Operational calculus for rows, columns, and blocks of linear relations
This articles deals with so-called rows and columns of linear relations. Let \(\mathcal H_1, \mathcal H_2\) and \(\mathcal H\) be Hilbert spaces and for \(j=1,2\) let \(R_j\) be a linear relation from \(\mathcal H_j\) to \(\mathcal H\) and \(C_j\) a linear relation from \(\mathcal H\) to \(\mathcal H_j\). Then the {\textit row} \((R_1;R_2)\) of \(R_1\) and \(R_2\) is the linear relation from \(\mathcal H_1 \oplus \mathcal H_2\) to \(\mathcal H\) defined by \[ (R_1; R_2) = \left\{ \left\{ \begin{pmatrix} h_1 \\ h_2 \end{pmatrix},\ k_1+k_2\right\}:\{h_1,k_1\}\in R_1,\ \{h_2, k_2\}\in R_2 \right\} \] which corresponds to the componentwise sum of the two relations. The {\textit column} \((C_1;C_2)\) of \(C_1\) and \(C_2\) corresponds to the usual sum of the two relations; it is the linear relation from \(\mathcal H\) to \(\mathcal H_1 \oplus \mathcal H_2\) defined by \[ (C_1; C_2) = \left\{ \left\{ h,\ \begin{pmatrix} k_1 \\ k_2 \end{pmatrix}\right\} \ :\ \{h,k_1\}\in R_1,\ \{h, k_2\}\in R_2 \right\}. \] The authors prove various necessary and sufficient criteria for a linear relation from \(\mathcal H_1 \oplus \mathcal H_2\) to \(\mathcal H\) to be a row and for a linear relation from \(\mathcal H\) to \(\mathcal H_1 \oplus \mathcal H_2\) to be a column. The adjoints of rows and columns of linear relations are calculated. Next the authors deal with \(2\times 2\) block matrices formed by linear relations which can be either viewed as a row of columns or equivalently as a column of rows. They give criteria when a relation between the direct sum of two Hilbert spaces can be written as a \(2\times 2\) block matrix of relations and they show that the adjoint of a block matrix is an extension of the \(2\times 2\) block matrix whose entries are the adjoints of the entries of the original block matrix. Finally, the paper is concerned with the multiplication of rows and columns and with the multiplication of \(2\times 2\) block matrices of linear relations.
scientific article; zbMATH DE number 7225072
scientific article