Quasi-positive definite operators and matrices (Q1108355)

Let F be the real or complex field, let \(V_ 1,V_ 2,...,V_ m\) be finite dimensional inner product spaces over F, and let W be their tensor product. Relative to the natural inner product induced on W by the inner products on the \(V_ i\), a Hermitian linear operator T on W is said to be quasi-positive definite (semidefinite) if the Hermitian form \(<Tw,w>\) is positive (nonnegative) for all nonzero decomposition tensors w in W. Properties of such operators are studied with emphasis given to their negative inertia and the relation between their largest and smallest eigenvalues. The authors observe that a better understanding of quasi- positive semidefinite operators may lead to resolution of some conjectured inequalities for generalized matrix functions.