##
**Theory of representations of Banach algebras, and abelian groups and semigroups in the spectral analysis of linear operators.**
*(Russian)*
Zbl 1098.47005

Sovrem. Mat., Fundam. Napravl. 9, 3-151 (2004); translation in J. Math. Sci., New York 137, No. 4, 4885-5036 (2006).

This survey paper is based on 28 articles published by the author (four of them were written jointly with K. I.Tchernyshov) in the years 1970–2002 and his doctoral dissertation from 1987. The paper is devoted to the spectral theory of Banach modules over commutative Banach algebras and its applications to the spectral theory of linear operators on Banach spaces. As a borderline discipline, the spectral theory of Banach modules inherited in a natural way many problems from adjacent theories, i.e., problems of analysis and synthesis (from Banach algebras, function theory, and operator theory), ergodic theorems (from the theory of semigroups of linear operators), classical problems of harmonic analysis, etc.

The main results of the present paper are obtained by means of the regular representation associated with a considered Banach module and a partition of unity in a spectrally regular algebra. The paper consists of an introduction and six chapters. In the first chapter, “Spectral theory of Banach modules”, the authors give a classification of well-known spectra, such as the Beurling spectrum, the thin spectrum, the approximate point spectrum, and the Taylor spectrum, based on the axiomatic definition of spectrum in a Banach module. They introduce a class of spectrally regular Banach algebras, which contains the regular Banach algebras defined by Shilov, and develop a theory analogous to the theory of Shilov algebras. Then they investigate Banach modules over spectrally regular Banach algebras. In particular, it is proved that all known spectra coincide on such modules. Finally, the theory of Banach modules is used to study some classes of decomposable operators.

The second chapter, “Abstract ergodic theorems”, contains general ergodic theorems for Banach modules, which are generalizations of well-known classical results from the theory of semigroups of linear operators. Some criteria for complementability of subspaces of a Banach space are obtained. In the third chapter, “Harmonic analysis in Banach modules over spectrally regular algebras”, firstly the primary submodules and elements with a singleton spectrum are studied. Then there are proved theorems on spectra of operators of regular representations and relations between the norm and the spectral radius of such an operator obtained. Based on these results, a new Ditkin type condition for group algebras and an alternative theorem on convergence of Fourier series of almost periodic functions are obtained. These results have many applications in the perturbation theory of linear operators.

Next, problems of analysis and synthesis in Banach modules are investigated. This general approach allows the author to give a unified treatment of many known results. In particular, the theorems of Shilov on synthesis of ideals and the spectral criterion of almost periodicity of bounded functions of Loomis are special cases of this theory. It is defined and investigated a class of spectrally stable Banach algebras. The obtained results are applied to investigation of properties of solutions of functional equations with “constant” coefficients. In particular, almost periodicity of such solutions is studied. The authors also present some applications to approximation theory and the spectral theory of linear operators.

In the fourth chapter, “Estimates of elements of invertible matrices and spectral analysis of linear operators”, applications of some estimates of norms of elements of matrices of inverses of certain operators to the spectral analysis of linear operators are presented. In particular, there are given conditions sufficient for the decomposability (in the sense of Foiaş) of linear operators and estimates of Fourier coefficients of eigenvectors of linear operators. The fifth chapter, “Spectral analysis of linear relations and degenerate semigroups of operators”, is concerned with studying some problems of the spectral theory of linear relations and constructing solutions of linear differential inclusions by means of degenerate semigroups of linear bounded operators. The final sixth chapter, “On the uniform injectivity of linear differential operators”, is devoted to studying properties of some linear differential operators generating a strongly continuous semigroup of operators. Most of the obtained results can be applied to the operator \(-d/dt+A(t)\), where \(A\in C({\mathbb R}_+,\text{End}\,X)\) and \(\text{End}\,X\) denotes the Banach algebra of all bounded linear operators on a complex Banach space \(X\).

The main results of the present paper are obtained by means of the regular representation associated with a considered Banach module and a partition of unity in a spectrally regular algebra. The paper consists of an introduction and six chapters. In the first chapter, “Spectral theory of Banach modules”, the authors give a classification of well-known spectra, such as the Beurling spectrum, the thin spectrum, the approximate point spectrum, and the Taylor spectrum, based on the axiomatic definition of spectrum in a Banach module. They introduce a class of spectrally regular Banach algebras, which contains the regular Banach algebras defined by Shilov, and develop a theory analogous to the theory of Shilov algebras. Then they investigate Banach modules over spectrally regular Banach algebras. In particular, it is proved that all known spectra coincide on such modules. Finally, the theory of Banach modules is used to study some classes of decomposable operators.

The second chapter, “Abstract ergodic theorems”, contains general ergodic theorems for Banach modules, which are generalizations of well-known classical results from the theory of semigroups of linear operators. Some criteria for complementability of subspaces of a Banach space are obtained. In the third chapter, “Harmonic analysis in Banach modules over spectrally regular algebras”, firstly the primary submodules and elements with a singleton spectrum are studied. Then there are proved theorems on spectra of operators of regular representations and relations between the norm and the spectral radius of such an operator obtained. Based on these results, a new Ditkin type condition for group algebras and an alternative theorem on convergence of Fourier series of almost periodic functions are obtained. These results have many applications in the perturbation theory of linear operators.

Next, problems of analysis and synthesis in Banach modules are investigated. This general approach allows the author to give a unified treatment of many known results. In particular, the theorems of Shilov on synthesis of ideals and the spectral criterion of almost periodicity of bounded functions of Loomis are special cases of this theory. It is defined and investigated a class of spectrally stable Banach algebras. The obtained results are applied to investigation of properties of solutions of functional equations with “constant” coefficients. In particular, almost periodicity of such solutions is studied. The authors also present some applications to approximation theory and the spectral theory of linear operators.

In the fourth chapter, “Estimates of elements of invertible matrices and spectral analysis of linear operators”, applications of some estimates of norms of elements of matrices of inverses of certain operators to the spectral analysis of linear operators are presented. In particular, there are given conditions sufficient for the decomposability (in the sense of Foiaş) of linear operators and estimates of Fourier coefficients of eigenvectors of linear operators. The fifth chapter, “Spectral analysis of linear relations and degenerate semigroups of operators”, is concerned with studying some problems of the spectral theory of linear relations and constructing solutions of linear differential inclusions by means of degenerate semigroups of linear bounded operators. The final sixth chapter, “On the uniform injectivity of linear differential operators”, is devoted to studying properties of some linear differential operators generating a strongly continuous semigroup of operators. Most of the obtained results can be applied to the operator \(-d/dt+A(t)\), where \(A\in C({\mathbb R}_+,\text{End}\,X)\) and \(\text{End}\,X\) denotes the Banach algebra of all bounded linear operators on a complex Banach space \(X\).

Reviewer: Andrzej Sołtysiak (Poznań)

### MSC:

47A10 | Spectrum, resolvent |

46J25 | Representations of commutative topological algebras |

47A06 | Linear relations (multivalued linear operators) |

47D03 | Groups and semigroups of linear operators |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

01A60 | History of mathematics in the 20th century |

01A70 | Biographies, obituaries, personalia, bibliographies |