Inclusion of the spectrum of two differential operators defined on $L^2[-a,a]$ and $L^2[0, \infty)$

Let $ T$ be the formal operator defined by $ $ Tu:= \sum_{j=0}^{2n} a_j\frac{d^ju}{dx^j}$ $ where $ 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.