The Browder and Weyl spectra of an operator and its diagonal (Q2867536)

Let \(T\in B(X)\) be a Banach space operator and \(E\) be a \(T\)-invariant closed subspace of \(X\). Then the restriction map \(A=T| _E\) and the quotient map \(B=T| _{X/E}\) are well defined operators in \(B(E)\) and \(B(X/E)\), respectively. The pair \((A,B)\) is called the diagonal of \(T\). This paper is devoted to studying the relationship between the spectrum \(\sigma\), the approximate point spectrum \(\sigma_a\), the Fredholm spectrum \(\sigma_e\), the Browder spectrum \(\sigma_b\), the Weyl spectrum \(\sigma_w\) and the Weyl essential approximate point spectrum \(\sigma_{aw}\) of the operator \(T\) and its diagonal \((A,B)\). Many interesting results are proved. For instance, the authors show that (i)~if \(\sigma_w(T)=\sigma_w(A)\cup\sigma_w(B)\), then \(\sigma(T)=\sigma(A)\cup\sigma(B)\); (ii)~if \(\sigma_{aw}(T)=\sigma_{aw}(A)\cup\sigma_{aw}(B)\) and \(A^*\) has the single-valued extension property, then \(\sigma_a(T)=\sigma_a(A)\cup\sigma_a(B)\); (iii)~if \(\sigma_w(T)=\sigma_w(A)\cup\sigma_w(B)\), then Browder's theorem transfers from \(A\) and \(B\) to \(T\); (iv)~if \(\sigma_w(T)=\sigma_w(A)\cup\sigma_w(B)\) and \(A,B\) are isoloid, then Weyl's theorem transfers from \(A\) and \(B\) to \(T\).