4. Conclusion

In this case φλ ∈ N λ (defect space associated with λ, [3]).

58 Perturbation Methods with Applications in Science and Engineering

Q α ∘φλ

φλ ∈L<sup>ψ</sup>

� � <sup>¼</sup> λφλ

( (

<sup>N</sup> <sup>λ</sup> <sup>¼</sup> ker <sup>A</sup><sup>∗</sup>

�

⇔

⇔

�

(

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

k¼1

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

<sup>H</sup><sup>Θ</sup> <sup>¼</sup> <sup>f</sup> <sup>þ</sup>X<sup>m</sup>

N λ and N <sup>λ</sup> are unidimensional.

So it suffices to solve the system:

3.2. Case of defect indices mð Þ ; m

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

We assume that α ¼ α and we set

We have

that is,

■

Proof. We know (see [3]) that A<sup>α</sup> has the defect indices 1ð Þ ; 1 if and only if its defect subspaces

Bαφλ ¼ λφλ φλ ∈L<sup>ψ</sup>

α ∘φλ

φλ ∈L<sup>ψ</sup>

φλ ∈L<sup>ψ</sup>

ð Þ α � λ ∘φλ ¼ Θ

<sup>⇔</sup> φλ <sup>¼</sup> ð Þ <sup>α</sup> � <sup>λ</sup> �<sup>1</sup> <sup>∘</sup> <sup>Θ</sup> φλ ∈L<sup>ψ</sup>

μkΘk; f ∈Lψ; μ<sup>k</sup> ∈ C; k ¼ 1;…; m

( )

Bαf ¼ Qð Þ α ∘f f ∈ D Bð Þ<sup>α</sup> , D Bð Þ¼ <sup>α</sup> f ∈L<sup>ψ</sup> : α ∘f ∈ H<sup>Θ</sup>

<sup>A</sup><sup>α</sup> <sup>¼</sup> <sup>B</sup><sup>∗</sup>

� � <sup>¼</sup> λφλ <sup>þ</sup> <sup>μ</sup>Θ, <sup>μ</sup><sup>∈</sup> <sup>C</sup>

:

� � (40)

<sup>α</sup>: (41)

: (39)

<sup>α</sup> � <sup>λ</sup><sup>I</sup> � � <sup>¼</sup> kerð Þ <sup>B</sup><sup>α</sup> � <sup>λ</sup><sup>I</sup> :

We have seen the interest of multiplication operators in reducing Carleman integral operators and how they simplify the spectral study of these operators with some perturbation. In the same way, we can easily generalize this perturbation theory to the case of the non-densely defined Carleman operators:

$$\begin{aligned} H(\mathbf{x}, \mathbf{y}) &= \mathbf{K}(\mathbf{x}, \mathbf{y}) + \sum\_{j=1}^{m} b\_{j} \psi\_{j}(\mathbf{x}) \varphi\_{j}(\mathbf{y}), \\ &\quad \left( \varphi\_{j} \in \mathbf{L}^{2}(\mathbf{X}, \mu), \psi\_{j} \notin \mathbf{L}^{2}(\mathbf{X}, \mu), j = \overline{1, m} \right), \end{aligned} \tag{43}$$

with K xð Þ ; y a Carleman kernel.

It should be noted that this study allows the estimation of random variables.
