The generalised injection and surjection moduli

Let X and Y be normed linear spaces and T(X -> Y) a linear operator. We investigate operator quantities for unbounded linear operators determined by perturbation by precompact operators, namely the generalised injective and surjective moduli. We show that these quantities are densely invariant and characterises the operators of semi-Fredholm type. These operator quantities are also then compared with other known operator quantities that are generated by norms of restrictions and quotients, such as those the quantity Gamma, for example, introduced by Gramsch and suitably modified by Cross to unbounded linear operators.