A note on commutativity of the exponential spectrum in the operator algebra \(\mathcal{B}(\ell^p\oplus\ell^q)\)

Abstract

In a Banach algebra \(\mathcal{A}\), for a pair of elements, the spectrum always commutes:

$$\sigma(ab)\setminus\{0\}=\sigma(ba)\setminus\{0\},$$

whereas the exponential spectrum does not, as shown by Klaja and Ransford. In literature, the operator algebra \(\mathcal{B}(\ell^p\oplus\ell^q)\) was anticipated to be a potential algebra of the same type. But we have established that the exponential spectrum commutes in this algebra. This is achieved by proving that the exponential spectrum commutes in \(\mathcal{B(X)}\) if it commutes in the corresponding Calkin algebra.