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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.

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