Componentwise and Cartesian decompositions of linear relations

S. Hassi, H. S. V. de Snoo, and F. H. Szafraniec. Componentwise and Cartesian decompositions of linear relations. 2009. <http://eudml.org/doc/286028>.

@book{S2009,
abstract = {Let A be a, not necessarily closed, linear relation in a Hilbert space ℌ with a multivalued part mul A. An operator B in ℌ with ran B ⊥ mul A** is said to be an operator part of A when A = B +̂ (\{0\} × mul A), where the sum is componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for an operator part are established via the so-called canonical decomposition of A. In addition, conditions are developed for the above decomposition to be orthogonal (components defined in orthogonal subspaces of the underlying space). Such orthogonal decompositions are shown to be valid for several classes of relations. The relation A is said to have a Cartesian decomposition if A = U + iV, where U and V are symmetric relations and the sum is operatorwise. The connection between a Cartesian decomposition of A and the real and imaginary parts of A is investigated.},
author = {S. Hassi, H. S. V. de Snoo, F. H. Szafraniec},
keywords = {linear relation; multivalued linear operator; componentwise decomposition; Cartesian decomposition; orthogonal decomposition; operator part; multivalued part; adjoint relation; closable operator; regular elation; singular relation},
language = {eng},
title = {Componentwise and Cartesian decompositions of linear relations},
url = {http://eudml.org/doc/286028},
year = {2009},
}

TY - BOOK
AU - S. Hassi
AU - H. S. V. de Snoo
AU - F. H. Szafraniec
TI - Componentwise and Cartesian decompositions of linear relations
PY - 2009
AB - Let A be a, not necessarily closed, linear relation in a Hilbert space ℌ with a multivalued part mul A. An operator B in ℌ with ran B ⊥ mul A** is said to be an operator part of A when A = B +̂ ({0} × mul A), where the sum is componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for an operator part are established via the so-called canonical decomposition of A. In addition, conditions are developed for the above decomposition to be orthogonal (components defined in orthogonal subspaces of the underlying space). Such orthogonal decompositions are shown to be valid for several classes of relations. The relation A is said to have a Cartesian decomposition if A = U + iV, where U and V are symmetric relations and the sum is operatorwise. The connection between a Cartesian decomposition of A and the real and imaginary parts of A is investigated.
LA - eng
KW - linear relation; multivalued linear operator; componentwise decomposition; Cartesian decomposition; orthogonal decomposition; operator part; multivalued part; adjoint relation; closable operator; regular elation; singular relation
UR - http://eudml.org/doc/286028
ER -