Hassi, S.; De Snoo, H. S. V.; Szafraniec, F. H. Componentwise and Cartesian decompositions of linear relations. (English) Zbl 1225.47004 Diss. Math. 465, 59 p. (2009). A linear relation (multivalued linear operator) in a Hilbert space \(\mathfrak{H}\) is a linear subspace in the product space \(\mathfrak{H} \times \mathfrak{H}\). The multivalued part \(\text{mul\,} A\) of a linear relation \(A\) is defined as \(\{g \in \mathfrak{H}: \{0,g\} \in A\}\). If the operator \(J\) on \(\mathfrak{H} \times \mathfrak{H}\) is given as \(J\{f,f^{\prime}\} = \{f^{\prime},-f\}\), then the adjoint relation \(A^*\) is defined by \(A^* = JA^{\bot} = (JA)^{\bot}\). The second adjoint \(A^{**}\) is equal to the closure \(\overline{A}\) of \(A.\)An operator \(B\) in \(\mathfrak{H}\) with \(\text{ran\,} B \bot \text{mul\,} A^{**}\) is called an operator part of \(A\) provided that \(A = B \widehat{+} (\{0\} \times A)\), where the sum is componentwise (i.e., span of the graphs). The last relation is said to be a componentwise decomposition of \(A\). Existence and uniqueness results for an operator part are obtained via the so-called canonical decomposition of \(A\). Furthermore, sufficient conditions are presented for the above decomposition to be orthogonal.A Cartesian decomposition of a linear relation \(A\) is defined as \(A = A_1 + iA_2,\) where the relations \(A_1\) and \(A_2\) are symmetric, i.e., \(A_1 \subset A_1^*,\) \(A_2 \subset A_2^*\), and the above sum is operatorwise. The connection between the Cartesian decomposition of \(A\) and its real and imaginary parts is studied. Reviewer: Valerii V. Obukhovskij (Voronezh) Cited in 55 Documents MSC: 47A06 Linear relations (multivalued linear operators) 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A12 Numerical range, numerical radius Keywords:linear relation; multivalued linear operator; componentwise decomposition; Cartesian decomposition; orthogonal decomposition; operator part; multivalued part; adjoint relation; closable operator; regular elation; singular relation PDFBibTeX XMLCite \textit{S. Hassi} et al., Diss. Math. 465, 59 p. (2009; Zbl 1225.47004) Full Text: DOI arXiv