On eigenfunction expansions of first-order symmetric systems and ordinary differential operators of an odd order

We study general (not necessarily Hamiltonian) first-order symmetric systems $J{y}^{\mathrm{\prime }}-B(t)y=\text{D}(t)f(t)$ on an interval $\text{cI}=[a,b⟩$ with the regular endpoint $a$. It is assumed that the deficiency indices $n_{\pm }(\text{Tmi})$ of the minimal relation $\text{Tmi}$ satisfy $n_{+}(\text{Tmi})<n_{-}(\text{Tmi})$. We define $\text{l}$-depending boundary conditions which are analogs of separated self-adjoint boundary conditions for Hamiltonian systems. With a boundary value problem involving such conditions we associate an exit space self-adjoint extension $\text{wt}T$ of $\text{Tmi}$ and the $m$-function $m(\text{cd})$, which is an analog of the Titchmarsh-Weyl coefficient for the Hamiltonian system. By using $m$-function we obtain the eigenfunction expansion with the spectral function $\text{Si}(\text{cd})$ of the minimally possible dimension and characterize the case when spectrum of $\text{wt}T$ is defined by $\text{Si}(\text{cd})$. Moreover, we parametrize all spectral functions in terms of a Nevanlinna type boundary parameter. Application of these results to ordinary differential operators of an odd order enables us to complete the results by Everitt and Krishna Kumar on the Titchmarsh-Weyl theory of such operators.