Let $M_C=\left(\begin{array}{cc} A & C\\ 0 & B \end{array} \right) \in B(\chi \oplus \chi)$ be an upper triangular Banach space operator. The relationship between the spectra of $M_C$ and $M_0$, and their various distinguished parts, has been studied by a large number of authors in the recent past. This paper brings forth the important role played by SVEP, the single-valued extension property, in the study of some of these relations. Operators $M_C$ and $M_0$ satisfying Browder's, or a-Browder's, theorem are characterized, and we prove necessary and sufficient conditions for implications of the type ``$M_0$ satisfies $a$-Browder's (or $a$-Weyl's) theorem $\Leftrightarrow$ $M_C$ satisfies $a$-Browder's (resp., $a$-Weyl's) theorem" to hold.