Compact \(AC(\sigma )\) operators (Q2380354)

Let \(\sigma\subset\mathbb{C}\) be a compact, nonempty set and \(BV(\sigma)\) be the Banach algebra of functions of bounded variation on \(\sigma\) with the norm defined by \[ \|f\|_{BV(\sigma)}\equiv\|f\|_{\infty}+\text{var}(f,\sigma) =\|f\|_{\infty}+\sup_{\gamma}\text{cvar}(f,\gamma)\,\rho(\gamma), \] where the supremum is taken over all piecewise linear curves \(\gamma:\,[0,1]\to \mathbb{C}\) in the plane, the term \(\text{cvar}(f,\gamma)\) measures the variation of \(f\) as one travels along the curve \(\gamma\), and \([\rho(\gamma)]^{-1}\) measures how sinuous the curve is. In this paper, the authors first estimate both \(\|f(T)\|\) and \(\|f\|_{BV(\sigma)}\), where \(f\in BV(\sigma)\), \(T\) has the expansion \(\sum_{j=1}^{\infty}\mu_{j}P_j\), and \(P_j\) for all \(j\in{\mathbb N}\) is the Riesz projection onto the eigenspace corresponding to the eigenvalue \(\mu_j\). Since all compact \(AC(\sigma)\) operators have a representation analogous to that for compact normal operators, as a partial converse the authors then obtain conditions which allow one to construct a large number of such operators and give examples to show that there are many ways of producing compact \(AC(\sigma)\) operators whose spectra do not lie in a finite number of lines through the origin. Finally, using the results in this paper, the authors answer a number of questions about the decomposition of a compact \(AC(\sigma)\) operators into real and imaginary parts.