The Gohberg lemma, compactness, and essential spectrum of operators on compact Lie groups (Q279755)

A version of Gohberg's lemma for compact Lie groups which gives an estimate from below from a given operator to the set of compact operators is obtained. Let \(G\) be a compact Lie group and \(T:C^\infty(G) \to C^\infty(G)\) is an operator. Then, the symbol of \(T\) is defined to be \[ \sigma_T(x, \xi) = \xi(x)^\ast (T\xi)(x) \in \mathbb C^{d_\xi \times d_\xi}, \] where \(x \in G\) and \(\xi \in \widehat{G}\). It follows that \[ Tf(x) = \sum_{\xi \in \widehat{G}} d_\xi\mathrm{Tr}\left (\xi(x) \sigma_T(x,\xi) \widehat{f}(\xi) \right), \] and the correspondence \(\sigma \to T_\sigma\) (the above operator) is one-one. The matrix components of \(\xi(x)\) are eigenfunctions of the Casimir, with eigenvalue \(-\lambda_\xi^2\) (say). Define \(\langle \xi \rangle = (1+\lambda_\xi^2)^{1/2}.\) Let \(\Psi^0(G)\) be the usual class of operators that have symbols in Hörmander's class \(S_{1, 0}^0(\mathbb R^n).\) The version of the Gohberg's lemma established is the following: Let \(T_\sigma \in \Psi^0(G)\) and \(\sigma(x, \xi)\) be the matrix symbol of \(T_\sigma.\) Then, for all compact operators \(K\) on \(L^2(G),\) \[ \|T_\sigma - K \| \geq d_{\min}, \] where \[ d_{\min} = \limsup_{\langle \xi \rangle \to \infty} \left \{ \sup_{x \in G} \frac{\| \sigma(x, \xi) \sigma(x, \xi)^\ast \|_{\min}}{\| \sigma(x, \xi)\|_{op}} \right \} \] and \(\| \sigma(x, \xi) \sigma(x, \xi)^\ast \|_{\min}\) is the smallest eigenvalue of \(\sigma(x, \xi) \sigma(x, \xi)^\ast\).