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New class of operators where the distance between the identity operator and the generalized Jordan \(\ast\)-derivation range is maximal. (English) Zbl 1494.47062

Summary: A new class of operators, larger than \(\ast\)-finite operators, named generalized \(\ast\)-finite operators and noted by \(\mathcal{G}\mathcal{F}^{\ast}(\mathcal{H})\) is introduced, where \[ \mathcal{G}\mathcal{F}^{\ast}(\mathcal{H})=\{(A,B)\in\mathcal{B}(\mathcal{H})\times\mathcal{B}(\mathcal{H})):\Vert TA-B{T}^{\ast}-\lambda I\Vert \ge |\lambda|,\, \forall \lambda \in\mathbb{C},\, \forall T\in\mathcal{B}(\mathcal{H})\}. \] Basic properties are given. Some examples are also presented.

MSC:

47B47 Commutators, derivations, elementary operators, etc.
47A12 Numerical range, numerical radius
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References:

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