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On the Multiplicative Spectral Characterization of the Jacobson Radical

For a unital Banach algebra A over C, we give improvements on the well-known multiplicative spectral characterization of the Jacobson radical, (1), in terms of several spectral parameters. In particular, we show that every non-invertible element a ∈ A for which the number of elements in the spectrum of ax is less or equal to the number of elements in the spectrum of x, for all x in an arbitrary small neighborhood of the unit, must belong to the Jacobson radical.