Weyl's theorem and tensor products: a counterexample
For a bounded operator \(T \in B(X)\) on a Banach space \(X\), denote by \(\omega_{\text{ess}}(T)\) and \(\beta_{\text{ess}}(T)\) its Weyl spectrum and its Browder spectrum, respectively. The operator is said to satisfy Weyl's theorem if the complement in the spectrum of the Weyl spectrum coincides with the isolated eigenvalues of finite multiplicity. In particular, such an operator also satisfies Browder's theorem, that is, \(\omega_{\text{ess}}(T)=\beta_{\text{ess}}(T)\). The authors construct an example of two operators \(A,B \in B(\ell_2)\) which satisfy Weyl's theorem each, and therefore also Browder's theorem, but whose tensor product \(A \otimes B \in B(\ell_2 \otimes \ell_2)\) does not satisfy Browder's theorem. The counterexample comprises the tensor product of two direct sums, each of which involves forward and backward shifts as simple building blocks. As a byproduct, the authors find that the product of circles in the complex plane is a limaçon.
scientific article; zbMATH DE number 5865642