We present a norm estimate for the partial transpose map Theta on the tensor product M-m circle times M-nTheta(Sigma(k) A(k) circle times B-k) := Sigma(k) A(k) circle times B-k(T)with respect to a unitarily invariant norm. This is related to the norm estimates of the following maps on M-m,M-n in terms of the spectral norm of Sigma(k) A(k) circle times B-k:X bar right arrow Sigma(k) A(k) X B-k and X bar right arrow Sigma(k) A(k) X B-k(T).We show further that in the special case of A circle times I-n + I-m circle times B as well as AX + XB and AX + XBT those estimates are much improved and thatparallel to A circle times I-n + I-m circle times B-T parallel to(p) = parallel to A circle times I-n + I-m circle times B parallel to(p)for certain Schatten p-norms.