The upper Browder spectrum property

In this note we continue our study on the upper Browder spectrum initiated in Benjamin and Mouton (Quaest. Math. 39(5), 2016). Recall that, for an element a of an ordered Banach algebra A and w.r.t. a Banach algebra homomorphism T : A -> B, we have inclusions sigma(Ta) subset of beta(T)(a) subset of beta(+)(T)(a) subset of sigma(a), where sigma(Ta), beta(T)(a), beta(+)(T)(a) and sigma(a) denote the Fredholm, Browder, upper Browder and usual) spectra of a, respectively (Benjamin and Mouton in Quaest. Math. 39(5), 2016). This paper concerns the following natural question: given that the spectral radius of a positive element is not in the Fredholm spectrum of the element, when will it be outside the upper Browder spectrum of that element?