Decomposable embeddings, complete trajectories, and invariant subspaces

Ralph deLaubenfels; Phóng Vũ

Studia Mathematica (1996)

  • Volume: 119, Issue: 1, page 65-76
  • ISSN: 0039-3223

Abstract

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We produce closed nontrivial invariant subspaces for closed (possibly unbounded) linear operators, A, on a Banach space, that may be embedded between decomposable operators on spaces with weaker and stronger topologies. We show that this can be done under many conditions on orbits, including when both A and A* have nontrivial non-quasi-analytic complete trajectories, and when both A and A* generate bounded semigroups that are not stable.

How to cite

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deLaubenfels, Ralph, and Vũ, Phóng. "Decomposable embeddings, complete trajectories, and invariant subspaces." Studia Mathematica 119.1 (1996): 65-76. <http://eudml.org/doc/216286>.

@article{deLaubenfels1996,
abstract = {We produce closed nontrivial invariant subspaces for closed (possibly unbounded) linear operators, A, on a Banach space, that may be embedded between decomposable operators on spaces with weaker and stronger topologies. We show that this can be done under many conditions on orbits, including when both A and A* have nontrivial non-quasi-analytic complete trajectories, and when both A and A* generate bounded semigroups that are not stable.},
author = {deLaubenfels, Ralph, Vũ, Phóng},
journal = {Studia Mathematica},
keywords = {closed nontrivial invariant subspaces; decomposable operators; non-quasi-analytic complete trajectories; bounded semigroups},
language = {eng},
number = {1},
pages = {65-76},
title = {Decomposable embeddings, complete trajectories, and invariant subspaces},
url = {http://eudml.org/doc/216286},
volume = {119},
year = {1996},
}

TY - JOUR
AU - deLaubenfels, Ralph
AU - Vũ, Phóng
TI - Decomposable embeddings, complete trajectories, and invariant subspaces
JO - Studia Mathematica
PY - 1996
VL - 119
IS - 1
SP - 65
EP - 76
AB - We produce closed nontrivial invariant subspaces for closed (possibly unbounded) linear operators, A, on a Banach space, that may be embedded between decomposable operators on spaces with weaker and stronger topologies. We show that this can be done under many conditions on orbits, including when both A and A* have nontrivial non-quasi-analytic complete trajectories, and when both A and A* generate bounded semigroups that are not stable.
LA - eng
KW - closed nontrivial invariant subspaces; decomposable operators; non-quasi-analytic complete trajectories; bounded semigroups
UR - http://eudml.org/doc/216286
ER -

References

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  1. [A] A. Atzmon, On the existence of hyperinvariant subspaces, J. Operator Theory 11 (1984), 3-40. Zbl0583.47009
  2. [B] B. Beauzamy, Sous-espaces invariants de type fonctionnel, Acta Math. 144 (1980), 65-82. Zbl0438.47005
  3. [vC] J. A. van Casteren, Generators of Strongly Continuous Semigroups, Res. Notes Math. 115, Pitman, Boston, 1985. Zbl0576.47023
  4. [Co-F] I. Colojoăra and C. Foiaş, Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968. 
  5. [Da] E. B. Davies, One-Parameter Semigroups, London Math. Soc. Monographs 15, Academic Press, 1980. Zbl0457.47030
  6. [dL] R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations, Lecture Notes in Math. 1570, Springer, 1994. Zbl0811.47034
  7. [dL-Ka] R. deLaubenfels and S. Kantorovitz, Laplace and Laplace-Stieltjes spaces, J. Funct. Anal. 116 (1993), 1-61. Zbl0795.47026
  8. [Du-S] N. Dunford and J. T. Schwartz, Linear Operators, Part III, Interscience, New York, 1971. 
  9. [E-W] I. Erdelyi and S. Wang, A Local Spectral Theory for Closed Operators, London Math. Soc. Lecture Note Ser. 105, Cambridge Univ. Press, 1985. Zbl0577.47035
  10. [G] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Univ. Press, New York, 1985. Zbl0592.47034
  11. [H] S. Huang, Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems, Studia Math. 116 (1995), 23-41. Zbl0862.45014
  12. [Ka] S. Kantorovitz, The Hille-Yosida space of an arbitrary operator, J. Math. Anal. Appl. 136 (1988), 107-111. Zbl0675.47033
  13. [Kr-Lap-Cv] S. G. Krein, G. I. Laptev and G. A. Cvetkova, On Hadamard correctness of the Cauchy problem for the equation of evolution, Soviet Math. Dokl. 11 (1970), 763-766. 
  14. [Lan-W] R. Lange and S. Wang, New Approaches in Spectral Decomposition, Contemp. Math. 128, Amer. Math. Soc., Providence, 1992. Zbl0765.47009
  15. [Lau-Ne] K. B. Laursen and M. M. Neumann, Asymptotic intertwining and spectral inclusions on Banach spaces, Czechoslovak Math. J. 43 (1993), 483-497. Zbl0806.47001
  16. [Lyu-Mat] Yu. I. Lyubich and V. I. Matsaev, On operators with separable spectrum, Math. USSR-Sb. 56 (1962), 433-468. 
  17. [Mar] E. Marschall, On the functional calculus of non-quasianalytic groups of operators and cosine functions, Rend. Circ. Mat. Palermo 35 (1986), 58-81. Zbl0656.47032
  18. [Na] R. Nagel (ed.), One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184, Springer, 1986. Zbl0585.47030
  19. [Ne] M. M. Neumann, Local spectral theory for operators on Banach spaces and applications to convolution operators on group algebras, in: Louisiana State Univ. Seminar Notes in Funct. Anal. and PDEs, 1993-94, 287-308. 
  20. [P] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. Zbl0516.47023
  21. [Va] F. H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, D. Reidel, Bucharest-Dordrecht, 1982. 
  22. [Vũ1] Vũ Quôc Phóng, On the spectrum, complete trajectories, and asymptotic stability of linear semi-dynamical systems, J. Differential Equations 105 (1993), 30-45. 
  23. [Vũ2] Vũ Quôc Phóng, Semigroups with nonquasianalytic growth, Studia Math. 104 (1993), 229-241. 

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