Let $Tu:= \sum_{j=0}^{2n} a_j\frac{d^ju}{dx^j}$ with $a_j \in \mathbb{C}$. Consider the differential operators $T_a: D(T_a)\subseteq L^2[-a,a] \to L^2[-a,a]$ and $T_\infty: D(T_\infty)\subseteq L^2[0,\infty) \to L^2[0,\infty)$ defined by $$T_af:=Tf, \ T_bg:=Tg, \ f \in D(T_a), g \in D(T_\infty),$$ where $$D(T_a):=\{ f \in L^2[-a,a] : Tf \in L^2[-a,a], f^{(j)}(-a)=f^{(j)}(a)=0 \mbox{ for } 0 \leq j \leq n-1\}$$ and $$D(T_\infty):=\{ f \in L^2[0,\infty) : Tf \in L^2[0,\infty) , f^{(j)}(0)=0 \mbox{ for } 0 \leq j \leq n-1\}.$$
Can we say that $\sigma(T_a) \subseteq \sigma(T_\infty)$?. I know that the inclusion is true if we take $Tu:=u''$ or $Tu:=-u''-2u'$, for example.
Thanks in advance for any help you are able to provide.