Toeplitz spectral inclusion and generalized in modulus property

Abstract:

Let $A \subset {C_b}(X)$ be a function algebra and let $\mu$ be a Borel measure on $X$ and ${H^2}(\mu ) = \bar A$—the closure in ${L^2}(\mu )$. It turns out that the spectral inclusion theorem for Toeplitz operators, defined in the above context, implies the density of finite sums $\sum \nolimits _i {{{\left | {{u_i}} \right |}^2},{u_i} \in A}$, in (i) the cone of positive functions in ${L^1}(\mu )$, (ii) the cone of positive functions in ${L^p}(\mu ),p \leq 2$, if $\mu (X){\text { < }}\infty$, (iii) the cone of positive functions in $C(X)$, if $X$ is compact.