Generalizing the Fischer inequality

AbstractIf q ⩾ 1, let Aq denote the complex vector space whose elements are the complex-valued alternating q-linear functions on Cm. If A ∈ An, B ∈ Ap are decomposable, then the classical Fischer inequality for partitioned positive semidefinite Hermitian matrices implies that n + pn‖ A ⋀B‖2⩽‖A‖2·‖B‖2. We show that this inequality holds if only one of A and B is decomposable or if A ∧ B is decomposable. We introduce the concept of relative decomposability and show that the above inequality continues to hold if A is decomposable relative to B. Also, an example is presented to demonstrate that the above inequality does not hold in general