Mohammed Hichem Mortad
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View article: On the reduction of powers of self-adjoint operators
On the reduction of powers of self-adjoint operators Open
Let $T\in B(H)$ be such that $T^n$ is self-adjoint for some $n\in\mathbb{N}$ with $n\geq 3$. The paper's primary aim is to establish the conditions that lead to the self-adjointness of $T$. We pay particular attention to the case where $T^…
View article: Unbounded operators having self‐adjoint, subnormal, or hyponormal powers
Unbounded operators having self‐adjoint, subnormal, or hyponormal powers Open
We show that if a densely defined closable operator A is such that the resolvent set of A 2 is nonempty, then A is necessarily closed. This result is then extended to the case of a polynomial . We also generalize a recent result by Sebesty…
View article: The square root theorem for positive semidefinite matrices and their monotonicity
The square root theorem for positive semidefinite matrices and their monotonicity Open
In this note, simple proofs of certain well-known results involving the positive square root of positive matrices are given.
View article: Certain properties involving the unbounded operators $p(T)$, $TT^*$, and $T^*T$; and some applications to powers and $nth$ roots of unbounded operators
Certain properties involving the unbounded operators $p(T)$, $TT^*$, and $T^*T$; and some applications to powers and $nth$ roots of unbounded operators Open
In this paper, we are concerned with conditions under which $[p(T)]^*=\bar{p}(T^*)$, where $p(z)$ is a one-variable complex polynomial, and $T$ is an unbounded, densely defined, and linear operator. Then, we deal with the validity of the i…
View article: Some properties of the adjoint of the unbounded operators $TT^*$ and $T^*T$
Some properties of the adjoint of the unbounded operators $TT^*$ and $T^*T$ Open
In this note, we mainly investigate the validity of the identities $(TT^*)^*=TT^*$ and $(T^*T)^*=T^*T$, where $T$ is a densely defined closable (or symmetric) operator.
View article: WHEN NILPOTENCE IMPLIES THE ZERONESS OF LINEAR OPERATORS
WHEN NILPOTENCE IMPLIES THE ZERONESS OF LINEAR OPERATORS Open
In this paper, we give conditions forcing nilpotent operators (everywhere bounded or closed) to be null. More precisely, it is mainly shown any closed or everywhere defined bounded nilpotent operator with a positive (self-adjoint) real par…
View article: On Generalized Powers of Operators
On Generalized Powers of Operators Open
In this note, we introduce generalized powers of linear operators. More precisely, operators are not raised to numbers but to other operators. We discuss several properties as regards this notion.
View article: Unbounded generalizations of the Fuglede-Putnam theorem and applications to the commutativity of self-adjoint operators
Unbounded generalizations of the Fuglede-Putnam theorem and applications to the commutativity of self-adjoint operators Open
In this article, we prove and disprove several generalizations of unbounded versions of the Fuglede-Putnam theorem. As applications, we give conditions guaranteeing the commutativity of a bounded self-adjoint operator with an unbounded clo…
View article: Unbounded generalizations of the Fuglede-Putnam theorem
Unbounded generalizations of the Fuglede-Putnam theorem Open
In this paper, we prove and disprove several generalizations of unbounded versions of the Fuglede-Putnam theorem.
View article: Counterexamples related to unbounded paranormal operators
Counterexamples related to unbounded paranormal operators Open
In this paper, we give an example of a densely defined non-closable paranormal operator. Then, we give another example of a densely defined closable paranormal operator whose closure fails to be paranormal. It is worth noticing that our fi…
View article: A Squeeze (Sandwich) Rule for Sequences of Self-adjoint Bounded Linear Operators
A Squeeze (Sandwich) Rule for Sequences of Self-adjoint Bounded Linear Operators Open
The squeeze theorem (or sandwich rule) for convergent sequences is a well known useful result when dealing with convergent real sequences. In this note, we show that this result can be carried over to the set of convergent sequences of sel…
View article: The sandwich rule for sequences of self-adjoint operators and some applications
The sandwich rule for sequences of self-adjoint operators and some applications Open
In this paper, we mainly deal with sequences of bounded linear operators on Hilbert space. The main result is the so-called squeeze theorem (or sandwich rule) for convergent sequences of self-adjoint operators. We show that this theorem re…
View article: When Nilpotence Implies the Zeroness of Linear Operators
When Nilpotence Implies the Zeroness of Linear Operators Open
In this paper, we give conditions forcing nilpotent operators (everywhere bounded or closed) to be null. More precisely, it is mainly shown any closed or everywhere defined bounded nilpotent operator with a positive (self-adjoint) real par…
View article: On Nilpotence of Bounded and Unbounded Linear Operators
On Nilpotence of Bounded and Unbounded Linear Operators Open
In this paper, we give conditions forcing nilpotent operators (everywhere bounded or closed) to be null. More precisely, it is mainly shown any closed or everywhere defined bounded nilpotent operator with a positive (self-adjoint) real par…
View article: Unbounded operators having self-adjoint or normal powers and some related results
Unbounded operators having self-adjoint or normal powers and some related results Open
We show that a densely defined closable operator $A$ such that the resolvent set of $A^2$ is not empty is necessarily closed. This result is then extended to the case of a polynomial $p(A)$. We also generalize a recent result by Sebestyén-…
View article: Operators having closed or self-adjoint powers and a generalization of some reversed von Neumann theorem
Operators having closed or self-adjoint powers and a generalization of some reversed von Neumann theorem Open
In this paper, we show that a densely closable operator $A$ such that the resolvent set of $A^2$ is not empty is necessarily closed. In particular, a closable operator with a self-adjoint square is automatically closed. These two results a…
View article: Unbounded operators: (square) roots, nilpotence, closability and some related invertibility results
Unbounded operators: (square) roots, nilpotence, closability and some related invertibility results Open
In this paper, we are mainly concerned with studying arbitrary unbounded square roots of linear operators as well as some of their basic properties. The paper contains many examples and counterexamples. As an illustration, we give explicit…
View article: Yet another generalization of the Fuglede-Putnam theorem to unbounded operators
Yet another generalization of the Fuglede-Putnam theorem to unbounded operators Open
In this note, we give the most natural (perhaps the simplest ever) generalization of the Fuglede-Putnam theorem where all operators involved are unbounded.
View article: A closed densely defined operator $T$ such that both $T$ and $T^*$ are injective and paranormal yet $T$ is not normal
A closed densely defined operator $T$ such that both $T$ and $T^*$ are injective and paranormal yet $T$ is not normal Open
In this note, we give an example of a densely defined closed one-to-one paranormal operator $T$ whose adjoint is also injective and paranormal, but $T$ fails to be normal.
View article: Simple examples of non closable paranormal operators
Simple examples of non closable paranormal operators Open
In this note, we give an example of a densely defined non-closable paranormal operator. Then, we give another example of a densely defined closable paranormal operator whose closure fails to be paranormal. These two examples are simpler th…
View article: On the Normality of the Product of Tow Operators in Hilbert Space
On the Normality of the Product of Tow Operators in Hilbert Space Open
In this paper we present the results of the maximality of operators not nec-essarily bounded. For that, we will see the results obtained by operators in situation ofextension. Regarding the normal product of normal operators we seem to be …
View article: CHERNOFF-LIKE COUNTEREXAMPLES RELATED TO UNBOUNDED OPERATORS
CHERNOFF-LIKE COUNTEREXAMPLES RELATED TO UNBOUNDED OPERATORS Open
In this paper, we give an example of a closed unbounded operator whose square's domain and adjoint's square domain are equal and trivial. Then, we come up with an essentially self-adjoint whose square has a trivial domain.
View article: Maximalité d'opérateurs linéaires
Maximalité d'opérateurs linéaires Open
This thesis is divided into three chapters:Preliminaries: in this chapter, some general mathematical reminders are given which are necessary for a good understanding of the methods which are implemented in the following. First, we recall w…
View article: On the Operator Equations $A^n=A^*A$
On the Operator Equations $A^n=A^*A$ Open
Let $n\in\mathbb{N}$ and let $A$ be a closed linear operator (everywhere bounded or unbounded). In this paper, we study (among others) equations of the type $A^*A=A^n$ where $n\geq2$ and see when they yield $A=A^*$ (or a weaker class of op…