On The Symmetric Quaternionic Banach Algebras, I (Geljfand Theory)
Let be the algebra of all bounded linear operators in a complex Banach space . We consider operators satisfying the relation for any vector , where denotes the local spectrum of at the point . We say then that and have the same local spectra. We prove that then, under some conditions, is a quasinilpotent operator, that is as . Without these conditions, we describe the operators with the same local spectra only in some particular cases.
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