For the scalar holomorphic discrete series representations of $\mathrm{SU}(2,2)$ and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside $\mathrm{SU}(2,2)$. We construct a Cayley transform between the Ol’shanskiĭ semigroup having $\mathrm{U}(1,1)$ as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for $\mathrm{SU}(2,2)$. This allows calculating the composition series in terms of harmonic analysis on $\mathrm{U}(1,1)$. In particular we show that the Ol’shanskiĭ Hardy space for $\mathrm{U}(1,1)$ is different...