The Krein formula in almost Pontryagin spaces. A proof via orthogonal coupling

A new proof is provided for the Krein formula for generalized resolvents in the context of symmetric operators or relations with defect numbers (1,1) in an almost Pontryagin space. The new proof is geometric and uses the orthogonal coupling of the almost Pontryagin spaces induced by the Q-function and the parameter function in the Krein formula.