Decomposition of complex hyperbolic isometries by involutions

A $k$-reflection of the $n$-dimensional complex hyperbolic space ${\rm H}{\C}^n$isanelementin${\rm U}(n,1)$withnegativetypeeigenvalue$\lambda$,$|\lambda|=1$,ofmultiplicity$k+1$andpositivetypeeigenvalue$1$ofmultiplicity$n-k$\mathrm{.}Weprovethataholomorphicisometryof${\rm H}{\C}^n$isaproductofatmostfourinvolutionsandacomplex$k$-reflection,$k \leq 2$\mathrm{.}Alongtheway,weprovethateveryelementin${\rm SU}(n)$isaproductoffourorfiveinvolutionsaccordingas$n \neq 2 \mod 4$or$n = 2 \mod 4$. We also give an easy proof of the result that every holomorphic isometry of ${\mathrm{H}}_{\text{C}}^n$ is a product of two anti-holomorphic involutions.