Determining elements in Banach algebras through spectral properties

Let A be a Banach algebra. By σ ( x ) and r ( x ) , we denote the spectrum and the spectral radius of x ∈ A , respectively. We consider the relationship between elements a , b ∈ A that satisfy one of the following two conditions: (1) σ ( a x ) = σ ( b x ) for all x ∈ A , (2) r ( a x ) ≤ r ( b x ) for all x ∈ A . In particular, we show that (1) implies that a = b if A is a C ∗ -algebra, and (2) implies that a ∈ 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.