| Abstrakt: |
Abstract.  Let A be a Banach algebra which does not contain any nonzero idempotent element, let γ > 0, and let $$x \in A$$ . We show that if $$\parallel x \parallel \geq \frac{log(\gamma 1)}{\gamma},$$ then $$\parallel e^{x} - e^{(\gamma)x} \parallel \geq \frac{\gamma} {{(\gamma)^{1\frac{1}{\gamma}}}}$$ . We also show, assuming a suitable spectral condition on x, that if $$\parallel x \parallel \geq 1 - \frac{1}{(\gamma)^{1\frac{1}{\gamma}}}$$ , then $$\parallel 1 - (1)^{\gamma}\parallel \geq \frac{\gamma} {(\gamma)^{1\frac{1}{\gamma}}}$$ [ABSTRACT FROM AUTHOR] |
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