Domain and Range Descriptions for Adjoint Relations, and Parallel Sums and Differences of Forms
Seppo Hassi, Zoltán Sebestyén, Henk S. V. de Snoo · Birkhäuser Basel eBooks · 2009
The adjoint of a linear operator or relation from a Hilbert space $$ \mathfrak{H} $$ to a Hilbert space $$ \mathfrak{K} $$ is a closed linear relation. The domain and the range of the adjoint are characterized in terms of certain mappings defined on K and $$ \mathfrak{H} $$ , respectively. These characterizations are applied to contractions between Hilbert spaces and to the form domains and ranges of the Friedrichs and Kreĭn-von Neumann extensions of a nonnegative operator or relation. Furthermore these characterizations are used to introduce and derive properties of the parallel sum and the parallel difference of a pair of forms on a linear space.