Closedness and normal solvability of an operator generated by a degenerate linear differential equation with variable coefficients

Abstract

For a linear operator $\mathcal{D}:\mathbb{W}_2^F \subset \mathbb{L}_2^n \to \mathbb{L}_2^m \times \mathbb{R}^m$ generated by the differential equation

$$ \frac{d} {{dt}}Fx(t) - C(t)x = f(t), Fx(t_0 ) = f_0 , $$

we prove that its graph is closed and determine the adjoint operator $\mathcal{D}*:\mathbb{W}_2^{F'} \subset \mathbb{L}_2^m \times \mathbb{R}^m \to \mathbb{L}_2^n$. For elements of the linear manifolds $\mathbb{W}_2^F$ and $\mathbb{W}_2^{F'}$, we propose an analog of the formula of integration by parts. We establish a criterion for the existence of a pseudosolution of the operator equation $\mathcal{D}x( \cdot ) = (f)( \cdot ),f_0 )$ and formulate sufficient conditions for the normal solvability of the operator $\mathcal{D}$ in terms of relations for blocks of the matrix C(t). The results obtained are illustrated by examples.