Joint normality of operators in semi-Hilbertian spaces

Davis–Wielandt shells of semi-Hilbertian space operators and its applications

“…So, by combining (2.10) together with (2.9) we get the desired result. (6) It is well known that ω A (T ) ≤ T A for all T ∈ B A 1/2 (H). This implies, by using (2.10), that…”

Spectral radius of semi-Hilbertian space operators and its applications

“…Thus T is A-normaloid by applying Proposition 2.4. Secondly, it was shown in [6,Theorem 5.1. ], that if T is an A-self-adjoint operator then, T n A = T n A for all n ∈ N * .…”

“…Also, G(n, d) denotes the set of all functions from {1, · · · , n} into {1, · · · , d} and T g := n k=1 T g(k) , for g ∈ G(n, d). Moreover, the following theorem was proved in [6].…”

“…Recently, there are many papers that study operators defined on semi-Hilbertian spaces. One may see [5,6,21,18] and their references.…”

“…Notice also that in [19], Saddi generalized the concepts of numerical radius and spectral radius of an operator and defined the A-numerical radius and A-spectral radius of an operator T ∈ B(H) by The study of the spectral radius and the joint spectral radius of operators has been the subject of intense research during last decades. For more details, the interested reader is referred to [4,6] and the references therein. Recently, the concept of of A-joint spectral radius associated with a d-tuple of operators T = (T 1 , · · · , T d ) ∈ B A (H) d (not necessarily to be commuting) was introduced by H.Baklouti et al in [6] as follows.…”