I am very hesitant to suggest that mathematicians as eminent as Kahane and Żelazko would make a mistake in their paper. But it looks as though the formula should read . The reason is that if then will not be defined (because x(t) is only defined for ).

In this algebra (as in any algebra of functions), the spectrum of a function in the algebra is the set of values of the function. For the function , for example, the spectrum of x is the whole unit disc. But the function Tx given by only takes the values . On the other hand, if x is an invertible function then 0 will not be in its spectrum, but 0 is in the spectrum of Tx.