d\mu\). In these notations the following two theorems should be counted to be main ones in the paper. Theorem 3.3. Let \((X, S^{-1}(\Sigma),\mu)\) be a \(\sigma\)-finite measure space. Then the following statements are equivalent: (a) \(P_ S\) is one-to-one; (b) \(U_ S\) is onto; (c) \(U_ S(L^ \infty(\mu))\) is dense in \(L^ \infty(\mu)\); (d) \(S^{-1}(\Sigma)= \Sigma\); (e) \(E(\cdot| S^{-1}(\Sigma))\) is the identity operator; (f) \(P_ S\) is isometric. Theorem 3.6. Let \((X,\Sigma,\mu)\) be the segment \([0,1]\) equipped with Lebesgue measure. Then \(P_ S\) is an isometry if and only if \(S\) is measure-theoretically one-to-one in the sense that there exists a non- singular transformation \(T\) such that \(T\circ S(x)= x\) a.e. on \([0,1]\). Furthermore, \(P_ T(h)= 1\). At the end a Corollary of spectral type is given.
scientific article; zbMATH DE number 703376