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Motivated by the study of existence, uniqueness and regularity of solutionsto stochastic partial differential equations driven by jump noise, we proveIt\^{o} isomorphisms for $L^p$-valued stochastic integrals with respect to acompensated Poisson random measure. The principal ingredients for the proof arenovel Rosenthal type inequalities for independent random variables takingvalues 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 operatornorm of a sum of independent random matrices.

It\^{o} isomorphisms for $L^{p}$-valued Poisson stochastic integrals