References
N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow (1965).
E. Albrecht, “Functionalkalküle in mehreren Veränderlichen für stetige Lineare Operatoren auf Banachräumen,” Manuscripta Math., 14, 1–40 (1974).
E. Albrecht, “Der Spectrale Abbiuldugsatz für nichtanalytische Functionalkalküle in mehreren Veränderlichen,” Manuscripta Math., 14, 263–277 (1974).
E. Albrecht, “An example of a weakly decomposable operator which is not decomposable,” Rev. Roum. Math. Pures Appl., 20, 855–861 (1975).
E. Albrecht, “On two questions of I. Colojoara and C. Foias,” Manuscripta Math., 25, 1–15 (1978).
E. Albrecht, “Spectral decomposition for systems of commuting operators,” Proc. Roy. Irish Acad., A 81, No. 1, 81–98 (1981).
E. Albrecht and S. Frunze, “Non-analytic functional calculi in several variables,” Manuscripta Math., 18, 327–336 (1976).
C. D. Antoni, R. Longo, and L. Zsido, “A spectral mapping theorem for locally compact groups of operators,” Pac. J. Math., 103, 17–24 (1982).
W. Arendt, “Spectral mapping theorems for compact and locally compact Abelian groups of operators, ” Rend. Circ. Mat. Palermo, 29, No. 1, 105–106 (1981).
W. Arendt, “Approximation of degenerate semigroups,” Tübinger Berichte fur Funktionalanalysis, 9, 33–46 (1999/2000).
W. Arendt and C. J. K. Batty, “Almost periodic solutions of first and second order Cauchy problems, ” J. Differential Equations, 137, 363–383 (1997).
W. Arendt, C. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser Verlag, Basel, (2001).
W. Arveson, “On groups of automorphisms of operator algebras,” J. Funct. Anal., 15, 217–243 (1974).
R. E. Atalla, “On the ergodic theory of contractions,” Revista Colombina de Mathematicas, 10, 75–81 (1976).
A. Atzmon, “Operators which are annihilated by analytic functions and invariant subspaces,” Acta Math., 44, No. 1–2, 27–63 (1980).
H. Bart and S. Goldberg, “Characterizations of almost periodic strongly continuous groups and semigroups,” Math. Ann., 236, 105–116 (1978).
B. Basit, “Harmonic analysis and asymptotic behavior of solutions of the abstract Cauchy problem, ” Semigroup Forum, 54, 58–74 (1997).
A. G. Baskakov, “On the Levitan almost periodic functions,” Coll. Stud. Works [in Russian], Voronezh State Univ., Voronezh, 91–94 (1970).
A. G. Baskakov, Some Questions on the Theory of Vector Almost Periodic Functions, Ph.D. thesis, Voronezh (1973).
A. G. Baskakov, “Spectral analysis of the representations of commutative Banach algebras,” Voronež. Gos. Univ. Trudy Naučn.-Issled. Inst. Mat. VGU, 14, 1–6 (1974).
A. G. Baskakov, “Spectral synthesis in commutative strongly regular Banach algebras,” Voronež. Gos. Univ. Trudy Naučn.-Issled. Inst. Mat. VGU, 20, 3–7 (1975).
A. G. Baskakov, On spectral analysis in Banach modules over commutative Banach algebras [in Russian], Preprint, Voronezh (1977).
A. G. Baskakov, “Spectral tests for the almost periodicity of the solutions of functional equations, ” Mat. Zametki, 24, No. 2, 195–206 (1978).
A. G. Baskakov, “Spectral mappings of the Banach modules,” Theory of operator equations, Work Collect. [in Russian], Voronezh, 7–12 (1978).
A. G. Baskakov, “On spectral functors,” School on Operator Theory in Function Spaces [in Russian], Minsk, 17–18 (1978).
A. G. Baskakov, “Locally regular commutative Banach algebras,” Theory of Operator Equations, Work Collect. [in Russian], Voronezh, 16–22 (1979).
A. G. Baskakov, “Inequalities of Bernshtein type in abstract harmonic analysis,” Sib. Math. J., 20, 665–672 (1980).
A. G. Baskakov, Krylov-Bogolyubov substitution in the theory of nonlinear perturbations of linear operators [in Russian], Preprint, Kyiv (1980).
A. G. Baskakov, “General ergodic theorems in Banach modules,” Funct. Anal. Appl., 14, 215–217 (1981).
A. G. Baskakov, “On Ditkin’s condition in certain Beurling algebras,” Sov. Math., 26, No. 1, 1–4 (1982).
A. G. Baskakov, “On complementability of subspaces of Banach spaces,” VII School on Operator Theory in Function Spaces [in Russian], Minsk, 20–21 (1982).
A. G. Baskakov, “Methods of abstract harmonic analysis in the perturbation of linear operators, ” Sib. Math. J., 24, 17–32 (1983).
A. G. Baskakov, “Spectral synthesis in Banach modules over commutative Banach algebras,” Math. Notes, 34, 776–782 (1983).
A. G. Baskakov, “Harmonic analysis of cosine and exponential operator functions,” Sb. Math., 124, No. 1, 68–95 (1984).
A. G. Baskakov, “A theorem on decomposition of an operator, and some related questions in the analytic theory of perturbations,” Math. USSR, Izv., 28, 421–444 (1987).
A. G. Baskakov, Harmonic Analysis of Linear Operators, Doctoral thesis, Kyiv (1987).
A. G. Baskakov, Harmonic Analysis of Linear Operators [in Russian], Voronezh State Univ., Voronezh (1987).
A. G. Baskakov, “Operator ergodic theorems and complementability of subspaces of Banach spaces, ” Sov. Math. (Iz. VUZ), No. 11, 1–14 (1988).
A. G. Baskakov, “Diagonalization of operators and complementability of subspaces of Banach spaces, ” Ukr. Math. J., 42, No. 7, 763–768 (1991).
A. G. Baskakov, “The Beurling spectrum in the investigation of some classes of Banach algebras, ” Russian Math. Surveys, 49, No. 4, 151–152 (1994).
A. G. Baskakov, “Linear differential operators with unbounded operator coefficients, and semigroups of difference operators,” Math. Notes, 59, No. 5–6, 586–593 (1996).
A. G. Baskakov, “Semigroups of difference operators in the spectral analysis of linear differential operators,” Funct. Anal. Appl., 30, No. 3, 149–157 (1996).
A. G. Baskakov and K. I. Chernyshov, “Ordered pairs of operators and semigroups,” Izv. Ross. Akad. Estestv. Nauk Mat. Mat. Model. Inform. Upr., 2, No. 3, 39–69 (1998).
A. G. Baskakov and K. I. Chernyshov, “Compactness conditions for the spectrum of ordered pairs of linear operators,” Izv. Ross. Akad. Estestv. Nauk Mat. Mat. Model. Inform. Upr., 3, No. 3, 5–24 (1999).
A. G. Baskakov and K. I. Chernyshov, “Construction of the phase space and solutions of linear equations not solved with respect to the derivative,” Dokl. Akad. Nauk, 371, No. 3, 295–298 (2000).
A. G. Baskakov and K. I. Chernyshov, “Spectral analysis of linear relations and degenerate semigroups of operators,” Sb. Math., 193, No. 11-12, 1573–1610 (2002).
C. J. K. Batty, W. Hutter, and F. Rabiger, “Almost periodicity of mild solutions of inhomogeneous periodic Cauchy problems,” J. Differential Equations, 156, 309–327 (1999).
R. Bhatia and P. Rosenthal, “How and why to solve the operator equation AX-XB = Y,” Bull. London Math. Soc., 29, 1–21 (1997).
A. Beurling, “Un theoreme sur les fonctions bornees et uniformement continues sur l’axe reel,” Acta Math., 77, 127–136 (1945).
W. Bloom, “Bernstein’s inequality for locally compact abelian groups,” J. Austral. Math. Soc., 17, No. 1, 88–101 (1974).
S. Bochner, “Beiträge zur Theorie der fastperiodischen Functionen. I Teil,” Math. Ann., 96, 119–147 (1926).
R. Boles Basit, “Generalization of two theorems of M. I. Kadets concerning the indefinite integral of abstract almost periodic functions,” Math. Notes, 9, 181–186 (1971).
B. Bollobas, “The spectral decomposition of compact Hermitian operators on Banach spaces,” Bull. London Math. Soc., 5, No. 1, 29–36 (1973).
C. De Boor, “Dichotomies for band matrices,” SIAM. J. Numer. Anal., 17, 894–907 (1980).
J. Bracic, “Unital strong harmonic commutative Banach algebras,” Studia Math., 149, No. 2. 253–266 (2002).
O. Bratteli and D. Robinson, Operator Algebras and Quantum Statistical Mechanics [Russian translation], Mir, Moscow (1982).
N. Bourbaki, Spectral Theory [Russian translation], Mir, Moscow (1972).
K. I. Chernyshov, On Operator Differential Equations Not Solved with Respect to the Derivative, Ph.D. thesis, Kyiv (1979).
P. Clement, H. Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroups [Russian translation], Mir, Moscow (1992).
I. Colojoara and C. Foias, Theory of Generalized Spectral Operators, Gordon and Breach, New York (1968).
A. Connes, “Une classification des facteurs de type III,” Ann. Sci Ecole Norm. Sup., 6, 133–152 (1973).
B. J. Crabb and J. Duncan, “Some inequalities for norm unitaries in Banach algebras,” Proc. Edinburgh Math. Soc., 21, 17–23 (1978).
R. Cross, Multivalued Linear Operators, Marcel Dekker, New York (1998).
Yu. I. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space [in Russian], Nauka, Moscow (1970).
C. Datry and G. Muraz, “Analyse harmonique dans les modules de Banach,” Part I: “ Propriétés générales,” Bull. Sci. Math. Paris, 119, 299–337 (1995), Part II: “Presque-periodicites et ergadicites,” Bull. Sci. Math. Paris, 120, 493–536 (1996).
S. Demko, “Spectral bounds for ∥A −1∥∞,” J. Approxim. Theory., 48, 207–212 (1986).
V. V. Ditkin, “Certain spectral properties of a pencil of linear bounded operators,” Math. Notes, 31, 39–41 (1982).
Y. Domar, “Harmonic analysis based in certain commutative Banach algebras,” Acta Math., 96, 1–66 (1956).
Y. Domar, “Some results on narrow spectral analysis,” Math. Scand., 20, 5–18 (1967).
Y. Domar and L.-A. Lindahl, “Three spectral notions for representations of commutative Banach algebras,” Ann. Inst. Fourier, 25, No. 2, 1–32 (1975).
H. R. Dowson, Spectral Theory of Linear Operators, Academic Press, London (1978).
N. Dunford and J. T. Schwartz, Linear Operators, John Wiley & Sons, New York (1988).
C. F. Dunkl, “Modules over commutative Banach algebras,” Monatsch. Math., 74, 6–14 (1970).
E. M. Dyn’kin “An operator calculus based on the Cauchy-Green formula, and the quasi analyticity of the classes D(h),” Semin. Math., V. A. Steklov Math. Inst., Leningrad, 19, 128–131 (1972).
E. M. Dyn’kin “Theorems of Wiener-Lévy type, and estimates for Wiener-Hopf operators,” Mat. Issled., 8, No. 3(29), 14–25 (1973).
W. F. Eberlein, “Abstract ergodic theorems and weak almost periodic functions,” Trans. Amer. Math. Soc., 67, 217–240 (1949).
W. F. Eberlein, “Mean ergodic flows,” Adv. Math., 21, No. 2, 229–232 (1976).
S. L. `Édel’shtein, “Asymptotic splitting of boundary value problems for abstract differential equations,” Sib. Math. J., 35, No. 6, 1244–1261 (1994).
R. E. Edwards, Functional Analysis. Theory and Applications [Russian translation], Mir, Moscow (1969).
Ed-Dari Elmouloudi, “On the (C, α)-uniform ergodic theorem,” Studia Math., 156, No. 1, 3–13 (2003).
K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, Berlin (2000).
I. Erdely and R. Lange, “Spectral decomposition on Banach spaces,” Lect. Notes Math., 623 (1977).
E. Fasangova, “A Banach algebra approach to the weak spectral mapping theorem for C 0-groups, ” Ulmer Seminare. Functionalanalysis und Differentialgleichungen, 5, 174–181 (2000).
A. Favini and A. Yagi, “Multivalued linear operators and degenerate evolution equations,” Ann. Math. Pura Appl. (4), CLXIII, 353–384 (1993).
A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker, New York (1998).
V. E. Fedorov, “Degenerate strongly continuous semigroups of operators,” St. Petersburg Math. J., 12, No. 3, 471–489 (2001).
K. Feis, Algebra: Rings, Modules and Categories. I [Russian translation], Mir, Moscow (1977).
G. M. Fel’dman, “On isometric representations of locally compact Abelian groups,” Sov. Math. Dokl., 13, 1688–1692 (1972).
G. M. Fel’dman, “Spectral subspaces of a nonquasianalytic operator,” Mat. Fiz. i Funkcional. Anal., No. 3, 81–87 (1972).
G. M. Fel’dman, “The semisimplicity of an algebra generated by an isometric operator,” Funct. Anal. Appl., 8, 182–183 (1974).
H. Flaschka, “Invariant subspaces of abstract multiplication operators,” Indiana Univ. Math. J., 21, No. 3, 413–418 (1971).
C. Foias, “Spectral maximal spaces and decomposable operators in Banach space,” Arc. Math., 14, 341–349 (1963).
S. Frunza, “The Taylor spectrum and spectral decomposition,” J. Funct. Anal., 19, 390–421 (1975).
I. M. Gel’fand, D. A. Rajkov, and G. E. Shilov, Commutative Normed Rings [in Russian], Fizmatgiz, Moscow (1960).
J. E. Gilbert, “On projections of L ∞(G) onto translation-invariant subspaces, ” Proc. London Math. Soc., 19, No. 1. 69–88 (1969).
H. A. Gindler and A. E. Taylor, “The minimum modules of a linear operator and its use in spectral theory,” Studia Math., 22, 15–41 (1962/63).
I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Amer. Math. Soc., Providence (1969).
E. A. Gorin, “Spectral stability of some Banach algebras,” Funct. Anal. Appl., 98, 158–159 (1974).
E. A. Gorin, “On the research of G. E. Shilov in the theory of commutative Banach algebras and their subsequent development,” Russian Math. Surveys, 33, 193–217 (1978).
E. A. Gorin, “Bernstein inequalities from the perspective of operator theory,” Vestnik Khar’kov. Gos. Univ., No. 205, 77–105 (1980).
F. Gramain and Y. Meyer, “Ensembles de frequences et fonctions presque periodiques,” Colloq. Math., 30, No. 2, 269–275 (1974).
F. Greenleaf, Invariant Means on Topological Groups and Their Applications [Russian translation], Mir, Moscow (1973).
V. P. Gurarii, “Harmonic analysis in spaces with weight,” Trudy Mosk. Mat. Obs., 35, 21–76 (1976).
V. P. Gurarii, “Harmonic analysis of functions that are bounded on the right semiaxis and growing on the left semiaxis,” Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 113, 225–230 (1981).
A. E. Hamza and G. Muraz, “Existence of abstract solutions of integro-differential operator equations,” Acta Math. Vietnamica 28, No. 1, 101–109 (2003).
S. Hartman and C. Ryll-Nardzewski, “Über die Spaltung von Fourierreichen fastperiodische Functionen,” Studia Math., 19, 287–295 (1960).
A. Ja. Helemskii, “Annihilator extensions of commutative Banach algebras,” Izv. Akad. Nauk SSSR Ser. Mat., 29, No. 4, 945–956 (1965).
A. Ja. Helemskii, “Singular extensions of the algebra of all continuous functions on a compactum, ” Sib. Mat. Ž., 10, No. 3, 671–684 (1969).
A. Ja. Helemskii, “Homological methods in the holomorphic calculus of several operators in Banach space, after Taylor,” Usp. Mat. Nauk, 36, No. 1(217), 127–172 (1981).
C. S. Herz, “The spectral theory of bounded functions,” Trans. Amer. Math. Soc., 94, 181–232 (1960).
E. Hille and R. Phillips, Functional Analysis and Semigroups [Russian translation], Inostr. Lit., Moscow (1962).
E. Hewitt and K. Ross, Abstract Harmonic Analysis. Vol. I [Russian translation], Nauka, Moscow (1975).
S.-Z. Huang, Spectral Theory for Nonquasianalytic Representations of Locally Compact Abelian Groups, Ph.D. thesis, Tubingen (1995).
S.-Z. Huang, “A spectral theory for locally compact abelian groups of automorphisms of commutative Banach algebras,” Studia Math., 132, No. 1, 37–69 (1999).
L. K. Jones and M. Lin, “Unimodular eigenvalues and weak mixing,” J. Funct. Anal., 35, No. 1, 42–48 (1980).
J.-P. Kahane, Absolutely Convergent Fourier Series [Russian translation], Mir, Moscow (1976).
L V. Kantorovich and G. P. Akilov, Functional Analysis [in Russian], Nauka, Moscow (1977).
D. Henry, Geometric Theory of Semilinear Parabolic Equations [Russian translation], Mir, Moscow (1985).
K. Koh, “On a representation of a strongly harmonic ring by sheaves,” Pac. J. Math., 41, 459–468 (1972).
S. G. Krein (Ed.), Functional Analysis [in Russian], Nauka, Moscow (1972).
S. G. Krein and K. I. Chernyshov, “Singularly perturbed differential equations in a Banach space, ” Ninth International Conference on Nonlinear Oscillations, Vol. 1 [in Russian], Kyiv, 193–197 (1984).
V. G. Kurbatov, Linear Differential-Difference Equations [in Russian], Voronezh State Univ., Voronezh (1990).
R. Lange, “Equivalent conditions for decomposable operators,” Proc. Amer. Math. Soc., 82, No. 3, 401–406 (1981).
K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, Clarendon Press, Oxford (2000).
G. Leaf, “A spectral theory for a class of linear operators,” Pac. J. Math., 13, 141–145 (1963).
I. Lehmann and H.-J. Rossberg, “Elementary generalization of a theorem of Pragmen-Lindelof with application,” Math. Nachr., 70, 87–93 (1975).
B. M. Levitan, “Über eine Verallgemeinerung der Ungleichungen von S. Bernstein und H. Bohr, ” C. R. (Dokl.) Acad. Sci. URSS, n. Ser., 15, 169–172 (1937).
B. M. Levitan, Almost Periodic Functions [in Russian], Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow (1953).
B. M. Levitan and V. V. Žikov, Almost-Periodic Functions and Di.erential Equations [in Russian], Moskov. Gos. Univ., Moscow (1978).
S. Lloyd, “On the mean ergodic theorem of Sine,” Proc. Amer. Math. Soc., 56, 121–126 (1976).
L. H. Loomis, Introduction to Abstract Harmonic Analysis [Russian translation], Inostr. Lit., Moscow (1956).
L. H. Loomis, “Spectral characterization of almost periodic functions,” Ann. Math., 72, No. 2, 362–368 (1960).
V. I. Lomonosov, Some Questions of the Theory of Invariant Subspaces, Ph.D. thesis, Khar’kov (1973).
V. I. Lomonosov, “Joint approximate spectrum of a commutative family of operators,” Teor. Funkts. Funkts. Anal. Prilozhen., No. 32, 39–48 (1979).
V. I. Lomonosov, Yu. I. Lyubich, and V. I. Matsaev, “Duality of spectral subspaces and conditions for the separation of the spectrum of a bounded linear operator,” Sov. Math. Dokl., 15, 878–881 (1974).
Yu. I. Lyubich, “On conditions for the completeness of a system of eigenvectors of a correct operator,” Usp. Mat. Nauk, 18, No. 1(109), 165–171 (1963).
Yu. I. Lyubich, “On one class of operators in Banach spaces,” Usp. Mat. Nauk, 20, No. 6(126), 131–133 (1965).
Yu. I. Lyubich, “On the spectrum of a representation of an abelian topological group,” Sov. Math. Dokl., 12, 1482–1486 (1971).
Yu. I. Lyubich, Introduction to the Theory of Banach Representations of Groups [in Russian], Vishcha Shkola, Khar’kov (1985).
Yu. I. Lyubich, Introduction to the Theory of Banach Representations of Groups, Birkhauser Verlag, Basel (1988).
Yu. I. Lyubich and V. I. Matsaev, “On operators with a separable spectrum,” Am. Math. Soc., Transl., II. Ser., 47, 89–129 (1962).
Yu. I. Lyubich, V. I. Machaev, and G. M. Fel’dman, “On representations with a separable spectrum, ” Funct. Anal. Appl., 7, 129–136 (1973).
A. S. Markus, “A spectral synthesis problem for operators with point spectrum,” Izv. Akad. Nauk SSSR Ser. Mat., 34, 662–688 (1970).
A. S. Markus and V. I. Macaev, “Convergence of expansions in eigenvectors of an operator that is nearly self-adjoint,” Mat. Issled., 61, 104–129 (1981).
E. Marshall, “On the functional calculus of non-quasianalytic groups of operators and cosine functions,” Rend. Circ. Math. Palermo, 35, 58–81 (1986).
V. P. Maslov, Operator Methods [in Russian], Nauka, Moscow (1973).
K. Mattila, “On proper boundary points of the spectrum and complemented eigenspaces,” Math. Scand., 43, 363–368 (1978).
A. E. Mayer, “Graste Polygone mit gegebenen Seitenvectoren,” Comm. Math. Helvetici, 10, 288–301 (1938).
I. V. Mel’nikova and A. V. Gladchenko, “Well-posedness of the Cauchy problem for inclusions in Banach spaces,” Dokl. Math., 58, No. 1, 123–126 (1998).
Y. Meyer, Algebraic Numbers and Harmonic Analysis, North Holland, Amsterdam (1972).
H. Milne, “Banach space properties of uniform algebras,” Bull. London Math. Soc., 4, 323–326 (1972).
H. Mirkil, “A counterexample to discrete spectral synthesis,” Composite Math., 14, 269–273 (1960).
E. Mukhamadiev, “On the inversion of functional operators in a space of functions bounded on the axis,” Math. Notes, 11, 169–172 (1972).
R. J. Nagel, “Mittelergodische Halbgruppen linearer Operatoren,” Ann. Inst. Fourier, 23, No. 4, 75–87 (1973).
R. J. Nagel, One-Parameter Semigroups of Positive Operators, Springer-Verlag, Berlin (1986).
R. J. Nagel and S.-Z. Huang, “Spectral mapping theorems for C 0-groups satisfying non-quasianalytic growth conditions,” Math. Nachr., 169, 207–218 (1994).
B. Nagy, “Operators with spectral decomposition property are decomposable,” Stud. Sci. Math. Hung., 13, No. 3–4, 429–432 (1978).
M. A. Naimark, Normed Rings [in Russian], Nauka, Moscow (1968).
J. Von Neumann, “Uber adjungierte Functionaloperatoren,” Ann. Math. 2, 33, 294–310 (1932).
N. K. Nikol’skii, “Selected problems of weighted approximation and spectral analysis,” Trudy Mat. Inst. Steklov, 120, 1–270 (1974).
N. K. Nikol’skii, “Invariant Subspaces in Operator Theory and Function Theory,” Mathematical Analysis, Vol. 12 [in Russian], Akad. Nauk SSSR Vsesojuz. Inst. Naučn. i Tehn. Informacii, Moscow (1974).
N. K. Nikol’skii, “The current state of the problem of spectral analysis-synthesis. I,” Operator Theory in Function Spaces [in Russian], Nauka Sibirsk. Otdel., Novosibirsk, 240–282 (1977).
D. Olesen, “On norm-continuity and compactness of spectrum,” Math. Scand., 35, 223–236 (1974).
T. W. Palmer, Banach Algebras and the General Theory of *-Algebras, Cambridge Univ. Press, Cambridge (1994, 2002).
A. Pietsch, Operator Ideals [Russian translation], Mir, Moscow (1982).
N. I. Radbel, Ordered Pairs of Linear Operators and the Cauchy Problem for the Equation Aẋ(t)+Bx(t)=0 in Banach Spaces, Ph.D. thesis, Donetsk (1984).
H. Reiter, Classical Harmonic Analysis and Locally Compact Groups, Calderon Press, Oxford (1968).
V. S. Ritsner, Theory of Linear Relations [in Russian], Preprint, Ul’yanovsk (1982).
J. G. Romo, “Spectral synthesis in Banach modules. I,” Tamkang J. Math., 11, No. 1, 91–109 (1980).
J. G. Romo, “Spectral synthesis in Banach modules. II,” Tamkang J. Math., 11, No. 2. 191–201 (1980).
W. Rudin, Fourier Analysis on Groups, Wiley-Interscience, New York (1962).
W. Rudin, Functional Analysis [Russian translation], Mir, Moscow (1975).
A. G. Rutkas, “Cauchy’s problem for the equation Ax′(t)+Bx(t) = ƒ (t),” Differ. Equations, No. 11, 1486–1497 (1975).
R. Sato, “On abstract mean ergodic theorems,” Tohoku Math. J., 30, No. 4, 575–581 (1978).
L. Schwartz, “Theorie generale des fonctions mogenne-periodiques,” Ann. Math., 48, No. 4, 857–929 (1947).
H. Seferoglu, “Spectral mapping theorem for representations of measure algebras,” Proc. Edinburgh Math. Soc., 40, 261–266 (1997).
H. Seferoglu, “A spectral mapping theorem for Banach modules,” Studia Math., 152, No. 2, 99–103 (2003).
M. V. Šeinberg, “Homological properties of closed ideals with a bounded approximate unit, ” Mosc. Univ. Math. Bull., 27, No. 3–4, 103–108 (1973).
G. Shilov, “On regular normed rings,” Trav. Inst. Math. Steklo., 21, 1–118 (1947).
M. A. Shubin, “The Favard-Muhamadiev theory and pseudodifferential operators,” Sov. Math. Dokl., 16, 1646–1649 (1976).
R. Sine, “A mean ergodic theorem,” Proc. Amer. Math. Soc., 24, No. 3, 438–439 (1970).
Z. Slodkowski, “An infinite family of joint spectra,” Studia Math., 61, 239–255 (1977).
Z. Slodkowski and W. Zelazko, “On joint spectra of commuting families of operators,” Studia Math., 48, 83–88 (1973).
Ju. A. Šreider, “Banach functionals and ergodic theorems,” Mat. Zametki, 2, No. 4, 385–394 (1967).
J. G. Stampfli, A local theory for operators; Invariant subspaces,” Indiana Univ. Math. J., 22, No. 2, 159–167 (1972).
G. A. Sviridyuk, “On the general theory of operator semigroups,” Russian Math. Surveys, 49, No. 4, 45–74 (1994).
G. A. Sviridyuk and V. E. Fedorov, “On the identities of analytic semigroups of operators with kernels,” Sib. Math. J., 39, No. 3, 522–533 (1998).
S.-E. Takahasi and J. Inoue, “A spectral mapping theorem for some representations of compact abelian groups,” Proc. Edinburgh Math. Soc., 35, 47–52 (1992).
J. L. Taylor, “A joint spectrum for several commuting operators,” J. Funct. Anal., 6, No. 2, 172–191 (1970).
J. L. Taylor, “The analytic functional calculus for several commuting operators,” Acta Math., 125, 1–38 (1970).
S. Teleman, “Analyse harmonique dans les algebras regulieres,” Rev. Roumaine Math. Pures Appl., 13, 691–750 (1968).
V. M. Tyurin, “Invertibility of linear di.erential operators in certain functional spaces,” Sib. Math. J., 32, No. 3, 485–490 (1991).
J. Wermer, “The existence of invariant subspaces,” Duke Math. J., 19, 615–622 (1952).
K. Yosida, Functional Analysis [Russian translation], Mir, Moscow (1967).
W. Zelazko, “On a certain class of non-removable ideals in Banach algebras,” Studia Math., 44, 87–92 (1972).
W. Zelazko, “An axiomatic approach to joint spectra. I,” Studia Math., 64, 249–261 (1979).
V. V. Žikov “Some questions of admissibility and dichotomy. The averaging principle,” Izv. Akad. Nauk SSSR Ser. Mat., 40, No. 6, 1380–1408 (1976).
L. Zsido, “On spectral subspaces associated to a locally compact abelian group of operators,” Adv. Math., 36, 213–276 (1980).
Author information
Authors and Affiliations
Additional information
__________
Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 9, Functional Analysis, 2004.
Rights and permissions
About this article
Cite this article
Baskakov, A.G. Representation theory for Banach algebras, Abelian groups, and semigroups in the spectral analysis of linear operators. J Math Sci 137, 4885–5036 (2006). https://doi.org/10.1007/s10958-006-0286-4
Issue Date:
DOI: https://doi.org/10.1007/s10958-006-0286-4