  A two-point algebra is a set of bounded analytic functions on the unit disk
that agree at two distinct points $a,b \in \mathbb{D}$. This algebra serves as
a multiplier algebra for the family of Hardy Hilbert spaces $H^2_t := \{ f\in
H^2 : f(a)=tf(b)\}$, where $t\in \mathbb{C}\cup\{\infty\}$. We show that
various spectra of certain Toeplitz operators acting on these spaces are
connected.
