Cyclic vectors for multiplication operators (Q1122117)

If \(E\) is a complex topological vector space and \(T\) a continuous linear transformation on \(E\), a vector \(x\in E\) is a cyclic vector for \(T\) if \(\{p(T)x: p\) a polynomial\(\}\) is dense in \(E\). If \(\mu\) is a compactly supported Borel measure on \(\mathbb C\), let \(X\) be the closed unit ball of \(L^{\infty}(\mu)\) and let \(X_ p\) be \(X\) equipped with the metric from \(L^ p(\mu)\). If \(T\) is the operator on \(L^ p(\mu)\) of multiplication by \(z\), the author shows that the set of cyclic vectors in \(L^ p(\mu)\) is a residual set and the subset of cyclic vectors that lie in \(X\) is a residual subset of \(X_ p\). A similar theorem concerning spaces of analytic functions is established.