Stability theory for nullity and deficiency of linear relations (Q2664852)

If \(\mathcal{A}\) is a closed linear relation (multivalued linear operator) between Banach spaces \(X\) and \(Y\), then the nullity \(\alpha (\mathcal{A})\) is the dimension of the kernel of \(\mathcal{A}\) and the deficiency \(\beta (\mathcal{A})\) is the dimension of \(Y/R(\mathcal{A})\), where \(R(\mathcal{A})\) is the range of \(\mathcal{A}\). The authors prove the stability of \(\alpha (\mathcal{A} - \lambda \mathcal{B})\) and \(\beta (\mathcal{A} - \lambda \mathcal{B})\), where \(\mathcal{B}\) is a closed linear relation such that \(D(\mathcal{B}) \supset D(\mathcal{A})\), \(\mathcal{B}(0) \subset \mathcal{A}(0)\) and \(\lambda\) is a complex number with sufficiently small \(|\lambda|\).