Suppose $T$ is a differential operator of order $m$ on $L^2(\mathbb{R})$, that is: $$ \big(Tf\big)(x) = \sum_{k \, \leq \, m} \alpha_k(x) \, f^{(k)}(x) \quad \text{ for all } \quad f \in \mathrm{D}(T) \subset L^2(\mathbb{R}) \: . $$ Deficiency indices $n_+$ and $n_-$ are defined as follows: $$ n_\pm = \dim \ker (T^* \pm \mathrm{i} I) \: . $$ Is is possible for $n_\pm$ to be infinite? And if the aswer is yes, what conditions have to be satisfied in order for $n_\pm < \infty$?
From the definition of the nulity indices, they should be equal to the number of independent solutions of the equation $$ T^* f(x) = \pm \mathrm{i} \, f(x) \: , $$ which is an ODE of order $m$. Therefore $n_\pm$ should be less or equal to $m$, is that right?