  We prove that the squared singular values of a fixed matrix multiplied with a
truncation of a Haar distributed unitary matrix are distributed by a polynomial
ensemble. This result is applied to a multiplication of a truncated unitary
matrix with a random matrix. We show that the structure of polynomial ensembles
and of certain Pfaffian ensembles is preserved. Furthermore we derive the joint
singular value density of a product of truncated unitary matrices and its
corresponding correlation kernel which can be written as a double contour
integral. This leads to hard edge scaling limits that also include new finite
rank perturbations of the Meijer G-kernels found for products of complex
Ginibre random matrices.
