New class of operators where the distance between the identity operator and the generalized Jordan ∗-derivation range is maximal - Messaoudene Hadia (De Gruyter, 2021)
A new class of operators, larger than ∗* -finite operators, named generalized ∗* -finite operators and noted by Gℱ∗(ℋ){{\mathcal{G {\mathcal F} }}}^{* }\left({\mathcal{ {\mathcal H} }}) is introduced, where: Gℱ∗(ℋ)={(A,B)∈ℬ(ℋ)×ℬ(ℋ):∥TA−BT∗−λI∥≥∣λ∣,∀λ∈C,∀T∈ℬ(ℋ)}.{{\mathcal{G {\mathcal F} }}}^{* }\left({\mathcal{ {\mathcal H} }})=\{(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})× {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}):\parallel TA-BT^{* }-λ I\parallel \ge | λ | ,\hspace{0.33em}\forall λ \in {\mathbb{C}},\hspace{0.33em}\forall T\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\}. Basic properties are given. Some examples are also presented.