In Chap. 5 the Kreı̆n theory of selfadjoint extensions of a Hermitian contraction is presented. In Sects. 5.2–5.3 we recall the notion of Kreı̆n’s shorted operator and present its properties. Next, in Sect. 5.4, it is shown that the set of all contractive selfadjoint extensions of a Hermitian contraction T forms an operator segment with two endpoints \(T_{\min }\) and \(T_{\max }\) being the extremal extensions of T. Our exposition of this result as well as the whole Kreı̆n’s theory in Sects. 5.4–5.7 is substantially relied on the block-matrix representations of operators \(T_{\min }\) and \(T_{\max }\) . In Sect. 5.5 we establish an important identity relating the top part \(T_{11}\) of the operator T and the reduced operator \(T_{\max }^{\prime }\) and show that \(T_{11}\) and \(T_{\max }^{\prime }\) have similar compactness properties. In Sects. 5.6–5.7 we apply these results to a non-negative symmetric operator A and show that certain compactness properties of the inverses of the reduced Kreı̆n extension \(A_{\mathrm {K}}^{\prime }\) and the operator A are similar. However it is not the case for the Friedrichs extension \(A_{\mathrm {F}}\) . In Sect. 5.8 we present an explicit counterexample to Birman’s conjecture, showing that the compactness of \(A^{-1}\) does not imply the compactness of the inverse to its Friedrichs extension.