Proper subspaces and compatibility

Esteban Andruchow, Eduardo Chiumiento, and María Eugenia Di Iorio y Lucero. "Proper subspaces and compatibility." Studia Mathematica 231.3 (2015): 195-218. <http://eudml.org/doc/285385>.

@article{EstebanAndruchow2015,
abstract = {Let 𝓔 be a Banach space contained in a Hilbert space 𝓛. Assume that the inclusion is continuous with dense range. Following the terminology of Gohberg and Zambickiĭ, we say that a bounded operator on 𝓔 is a proper operator if it admits an adjoint with respect to the inner product of 𝓛. A proper operator which is self-adjoint with respect to the inner product of 𝓛 is called symmetrizable. By a proper subspace 𝓢 we mean a closed subspace of 𝓔 which is the range of a proper projection. Furthermore, if there exists a symmetrizable projection onto 𝓢, then 𝓢 belongs to a well-known class of subspaces called compatible subspaces. We find equivalent conditions to describe proper subspaces. Then we prove a necessary and sufficient condition for a proper subspace to be compatible. The existence of non-compatible proper subspaces is related to spectral properties of symmetrizable operators. Each proper subspace 𝓢 has a supplement 𝒯 which is also a proper subspace. We give a characterization of the compatibility of both subspaces 𝓢 and 𝒯. Several examples are provided that illustrate different situations between proper and compatible subspaces.},
author = {Esteban Andruchow, Eduardo Chiumiento, María Eugenia Di Iorio y Lucero},
journal = {Studia Mathematica},
keywords = {projection; compatible subspace; proper operator},
language = {eng},
number = {3},
pages = {195-218},
title = {Proper subspaces and compatibility},
url = {http://eudml.org/doc/285385},
volume = {231},
year = {2015},
}

TY - JOUR
AU - Esteban Andruchow
AU - Eduardo Chiumiento
AU - María Eugenia Di Iorio y Lucero
TI - Proper subspaces and compatibility
JO - Studia Mathematica
PY - 2015
VL - 231
IS - 3
SP - 195
EP - 218
AB - Let 𝓔 be a Banach space contained in a Hilbert space 𝓛. Assume that the inclusion is continuous with dense range. Following the terminology of Gohberg and Zambickiĭ, we say that a bounded operator on 𝓔 is a proper operator if it admits an adjoint with respect to the inner product of 𝓛. A proper operator which is self-adjoint with respect to the inner product of 𝓛 is called symmetrizable. By a proper subspace 𝓢 we mean a closed subspace of 𝓔 which is the range of a proper projection. Furthermore, if there exists a symmetrizable projection onto 𝓢, then 𝓢 belongs to a well-known class of subspaces called compatible subspaces. We find equivalent conditions to describe proper subspaces. Then we prove a necessary and sufficient condition for a proper subspace to be compatible. The existence of non-compatible proper subspaces is related to spectral properties of symmetrizable operators. Each proper subspace 𝓢 has a supplement 𝒯 which is also a proper subspace. We give a characterization of the compatibility of both subspaces 𝓢 and 𝒯. Several examples are provided that illustrate different situations between proper and compatible subspaces.
LA - eng
KW - projection; compatible subspace; proper operator
UR - http://eudml.org/doc/285385
ER -