On the spectra of first order irregular operator equations

The distribution of the spectrum $\sigma L=P\sigma L\cup C\sigma L\cup R\sigma L$ of the operator $L=L(\mu ,\alpha ,a,A)$ in the complex plane $\mathbb C$ is studied. The operator $L$ is the closure in $H=\mathscr L_2(0,b)\otimes \mathfrak H$ of the operator $t^\alpha aD_t+A$ originally defined on smooth functions $u(t)\colon [0,b]\to \mathfrak H$ satisfying the condition $\mu u(0)-u(b)=0$, where $\alpha \in \mathbb R$, $a\in \mathbb C$, $D_t\equiv d/dt$, $A$ is a model operator in a Hilbert space $\mathfrak H$ and $\mu \in \overline {\mathbb C}$. Conditions (criteria) in terms of the parameter $\alpha$ ensuring that the eigenfunctions of the operator $L\colon H\to H$ make up a complete system, a minimal system, or a (Riesz) basis in the Hilbert space $H$ are obtained.