arXiv subjects
Publications and source records attributed to Mohammed Hichem Mortad.
We are interested in an open question raised by Fong-Tsui (dating back to the beginning of the eighties of last century) as to whether a bounded operator whose absolute value is less than the absolute value of its real part is self-adjoint. The analogue in the unbounded operators setting is also treated.
The present paper is mainly concerned with equations involving exponentials of bounded normal operators. Conditions implying commutativity of those normal operators are given. This is carried out without the known $2πi$-congruence-free hypothesis. It is also a continuation of a recent work by the corresponding author.
Let $A$ and $B$ be two -non necessarily bounded- normal operators. We give new conditions making their product normal. We also generalize a result by Deutsch et al on normal products of matrices.
In the present paper we give results on the closedness and the self-adjointness of the sum of two unbounded operators. We present a new approach to these fundamental questions in operator theory. We also prove a new version of the Fuglede theorem where the operators involved are all unbounded. This new Fuglede theorem allows us to prove (under extra conditions) that the sum of two unbounded normal operators remains normal. Also a result on the normality of the unbounded product of two normal operators is obtained as a consequence of this new "Fuglede theorem". Some interesting examples are also given.
We present a new approach to the question of when the commutativity of operator exponentials implies that of the operators. This is proved in the setting of bounded normal operators on a complex Hilbert space. The proofs are based on some similarities results by Berberian and Embry as well as the celebrated Fuglede theorem.
The present paper partly constitutes an "unbounded" follow-up of a paper by I. Kaplansky dealing with bounded products of normal operators. Results on the normality of unbounded products are also included.