A local spectral condition for strong compactness with some applications to bilateral weighted shifts (Q2862187)

Let \({\mathcal B}(X)\) be the algebra of all bounded linear operators on a complex Banach space \(X\). For an operator \(T\in{\mathcal B}(X)\) and a vector \(x\in X\), let \(r_T(x):=\limsup_{n\to\infty}\|T^nx\|^{\frac{1}{n}}\) be the local spectrum of \(T\) at \(x\), and let \(\sigma(T)\) and \(r(T)\) denote the spectrum and the spectral radius of \(T\), respectively. An operator \(T\in{\mathcal B}(X)\) is said to be strongly compact if the closed unit ball of the unital subalgebra of \({\mathcal B}(X)\) generated by \(T\) is precompact in the strong operator topology. The authors show that an operator \(T\in{\mathcal B}(X)\) for which \(0\) is in the interior of its full spectrum \(\eta(\sigma(T))\) is strongly compact provided that there exists a set \(S\subset X\) that generates a dense linear manifold in \(X\) such that \(r_T(x)\leq \text{dist}(0,\partial\eta(\sigma(T)))\) for all \(x\in S\). As an application of this result, they show, in particular, that if \(H\) is a complex Hilbert space with an orthonormal basis \((e)_{n\in\mathbb{Z}}\) and \(W\) is an injective bilateral weighted shift on \(H\) with a nonzero bounded weight \(\{\omega_n\}_{n\in\mathbb{Z}}\) (i.e., \(We_n=\omega_n e_{n+1},~(n\in\mathbb{Z})\)), then \(W\) is strongly compact provided that \(r_W(e_0)=\limsup_{n\to\infty}|\omega_0\dots\omega_{n-1}|^{\frac{1}{n}}<r(W)\). This is an immediate consequence of the main result once it is observed that \(r_W(e_0)=r_W(e_n)\) for all \(n\in\mathbb{Z}\).