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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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