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Let $A$ be a Banach algebra. By $\sigma(x)$ and $r(x)$ we denote the spectrumand the spectral radius of $x\in A$, respectively. We consider the relationshipbetween elements $a,b\in A$ that satisfy one of the following two conditions:(1) $\sigma(ax) = \sigma(bx)$ for all $x\in A$, (2) $r(ax) \le r(bx)$ for all$x\in A$. In particular we show that (1) implies $a=b$ if $A$ is a$C^*$-algebra, and (2) implies $a\in \mathbb C b$ if $A$ is a prime$C^*$-algebra. As an application of the results concerning the conditions (1)and (2) we obtain some spectral characterizations of multiplicative maps.

Determining elements in Banach algebras through spectral properties