Stable perturbations of operators and related topics
The present book consists of six chapters, two appendices, a bibliography, and an index.NEWLINENEWLINEChapter 1 contains background material from functional analysis (including Banach algebras and \(C^*\)-algebras) given partly without proofs, but with references.NEWLINENEWLINEChapter 2 deals with stable perturbation of closed densely defined linear operators. By a stable perturbation of a linear closed operator \(A\) by a linear bounded operator \(B\), the author means that \(\operatorname{Ran}(A+B)\cap \operatorname{Ker}A^+=\{0\}\), where \(A^+\) denotes the generalized inverse of \(A\), \(\operatorname{Ran}(A)\) denotes the range of \(A\), and \(\operatorname{Ker}A\) denotes the null-space of \(A\).NEWLINENEWLINEChapters 3 and 4 deal with the Moore-Penrose (MP), Drazin (D), and \(T-S\) generalized inverses and their stable perturbations.NEWLINENEWLINEChapter 5, Miscellaneous Applications, deals with the generalized Bott-Duffin inverse, some applications of MP inverses to the theory of frames, generalized Cowen-Douglas mappings, condition numbers for MP and D inverses, and with a local linearization of \(C^1\)-smooth nonlinear maps.NEWLINENEWLINEChapter 6, Some Related Topics, deals with the perturbation of the reduced minimum modulus, the density of MP invertible elements in some \(C^*\)-algebras, and other topics. The reduced minimum modulus of a linear operator \(T\) is denoted by \(\gamma(T)\) and is defined by the formula \(\gamma(T)=\inf_{x\in D(T), \operatorname{dist}(x, \operatorname{Ker}T)=1}\| Tx\|\).NEWLINENEWLINEAppendix A contains some results from topology and analysis, and Appendix~B contains some results from the theory of \(C^*\)-algebras.NEWLINENEWLINEThe book has some weak points which negatively affect its suitability as a textbook for students: (a) some definitions are not given (for example, the definition of the space \(X^*\) in Proposition 1.1.4 on p.\,3, is not given; the term ``unital Banach algebra'', p.\,30, is not defined); (b) some theorems and proofs are not clear (for example, on p.\,8, Theorem 1.1.8 claims that \(T\) is bounded, but the proof is given for the claim that \(T\) is bounded on a subspace; the proof of item (4) in Proposition 1.1.1 on p.\,2 is not clear because the infimum should have been taken with respect to \(z\in A\)); (c) some terminology is not standard (for example, on p.\,7, Definition 1.1.14 is for ``directed set'', and such a set is usually called an ordered set; the expression ``internal of \(A\)'' on p.\,1 should be replaced by interior of \(A\)); (d) there are quite a few linguistic errors and misprints (for example, ``liner'' on p.\,29; ``we call \(\mathcal{A}\) is a Banach algebra'', p.\,29; ``is called to be'', p.\,29; ``we give the prove'', p.\,33, ``preposition'', p.\,66, etc.); (e) in the index, many items are missing (for example, \(GL(\mathcal{A})\), unital Banach algebra, etc.); (f) the bibliography has obvious omissions (for example, references to the works of M.\,G.\thinspace Krein, M.\,A.\thinspace Krasnosel'skii, I.\,Gohberg, B.\,Sz.-Nagy, related to the subject matter of this book, are missing); (g) some topics usually discussed in the area of perturbation theory for linear operators are not mentioned (for example, the influence of a perturbation of a linear operator on its spectrum; the perturbation of eigenvalues, eigenspaces, and root spaces of a linear operator \(A\) caused by a perturbation of this operator by a linear operator \(B\)).NEWLINENEWLINENevertheless, this book may be of some interest to specialists.
scientific article; zbMATH DE number 6022909