Stable invariant subspaces for operators on Hilbert space

John B. Conway; Don Hadwin

Annales Polonici Mathematici (1997)

  • Volume: 66, Issue: 1, page 49-61
  • ISSN: 0066-2216

Abstract

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If T is a bounded operator on a separable complex Hilbert space ℋ, an invariant subspace ℳ for T is stable provided that whenever T n is a sequence of operators such that ‖ T n - T ‖ → 0 , there is a sequence of subspaces ℳ n , with ℳ n in L a t T n for all n, such that P ℳ n → P ℳ in the strong operator topology. If the projections converge in norm, ℳ is called a norm stable invariant subspace. This paper characterizes the stable invariant subspaces of the unilateral shift of finite multiplicity and normal operators. It also shows that in these cases the stable invariant subspaces are the strong closure of the norm stable invariant subspaces.

How to cite

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John B. Conway, and Don Hadwin. "Stable invariant subspaces for operators on Hilbert space." Annales Polonici Mathematici 66.1 (1997): 49-61. <http://eudml.org/doc/269992>.

@article{JohnB1997,
abstract = {If T is a bounded operator on a separable complex Hilbert space ℋ, an invariant subspace ℳ for T is stable provided that whenever $\{T_n\}$ is a sequence of operators such that $‖T_n - T‖ → 0$, there is a sequence of subspaces $\{ℳ_n\}$, with $ℳ_n$ in $Lat T_n$ for all n, such that $P_\{ℳ_n\} → P_\{ℳ\}$ in the strong operator topology. If the projections converge in norm, ℳ is called a norm stable invariant subspace. This paper characterizes the stable invariant subspaces of the unilateral shift of finite multiplicity and normal operators. It also shows that in these cases the stable invariant subspaces are the strong closure of the norm stable invariant subspaces.},
author = {John B. Conway, Don Hadwin},
journal = {Annales Polonici Mathematici},
keywords = {invariant subspace; stability; normal operators; strong operator topology; norm stable invariant subspace; unilateral shift of finite multiplicity},
language = {eng},
number = {1},
pages = {49-61},
title = {Stable invariant subspaces for operators on Hilbert space},
url = {http://eudml.org/doc/269992},
volume = {66},
year = {1997},
}

TY - JOUR
AU - John B. Conway
AU - Don Hadwin
TI - Stable invariant subspaces for operators on Hilbert space
JO - Annales Polonici Mathematici
PY - 1997
VL - 66
IS - 1
SP - 49
EP - 61
AB - If T is a bounded operator on a separable complex Hilbert space ℋ, an invariant subspace ℳ for T is stable provided that whenever ${T_n}$ is a sequence of operators such that $‖T_n - T‖ → 0$, there is a sequence of subspaces ${ℳ_n}$, with $ℳ_n$ in $Lat T_n$ for all n, such that $P_{ℳ_n} → P_{ℳ}$ in the strong operator topology. If the projections converge in norm, ℳ is called a norm stable invariant subspace. This paper characterizes the stable invariant subspaces of the unilateral shift of finite multiplicity and normal operators. It also shows that in these cases the stable invariant subspaces are the strong closure of the norm stable invariant subspaces.
LA - eng
KW - invariant subspace; stability; normal operators; strong operator topology; norm stable invariant subspace; unilateral shift of finite multiplicity
UR - http://eudml.org/doc/269992
ER -

References

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  1. [1] G. T. Adams, A nonlinear characterization of stable invariant subspaces, Integral Equations Operator Theory 6 (1983), 473-487. Zbl0562.47004
  2. [2] C. Apostol, L. A. Fialkow, D. A. Herrero, and D. Voiculescu, Approximation of Hilbert Space Operators, Vol. II, Pitman Res. Notes Math. 102, Pitman, Boston, 1984. Zbl0572.47001
  3. [3] C. Apostol, C. Foiaş, and N. Salinas, On stable invariant subspaces, Integral Equations Operator Theory 8 (1985), 721-750. Zbl0616.47005
  4. [4] H. Bart, I. Gohberg, and M. A. Kaashoek, Stable factorizations of monic matrix polynomials and stable invariant subspaces, Integral Equations Operator Theory 1 (1978), 496-517. Zbl0398.47011
  5. [5] S. Campbell and J. Daughtry, The stable solutions of quadratic matrix equations, Proc. Amer. Math. Soc. 74 (1979), 19-23. Zbl0403.15012
  6. [6] J. B. Conway, A Course in Functional Analysis, Springer, New York, 1990. Zbl0706.46003
  7. [7] J. B. Conway and P. R. Halmos, Finite-dimensional points of continuity of Lat, Linear Algebra Appl. 31 (1980), 93-102. Zbl0435.15005
  8. [8] Yu. P. Ginzburg, The factorization of analytic matrix functions, Dokl. Akad. Nauk SSSR 159 (3) (1964), 489-492 (in Russian). 
  9. [9] D. W. Hadwin, An addendum to limsups of lats, Indiana Univ. Math. J. 29 (1980), 313-319. Zbl0457.47010
  10. [10] P. R. Halmos, Limsups of lats, Indiana Univ. Math. J., 293-311. Zbl0404.47003
  11. [11] D. A. Herrero, Inner functions under uniform topology, II, Rev. Un. Mat. Argentina 28 (1976), 23-35. Zbl0296.30029
  12. [12] D. A. Herrero, Approximation of Hilbert Space Operators, I, Pitman, London, 1982. Zbl0494.47001

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