In this chapter, we introduce a multiplication operation that allows us to give to the Carleman integral operator of second class the form of a multiplication operator. Also we establish the formal theory of perturbation of such operators.
- Carleman kernel
- defect indices
- integral operator
- formal series
In this chapter, we shall assume that the reader is familiar with the fundamental results and the standard notation of the integral operators theory [1, 2, 3, 5, 6, 8, 9, 10, 11, 12]. Let be an arbitrary set, be a finite measure on (is defined on a algebra of subsets of ; we do not indicate this algebra), and the Hilbert space of square integrable functions with respect to . Instead of writing “measurable,” “almost everywhere,” and “(),” we write “measurable,” “a e,” and “.”
A linear operator : , where the domain is a dense linear manifold in , is said to be integral if there exist a measurable function on , a kernel, such that, for every
A kernel on is said to be Carleman, if for almost every fixed that is to say
where the overbar in (3) denotes the complex conjugation and is an orthonormal sequence in such that
and is a real number sequence verifying
We call a Carleman sequence.
Moreover, we assume that there exist a numeric sequence such that
Moreover, we have
2. Position operator
Let be a fixed Carleman sequence in . It is clear from the foregoing that is not a complete sequence in . We denote by the closure of the linear span of the sequence :
We start this section by defining some formal spaces.
2.1. Formal elements
Definition 1. (see ) We call formal element any expression of the form
where the coefficients are scalars.
The sequence is said to generate the formal element .
Definition 2. We say that is the zero formal element, and we note if for all
We say that two formal elements and are equal if for all .
If is a scalar function defined for each , we set
or in another form,
For example, let
If for all , then the formal element
is called inverse of the formal element .
Furthermore, we define the conjugate of a formal element by
Denote by the set of all formal elements (12).
On we define the following algebraic operations:
and the product
Hence, we obtain a complex vector space structure for .
2.2. Bounded formal elements
Definition 3. A formal element is bounded if its sequence is bounded.
We denote by the set of all bounded formal elements.
It is clear that is a subspace of .
We claim that:
is a subspace of
Furthermore we have the strict inclusions:
We define a linear form on by setting:
with the series converging on the right side of (21).
Proposition 4. The form (21) verifies the properties of scalar product.
Proof. Indeed, let
We have then:
Remark 5. On the scalar product coincides with the scalar product of .
2.3. The multiplication operation
Here, we introduce the crucial tool of our work.
Definition 6. We call multiplication with respect to the Carleman sequence , the operation denoted “” and defined by:
Definition 7. We call position operator in any unitary self-adjoint (see ) operator satisfying
The term “position operator” comes from the fact (as it will be shown in the following theorem) that for the elements of the sequence , the operator acts as operator of change of position of these elements.
2.4. Main results
Theorem 8. A linear operator defined on is a position operator if and only if there exist an involution (i.e., ) of the set such that for all
It is easy to see that if () holds, then is a position operator.
Let be a position operator. According to 1, we can write
On the other hand, we have
The equalities (26) lead to the resolution of the system:
We get then
Let us now consider the following application:
It’s clear that is injective.
Now let Since then
Finally is well defined as involution.
Notation In the sequel will be noted by . We write
Remark 9. The position operator can be extended over as follows:
If , then
3. Carleman operator in
3.1. Case of defect indices
Let ; we introduce the operator defined in by
It is clear that is a Carleman operator induced by the kernel
Moreover, if is self-adjoint.
Now and . We introduce the following set
which verifies the following properties.
Proposition 10. 1. is a subset of .
2. , i.e., direct sum of with
Proof. The first property is easy to establish. We show the uniqueness for the second.
Let and , two formal elements in Then
This last equality is verified only if Therefore, . ■
Denote by the projector of on , that is to say: if
We define the operator by:
It is clear that
Theorem 11 is a densely defined and closed operator.
and that is complete in then
2. Let be a sequence of elements in Checking:
We have then
This implies that
Or, when tends to , we have
Therefore, there exist such that
And as is a closed operator, then we can write
Finally and .
It follows from this theorem that the adjoint operator exists and
Let us denote by the operator adjoint of
In the case , the operator is symmetric and we have the following results:
Theorem 12. admits defect indices if and only if
In this case (defect space associated with ).
Proof. We know (see ) that has the defect indices if and only if its defect subspaces and are unidimensional.
So it suffices to solve the system:
3.2. Case of defect indices
In this section, we give the generalization for the case of defect indices
be formal elements not belonging to , and let
We consider the operator defined by
We assume that and we set
By analogy to the case of defect indices , we also have the following:
Theorem 13. The operator is densely defined and closed.
Theorem 14. The operator admits defect indices if and only if
In this case, the functions are linearly independent and generate the defect space .
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:
with a Carleman kernel.
It should be noted that this study allows the estimation of random variables.