Componentwise and Cartesian decompositions of linear relations

Volume 465 / 2009

S. Hassi, H. S. V. de Snoo, F. H. Szafraniec Dissertationes Mathematicae 465 (2009), 1-59 MSC: Primary 47A05, 47A06; Secondary 47A12. DOI: 10.4064/dm465-0-1


Let $A$ be a, not necessarily closed, linear relation in a Hilbert space $\got H$ with a multivalued part $\mathop{\rm mul} A$. An operator $B$ in $\got H$ with $\mathop{\rm ran} B\perp\mathop{\rm mul} A^{**}$ is said to be an operator part of $A$ when $A=B \mathbin{\widehat{+}} (\{0\}\times \mathop{\rm mul} A)$, 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 $A=U+\mathop{\rm i}V$, 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.


  • S. HassiDepartment of Mathematics and Statistics
    University of Vaasa
    P.O. Box 700
    65101 Vaasa, Finland
  • H. S. V. de SnooDepartment of Mathematics and Computing Science
    University of Groningen
    P.O. Box 407
    9700 AK Groningen, Nederland
  • F. H. SzafraniecInstytut Matematyki
    Uniwersytet Jagielloński
    Łojasiewicza 6
    30-348 Kraków, Poland

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image