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Title

Generalized eigenfunction expansions and spectral decompositions

Authors 1

Affiliations

  1. Department of Mathematics, University of California, Riverside, California 92521, U.S.A.

Abstract

The paper relates several generalized eigenfunction expansions to classical spectral decomposition properties. From this perspective one explains some recent results concerning the classes of decomposable and generalized scalar operators. In particular a universal dilation theory and two different functional models for related classes of operators are presented.

Bibliography

  1. Agler, J. (1985). Rational dilation in an annulus, Ann. of Math. 121, 537-564.
  2. Albrecht, E. (1978). On two questions of Colojoarǎ and Foiaş, Manuscripta Math. 25, 1-15.
  3. Albrecht, E. and Eschmeier, J. (1987). Functional models and local spectral theory, preprint.
  4. Berezanskiĭ, Yu. M. (1968). Expansions in Eigenfunctions of Selfadjoint Operators, Amer. Math. Soc. Transl., Providence, R.I.
  5. Bishop, E. (1959). A duality theorem for an arbitrary operator, Pacific J. Math. 9, 379-394.
  6. Colojoarǎ, I. and Foiaş, C. (1968). Theory of Generalized Scalar Operators, Gordon and Breach, New York.
  7. Dirac, P. A. M. (1930). The Principles of Quantum Mechanics, Clarendon Press, Oxford.
  8. Douglas, R. G. and Foiaş, C. (1976). A homological view in dilation theory, preprint.
  9. Douglas, R. G. and Paulsen, V. (1989). Hilbert modules over function algebras, Pitman Res. Notes in Math. 219, Harlow.
  10. Dunford, N. and Schwartz, J., Linear Operators, p. I. (1958), p. II. (1963), p. III. (1971), Wiley-Interscience, New York.
  11. Dynkin, E. M. (1972). Functional calculus based on Cauchy-Green's formula, in: Research in linear operators and function theory. III, Leningrad Otdel. Mat. Inst. Steklov, 33-39 (in Russian).
  12. Eschmeier, J. (1985). Spectral decompositions and decomposable multipliers, Manuscripta Math. 51, 201-224.
  13. Eschmeier, J. and Prunaru, B. (1990). Invariant subspaces for operators with Bishop's property (β) and thick spectrum, J. Funct. Anal. 94, 196-222.
  14. Eschmeier, J. and Putinar, M. (1984). Spectral theory and sheaf theory. III, J. Reine Angew. Math. 354, 150-163.
  15. Eschmeier, J. and Putinar, M. (1988). Bishop's condition (β) and rich extensions of linear operators, Indiana Univ. Math. J. 37, 325-348.
  16. Eschmeier, J. and Putinar, M. (1989). On quotients and restrictions of generalized scalar operators, J. Funct. Anal. 84, 115-134.
  17. Eschmeier, J. and Putinar, M. (1996). Spectral Decompositions and Analytic Sheaves, Oxford University Press, Oxford.
  18. Foiaş, C. (1963). Spectral maximal spaces and decomposable operators in Banach spaces, Arch. Math. (Basel) 14, 341-349.
  19. Frunzǎ, S. (1975). The Taylor spectrum and spectral decompositions, J. Funct. Anal. 19, 390-421.
  20. Gamelin, T. (1970). Localization of the corona problem, Pacific J. Math. 34, 73-81.
  21. Garnett, J. B. (1981). Bounded Analytic Functions, Academic Press, New York.
  22. Gohberg, I. and Krein, M. G. (1969). Introduction to the Theory of Linear Non-selfadjoint Operators, Transl. Math. Monographs 18, Amer. Math. Soc., Providence, R.I.
  23. Henkin, G. and Leiterer, J. (1984). Theory of Functions on Complex Manifolds, Birkhäuser, Basel-Boston-Stuttgart.
  24. Hörmander, L. (1965). L2-estimates and existence theorems for the ∂¯-operator, Acta Math. 113, 89-152.
  25. Lange, R. (1981). A purely analytic criterion for a decomposable operator, Glasgow Math. J. 21, 69-70.
  26. Levy, R. N. (1987). The Riemann-Roch theorem for complex spaces, Acta Math. 158, 149-188.
  27. Lyubich, Yu. I. and Matsaev, V. I. (1962). Operators with separable spectrum, Mat. Sb. 56, 433-468 (in Russian).
  28. MacLane, S. (1963). Homology, Springer, Berlin.
  29. Martin, M. and Putinar, M. (1989). Lectures on Hyponormal Operators, Birkhäuser, Basel.
  30. Putinar, M. (1983). Spectral theory and sheaf theory. I, in: Dilation Theory, Toeplitz Operators and Other Topics, Birkhäuser, Basel.
  31. Putinar, M. (1985). Hyponormal operators are subscalar, J. Operator Theory 12, 385-395.
  32. Putinar, M. (1986). Spectral theory and sheaf theory. II, Math. Z. 192, 473-490.
  33. Putinar, M. (1990). Spectral theory and sheaf theory. IV, in: Proc. Sympos. Pure Math. 51, part 2, Amer. Math. Soc., 273-293.
  34. Putinar, M. (1992). Quasi-similarity of tuples with Bishop's property (β), Integral Equations Operator Theory 15, 1040-1052.
  35. Radjabalipour, M. (1978). Decomposable operators, Bull. Iranian Math. Soc. 9, 1-49.
  36. Schwartz, L. (1955). Division par une fonction holomorphe sur une variété analytique complexe, Summa Brasil. Math. 3, 181-209.
  37. Sz.-Nagy, B. and Foiaş, C. (1967). Analyse harmonique des opérateurs de l'espace de Hilbert, Akad. Kiadó, Budapest.
  38. Taylor, J. L. (1970). A joint spectrum for several commuting operators, J. Funct. Anal. 6, 172-191.
  39. Taylor, J. L. (1972a). Homology and cohomology for topological algebras, Adv. in Math. 9, 147-182.
  40. Taylor, J. L. (1972b). A general framework for a multioperator functional calculus, Adv. in Math. 9, 184-252.
  41. Vasilescu, F. H. (1982). Analytic Functional Calculus and Spectral Decompositions, Reidel, Dordrecht.
Banach Center Publications
1997/38/1
Export to PBN, Opens in new tab
Pages:
265-286
Main language of publication
English
Published
1997
Exact and natural sciences