Summary: [For part I see ibid. 39, No. 1, 154-174 (1991; Zbl 0749.62037).]
A general easily checkable Cochran theorem is obtained for a normal random operator \(Y\). This result does not require that the covariance, \(\Sigma_ Y\), of \(Y\) is nonsingular or is of the usual form \(A\otimes\Sigma\); nor does it assume that the mean \(\mu\), of \(Y\) is equal to zero.
Indeed, \(\{Y'W_ i Y\}\) (with nonnegative definite \(W_ i\)’s) is a family of independent Wishart random operators \(Y'W_ i Y\) of parameter \((m_ i,\Sigma,\lambda_ i)\) if and only if for some nonnegative definite \(A\) and for all \(i\neq j\): \[ (W_ i\otimes I)(\Sigma_ Y- A\otimes \Sigma)(W_ i\otimes I)=0;\tag{a} \]
\[ AW_ i AW_ i=AW_ i,\;r(AW_ i)=m_ i,\tag{b} \]
\[ \lambda_ i=\mu' W_ i\mu=\mu'W_ i AW_ i\mu;\tag{c} \] and \[ (W_ i\otimes I)\Sigma_ Y(W_ j\otimes I)=0.\tag{d} \] The usual multivariate versions of Cochran’s theorem are contained in a special case of our result where \(\Sigma_ Y=A\otimes \Sigma\). The \(A\) in our version of Cochran’s theorem can actually be constructed from \(\Sigma\), \(\Sigma_ Y\), and the sum of the \(W_ i\)’s.