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Componentwise and Cartesian decompositions of linear relations. (English) Zbl 1225.47004
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.

47A06 Linear relations (multivalued linear operators)
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A12 Numerical range, numerical radius
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