Let \(\mathcal H\) and \(\mathcal K\) be Hilbert spaces. From the introduction: “The main objective of this paper is to introduce the notions of compact and finite rank perturbations of closed linear operators and, more generally, closed linear relations \(A\) and \(B\) acting between \(\mathcal H\) and \(\mathcal K\), and to give various equivalent characterizations. The key idea here is to use the orthogonal projections \(P_A\) and \(P_B\) in \(\mathcal H \oplus \mathcal K\) onto the closed graphs or subspaces \(A\) and \(B\) of \(\mathcal H \oplus \mathcal K\).”
Indeed, \(A\) is said to be a compact (resp., finite rank) perturbation of \(B\) if \(P_A-P_B\) is a compact (resp., finite rank) operator. It is proved that: (1) \(A\) is a finite rank perturbation of \(B\) if and only if \(A\) and \(B\) are both finite-dimensional extensions of their common part \(A \cap B\). (2) \(A\) is a compact perturbation of \(B\) if and only if for every \(\epsilon > 0\) there exists a closed linear relation \(F\) from \(\mathcal H\) in \(\mathcal K\) such that \(P_B-P_F\) is a finite-rank operator and \(\|P_A-P_F\| < \epsilon\). (3) These notions are natural generalizations of the usual concepts of compact and finite rank perturbations, i.e., for bounded linear operators \(A\) and \(B\) and if in the special case \(\mathcal H = \mathcal K\) the operators \(A\) and \(B\) have a common point in their resolvent set.
The proofs are based on a representation of \(P_A\) in terms of the operator part \(A_{\text{op}}\) of \(A\) which coincides in essence with the Stone-de Snoo formula, cf.[M. H. Stone, J. Indian Math. Soc. (N.S.) 15, 155–192 (1952; Zbl 0047.11102)] and [Y. Mezroui, Trans. Am. Math. Soc. 352, No. 6, 2789–2800 (2000; Zbl 0951.47003)].