Singular decomposition of a differential operator on a semiaxis (Q1335954)

Let \(A\) be an \(n\times n\)-matrix and let \(M_ 1\) be a \(k\times n\)-matrix \((1\leq k\leq n-1)\) with orthonormal rows: \(M_ 1 M^*_ 1= I_ k\). Define the operator \(T\) in the space \(L_ 2(\mathbb{R}_ +)\) by \[ D(T)= \{x\in W^ 1_ 2(\mathbb{R}_ +);\;M_ 1 x(0)= 0\},\;Tx(t)= \textstyle{{dx\over dt}} (t)- Ax(t),\;t\in \mathbb{R}_ +. \] The author proves that \(T\) can be represented as a product of three operators: an isometry, a diagonal nonnegative definite operator, and one more isometry.