3.2. Case of defect indices mð Þ ; m

In this section, we give the generalization for the case of defect indices ð Þ m; m , m > 1: Let Θ1, Θ2, …, Θm, m be formal elements not belonging to Lψ, and let

$$\mathcal{H}\_{\Theta} = \left\{ f + \sum\_{k=1}^{m} \mu\_k \Theta\_k, \quad f \in \mathcal{L}\_{\psi}, \mu\_k \in \mathbb{C}, \quad k = 1, \ldots, m \right\}. \tag{39}$$

We consider the operator B<sup>α</sup> defined by

$$\begin{aligned} B\_{\alpha}f &= Q(\alpha \circ f) & f \in D(B\_{\alpha}), \\ D(B\_{\alpha}) &= \left\{ f \in \mathcal{L}\_{\psi} : \alpha \circ f \in \mathcal{H}\_{\Theta} \right\} \end{aligned} \tag{40}$$

We assume that α ¼ α and we set

$$A\_{\alpha} = B\_{\alpha}^\*. \tag{41}$$

By analogy to the case of defect indices 1ð Þ ; 1 , we also have the following:

Theorem 13. The operator B<sup>α</sup> is densely defined and closed.

Theorem 14. The operator A<sup>α</sup> admits defect indices mð Þ ; m if and only if

$$
\varphi\_{\lambda}^{(k)} = (\alpha - \lambda) \bullet \Theta\_k \in \mathcal{L}\_{\psi}, k = 1, \dots, m. \tag{42}
$$

In this case, the functions φð Þ<sup>k</sup> <sup>λ</sup> ð Þ k ¼ 1;…; m are linearly independent and generate the defect space N λ.
