  Motivated by the study of existence, uniqueness and regularity of solutions
to stochastic partial differential equations driven by jump noise, we prove
It\^{o} isomorphisms for $L^p$-valued stochastic integrals with respect to a
compensated Poisson random measure. The principal ingredients for the proof are
novel Rosenthal type inequalities for independent random variables taking
values in a (noncommutative) $L^p$-space, which may be of independent interest.
As a by-product of our proof, we observe some moment estimates for the operator
norm of a sum of independent random matrices.
