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Stable invariant subspaces for operators on Hilbert space

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Conway J., Hadwin D.

Annales Polonici Mathematici

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1997

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tom 66

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nr 1

49-61

EN

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.

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Strictly singular operators and the invariant subspace problem

100%

Read C.

Studia Mathematica

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1999

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tom 132

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nr 3

203-226

EN

Properties of strictly singular operators have recently become of topical interest because the work of Gowers and Maurey in [GM1] and [GM2] gives (among many other brilliant and surprising results, such as those in [G1] and [G2]) Banach spaces on which every continuous operator is of form λ I + S, where S is strictly singular. So if strictly singular operators had invariant subspaces, such spaces would have the property that all operators on them had invariant subspaces. However, in this paper we exhibit examples of strictly singular operators without nontrivial closed invariant subspaces. So, though it may be true that operators on the spaces of Gowers and Maurey have invariant subspaces, yet this cannot be because of a general result about strictly singular operators. The general assertion about strictly singular operators is false.

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On the reflexivity of multigenerator algebras

53%

Ptak M.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS 1. Introduction...................................................................................................5 2. N-tuples of linear transformations in finite-dimensional space......................8 3. Toeplitz operators on the polydisc and the unit ball....................................18 4. Subspaces of weighted shifts.....................................................................23 5. Joint spectra for N-tuples of operators........................................................27 6. Algebras of operator weighted shifts...........................................................30 7. Functional calculus for N-tuples of contractions..........................................37 8. Dual algebras, invariant subspace problem and reflexivity..........................44 9. Reflexivity of jointly quasinormal operators and spherical isometries..........45 10. Reflexivity and existence of invariant subspaces......................................47 11. Questions and open problems..................................................................56 References.....................................................................................................58

Ograniczanie wyników

1 Annales Polonici Mathematici

1 Studia Mathematica

1 Conway J.

1 Hadwin D.

1 Ptak M.

1 Read C.

1 1999

1 1998

1 1997

Stable invariant subspaces for operators on Hilbert space

Strictly singular operators and the invariant subspace problem

On the reflexivity of multigenerator algebras