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1993 | 105 | 1 | 25-36
Tytuł artykułu

The decomposability of operators relative to two subspaces

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Let M and N be nonzero subspaces of a Hilbert space H satisfying M ∩ N = {0} and M ∨ N = H and let T ∈ ℬ(H). Consider the question: If T leaves each of M and N invariant, respectively, intertwines M and N, does T decompose as a sum of two operators with the same property and each of which, in addition, annihilates one of the subspaces? If the angle between M and N is positive the answer is affirmative. If the angle is zero, the answer is still affirmative for finite rank operators but there are even trace class operators for which it is negative. An application gives an alternative proof that no distance estimate holds for the algebra of operators leaving M and N invariant if the angle is zero, and an analogous result is obtained for the set of operators intertwining M and N.
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autor
  • Department of Mathematics, University of Athens, Panepistimiopolis, 15784 Athens, Greece
  • Department of Mathematics, University of Crete, 71409 Iraklion, Crete, Greece
  • Department of Mathematics, University of Western Australia, Nedlands, Western Australia 6009, Australia
Bibliografia
  • [1] S. Argyros, M. S. Lambrou and W. E. Longstaff, Atomic Boolean subspace lattices and applications to the theory of bases, Mem. Amer. Math. Soc. 445 (1991).
  • [2] W. B. Arveson, Interpolation problems in nest algebras, J. Funct. Anal. 20 (1975), 208-233.
  • [3] W. B. Arveson, Ten Lectures on Operator Algebras, CBMS Regional Conf. Ser. in Math. 55, Amer. Math. Soc., Providence 1984.
  • [4] J. A. Erdos, Operators of finite rank in nest algebras, J. London Math. Soc. 43 (1968), 391-397.
  • [5] J. A. Erdos, Reflexivity for subspace maps and linear spaces of operators, Proc. London Math. Soc. (3) 52 (1986), 582-600.
  • [6] J. A. Erdos and S. C. Power, Weakly closed ideals of nest algebras, J. Operator Theory 7 (1982), 219-235.
  • [7] P. R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381-389.
  • [8] A. Hopenwasser and R. Moore, Finite rank operators in reflexive operator algebras, J. London Math. Soc. (2) 27 (1983), 331-338.
  • [9] J. Kraus and D. R. Larson, Some applications of a technique for constructing reflexive operator algebras, J. Operator Theory 13 (1985), 227-236.
  • [10] J. Kraus and D. R. Larson, Reflexivity and distance formulae, Proc. London Math. Soc. (3) 53 (1986), 340-356.
  • [11] M. S. Lambrou and W. E. Longstaff, Unit ball density and the operator equation AX = YB, J. Operator Theory 25 (1991), 383-397.
  • [12] E. C. Lance, Cohomology and perturbations of nest algebras, Proc. London Math. Soc. (3) 43 (1981), 334-356.
  • [13] D. R. Larson and W. R. Wogen, Reflexivity properties of T ⊕ 0, J. Funct. Anal. 92 (1990), 448-467.
  • [14] C. Laurie and W. E. Longstaff, A note on rank one operators in reflexive algebras, Proc. Amer. Math. Soc. (2) 89 (1983), 293-297.
  • [15] W. E. Longstaff, Strongly reflexive lattices, J. London Math. Soc. (2) 11 (1975), 491-498.
  • [16] W. E. Longstaff, Operators of rank one in reflexive algebras, Canad. J. Math. 28 (1976), 19-23.
  • [17] M. Papadakis, On hyperreflexivity and rank one density for non-CSL algebras, Studia Math. 98 (1991), 11-17.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv105i1p25bwm
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