Contrary to the situation on
Hilbert space, the sum and product of two commuting scalar spectral operators on a
Banach space X need not be spectral, even if X is reflexive. This has been shown by
Kakutani and Mc Carthy. In this note, order-theoretic methods are used to discuss
Mc Carthy’s construction. To this end, a special class of scalar spectral operators is
introduced.