On Pseudo-cyclic Multipliers in Hilbert Function Spaces

Let $\mathcal{H}$ be a separable complete Pick space of continuous functions on a compact set $\Omega$ with multiplier algebra $\mathrm{M}(\mathcal{H})$. The notion of the pseudo-cyclicity is recently defined by Aleman et al. In this short paper, we first extend their definition of the pseudo-cyclic multipliers to all functions $f$ in $\mathcal{H}$. Then we show that whenever one-function corona theorem holds for $\mathrm{M}(\mathcal{H})$ then a function $f$ in $\mathcal{H}$ is in the pseudo-cyclic class $ \mathcal{C}_n(\mathcal{H})$ if and only if $1/f$ is in the corresponding Pick-Smirnov type class $N_n^+(\mathcal{H})$. Furthermore, we show that non-vanishing functions $f \in \mathcal{H}$ are in the class $\mathcal{C}_1(\mathcal{H})$. For functions $\varphi, \psi$ in $\mathrm{M}(\mathcal{H})$, with at least one being in $\mathcal{C}_1(\mathcal{H})$, we also show that the invariant subspace generated by $\varphi \psi$ is equal to the intersection of invariant subspaces generated by $\varphi$ and $ \psi$.