ON THE NORMALITY OF THE PRODUCT OF TOW OPERATORS IN HILBERT SPACE

In this paper we present the results of the maximality of operators not necessarily bounded. For that, we will see the results obtained by operators in situation of extension. Regarding the normal product of normal operators we seem to be the key to maximality.


Introduction
First, we assume that all operators operators are non necessarily bounded on a complex Hilbert space Recall too that the unbounded operator A, defined on D(A), is said to be invertible if there exists an everywhere defined (i.e. on the whole of H) bounded operator B, which then will be designated by A −1 , such that where I is the indentity operator on H. An operator A is said to be closed if its graph is closed in H ⊕ H.
The closing of the domain D(A) of A implies the closing of A if A is bounded on D(A). It is known that the product operators AB is closed if for instance A is closed and B ∈ B(H), or A −1 ∈ B(H) and B is closed.
We also recall that an operator S is said to be densely defined if its domain D(S) is dense in H. It is known that in such case its adjoint S * exists and is unique. If T ⊂ S, then S * ⊂ T * . Notice that if S, T and ST are all densely defined, then we are only sure of and a full equality occurring if e.g.
We say that S is normal if S is densely defined, and Sx = S * x for all x ∈ D(S) = D(S * ) (hence from known facts normal operators are automatically closed). Recall that the previous is equivalent to S is closed and SS * = S * S. Other classes of operators are defined in the usual fashion. Let us also agree that any operator is linear and non necessarily bounded unless we specify that it belongs to B(H). We also assume the basic theory of operators (see e.g. [1] or [20]). We do recall the celebrated Fuglede-Putnam Theorem though: One of the main objectives of this work is to impose conditions to obtain other results, starting from an extension. The following theorem and corollary result are a powerful tool to prove results on unbounded operators. For instance, Statement (3) of the next theorem is used in the proof of the "unbounded" version of the spectral theorem of normal operators (see e.g. [15]). For other uses, see e.g. [6] or [10].
(2) T is symmetric and S is self-adjoint (resp. normal). We then say that self-adjoint (resp. normal) operators are maximally symmetric.
(3) T and S are normal (we say that normal operators are maximally normal). Hence, self-adjoint (resp. normal) operators are maximally normal (resp. self-adjoint).
Commutativity of operators must be handled with care. First, recall the definition of two strongly commuting (normal) operators (see e.g. [16]):

Main Results
The normality of unbounded products of normal operators has been studied recently. See e.g. [5] and the references therein. We recall  The proof requires the following lemma whose proof is very akin to the one in [15].

Now we prove proposition 2.4
Proof. We denote the graphe of T by G r (T ). By lemma 2.1, we obtain G r (T ) is closed on D(T ) × H, i.e.
(G r (T )) C is open. We may write where I, J are arbitrary and U 1 i , U 2 j are open on D(T ), H respectively. Hence with F 1 i , F 2 j closed sets on H. Therefore G r (T ) becommes closed on H × H when D(T ) is closed on H (It's the induced topology).
Remark 2.1. In the previous proof, we did not use the linearity of T , we used only topological notions.

Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.