*(English)*Zbl 1093.47045

The author of this paper deals with a $2\times 2$ symmetric system of the differential operators of Dirac type $A{\left(x\right)}^{-1}[J\frac{d}{dx}+B\left(x\right)]$ in $\mathbb{R}$, where $J=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$, and $A\left(x\right)>0$ and $B\left(x\right)$ are $2\times 2$ Hermitian matrices with elements being real-valued, locally integrable functions in $\mathbb{R}$. It is considered as an operator in the Hilbert space ${L}_{A}(\mathbb{R};{\u2102}^{2})$ of the ${\u2102}^{2}$-valued measurable functions in $\mathbb{R}$ with the inner product $(y,z)={\int}_{\mathbb{R}}{(A\left(x\right)y\left(x\right),z\left(x\right))}_{{\u2102}^{2}}\phantom{\rule{0.166667em}{0ex}}dx$.

The main purpose of the present paper is, for the corresponding minimal operator with deficiency indices $(2,2)$, to investigate all its maximal dissipative, selfadjoint, and/or other extensions in terms of boundary conditions at infinity. The dilation theory is also employed.

##### MSC:

47E05 | Ordinary differential operators |

34L40 | Particular ordinary differential operators |

47A20 | Dilations, extensions and compressions of linear operators |

47A45 | Canonical models for contractions and nonselfadjoint operators |

47B25 | Symmetric and selfadjoint operators (unbounded) |

47B44 | Accretive operators, dissipative operators, etc. (linear) |