The decomposability of operators relative to two subspaces

Studia Mathematica (1993)

• Volume: 105, Issue: 1, page 25-36
• ISSN: 0039-3223

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Abstract

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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.

How to cite

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Katavolos, A., Lambrou, M., and Longstaff, W.. "The decomposability of operators relative to two subspaces." Studia Mathematica 105.1 (1993): 25-36. <http://eudml.org/doc/215981>.

@article{Katavolos1993,
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.},
author = {Katavolos, A., Lambrou, M., Longstaff, W.},
journal = {Studia Mathematica},
keywords = {decomposability of operators relative to two subspaces; distance estimate; algebra of operators},
language = {eng},
number = {1},
pages = {25-36},
title = {The decomposability of operators relative to two subspaces},
url = {http://eudml.org/doc/215981},
volume = {105},
year = {1993},
}

TY - JOUR
AU - Katavolos, A.
AU - Lambrou, M.
AU - Longstaff, W.
TI - The decomposability of operators relative to two subspaces
JO - Studia Mathematica
PY - 1993
VL - 105
IS - 1
SP - 25
EP - 36
AB - 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.
LA - eng
KW - decomposability of operators relative to two subspaces; distance estimate; algebra of operators
UR - http://eudml.org/doc/215981
ER -

References

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