Generalized F. and M. Riesz Theorems

Throughout this section, we adopt the notation of Section 5, specifically, the notation and assumptions of Theorem 5.1.

Suppose that $T$ is a sup path attaining representation of $\Bbb R$ by isomorphisms of $M(\Sigma)$. In [4], we proved the following result concerning bounded operators ${\cal P}$ from $M(\Sigma)$ into $M(\Sigma)$ that commute with the representation $T$ in the following sense:

$${\cal P}\circ T_t=T_t\circ {\cal P}$$

for all $t\in \Bbb R$.

Theorem 6.1   Suppose that $T$ is a representation of $\Bbb R$ that is sup path attaining, and that ${\cal P}$ commutes with $T$. Let $\mu\in M(\Sigma)$ be weakly analytic. Then ${\cal P}\mu$ is also weakly analytic.

To describe an interesting application of this theorem from [4], let us recall the following.

Definition 6.2   Let $T$ be a sup path attaining representation of $G$ in $M(\Sigma)$. A weakly measurable $\sigma$ in $M(\Sigma)$ is called quasi-invariant if $T_t\sigma$ and $\sigma$ are mutually absolutely continuous for all $t\in G$. Hence if $\sigma$ is quasi-invariant and $A\in \Sigma$, then $\vert\sigma\vert(A)=0$ if and only if $\vert T_t(\sigma)\vert(A)=0$ for all $t\in G$.

Using Theorem 6.1 we obtained in [4] the following extension of results of de Leeuw-Glicksberg [9] and Forelli [12], concerning quasi-invariant measures.

Theorem 6.3   Suppose that $T$ is a sup path attaining representation of $\Bbb R$ by isometries of $M(\Sigma)$. Suppose that $\mu\in M(\Sigma)$ is weakly analytic, and $\sigma$ is quasi-invariant. Write $\mu=\mu_a+\mu_s$ for the Lebesgue decomposition of $\mu$ with respect to $\sigma$. Then both $\mu_a$ and $\mu_s$ are weakly analytic. In particular, the spectra of $\mu_a$ and $\mu_s$ are contained in $[0,\infty)$.

Our goal in this section is to extend Theorems 6.1 above to representations of a locally compact abelian group $G$ with ordered dual group $\Gamma$. More specifically, we want to prove the following theorems.

Theorem 6.4   Suppose that $T$ is a sup path attaining representation of $G$ by isomorphisms of $M(\Sigma)$ such that $T_{\phi_\alpha}$ is sup path attaining for each $\alpha$. Suppose that ${\cal P}$ commutes with $T$ in the sense that

$${\cal P}\circ T_t=T_t\circ {\cal P}$$

for all $t\in G$. Let $\mu\in M(\Sigma)$ be weakly analytic. Then ${\cal P}\mu$ is also weakly analytic.

As shown in [4, Theorem (4.10)] for the case $G=\Bbb R$, an immediate corollary of Theorem 6.4 is the following result.

Theorem 6.5   Suppose that $T$ is a sup path attaining representation of $G$ by isometries of $M(\Sigma)$, such that $T_{\phi_\alpha}$ is sup path attaining for each $\alpha$. Suppose that $\mu\in M(\Sigma)$ is weakly analytic with respect to $T$, and $\sigma$ is quasi-invariant with respect to $T$. Write $\mu=\mu_a+\mu_s$ for the Lebesgue decomposition of $\mu$ with respect to $\sigma$. Then both $\mu_a$ and $\mu_s$ are weakly analytic with respect to $T$. In particular, the $T$-spectra of $\mu_a$ and $\mu_s$ are contained in $\overline{P}$.

Proof of Theorem 6.4.     Write

$$\mu = \mu_{\alpha_0}*_T\mu+ \sum_\alpha d_\alpha *_T\mu ,$$

as in (5.1), where the series converges unconditionally in $M(\Sigma)$. Then

$${\cal P}\mu={\cal P}(\mu_{\alpha_0}*_T\mu) +\sum_\alpha {\cal P}(d_\alpha *_T\mu).$$ (42)

It is enough to show that the $T$-spectrum of each term is contained in $\overline{P}$. Consider the measure $\mu_{\alpha_0}*_T\mu$. We have ${\rm spec}_T(\mu_{\alpha_0}*_T\mu)\subset S_{\alpha_0}$. Hence by Theorem 4.4, $\mu_{\alpha_0}*_T\mu$ is $T_{\phi_{\alpha_0}}$-analytic. Applying Theorem 6.1, we see that

$${\rm spec}_{T_{\phi_{\alpha_0}}} ({\cal P}(\mu_{\alpha_0}*_T\mu))\subset [0,\infty[.$$ (43)

Since ${\cal P}$ commutes with $T$, it is easy to see from Proposition 3.10 and Corollary 3.11 that

$${\rm spec}_T ({\cal P}(\mu_{\alpha_0}*_T\mu))\subset C_{\alpha_0}.$$

Hence by (43) and Theorem 4.4,

$${\rm spec}_T ({\cal P}(\mu_{\alpha_0}*_T\mu))\subset S_{\alpha_0},$$

which shows the desired result for the first term of the series in (42). The other terms of the series (42) are handled similarly.

Acknowledgments The second author is grateful for financial support from the National Science Foundation (U.S.A.) and the Research Board of the University of Missouri.