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ArticleOriginal scientific text

Title

The decomposability of operators relative to two subspaces

Authors 1, 2, 3

Affiliations

  1. Department of Mathematics, University of Athens, Panepistimiopolis, 15784 Athens, Greece
  2. Department of Mathematics, University of Crete, 71409 Iraklion, Crete, Greece
  3. Department of Mathematics, University of Western Australia, Nedlands, Western Australia 6009, Australia

Abstract

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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Studia Mathematica
1993/105/1
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Pages:
25-36
Main language of publication
English
Received
1991-06-27
Published
1993
Exact and natural sciences