New class of operators where the distance between the identity operator and the generalized Jordan ∗-derivation range is maximal

A new class of operators, larger than ∗\ast -finite operators, named generalized ∗\ast -finite operators and noted by Gℱ∗(ℋ){{\mathcal{G {\mathcal F} }}}^{\ast }\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} }}}^{\ast }\left({\mathcal{ {\mathcal H} }})=\{(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}):\parallel TA-B{T}^{\ast }-\lambda I\parallel \ge | \lambda | ,\hspace{0.33em}\forall \lambda \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.