Bounded operators on topological vector spaces and their spectral radii (Q2867614)

Let \(X\) be a topological vector space over the field of real or complex numbers, and let \(T:X\to X\) be a linear operator. Following \textit{V. G. Troitsky} [Panam. Math. J. 11, No. 3, 1--35 (2001; Zbl 1006.47001)], the authors use the following spectral radii of \(T\): (i)~\(r_{nb}(T)=\inf\{\nu>0:T^n/\nu^n\to0\text{ as }n\to\infty,\text{ uniformly on some zero neighborhood}\}\); (ii)~\(r_{bb}(T)=\inf\{\nu>0:T^n/\nu^n\to0\text{ as }n\to\infty,\text{ uniformly on every bounded set}\}\); (iii)~\(r_c(T)=\inf\{\nu>0:T^n/\nu^n\to0\text{ as }n\to\infty,\text{ equicontinuously}\}\). Several inequalities are proved for these spectral radii of sums and products of two operators. Further, the completeness of various classes of bounded operators on \(X\) is investigated under topologies also introduced in the cited paper. As an application, sufficient conditions for the invertibility of \(T\) are given in the terms of the spectral radii.