Multivariate versions of Cochran's theorems

Let E, V be n-, p-dimensional inner product spaces over the real field, let (V, E) be the set of all linear maps of V into E, and let Y be a normal random vector in (V, E) with mean [mu] = 0 and covariance [Sigma]Y such that S1 [square not subset] S2([not equal to] }0{) is the image set, Im [Sigma]Y of [Sigma]Y, where S1, S2 are linear subspaces of E, V, respectively, and [square not subset] is the outer product. Let }Wi{ be a family of self-adjoint operators in (E, E). Then (*): }Y'WiY{ is an independent family of Wishart random operators Y'WiY with parameter (mi, [Sigma], [lambda]i), each mi > 0 and [lambda]i = 0," if and only if Im [Sigma] = S2 and for any distinct i,j[set membership, variant]I, [Sigma]Y(Wi[circle times operator][Sigma]+)[Sigma]Y(Wi[circle times operator][Sigma]+)[Sigma]Y = [Sigma]Y(Wi[circle times operator][Sigma]+)[Sigma]Y, tr([Sigma]Y(Wi[circle times operator][Sigma]+)) [not equal to] 0, and [Sigma]Y(Wi[circle times operator][Sigma]+)[Sigma]Y(Wj[circle times operator][Sigma]+)[Sigma]Y = 0. A necessary and sufficient condition for (*) is also obtained for the general case where no condition whatever is imposed on ([mu], [Sigma]Y). This generalizes a recent result of Pavur who considered the case where [Sigma] is nonsingular and each Wi is nonnegative definite.