Spectral radius of weighted endomorphisms with applications on generalized countable Markov shifts - Rodriguez, Ivan Francisco Diaz Granados (2024)
This thesis is divided into two parts. In the first part, we consider an endomorphism \\alpha: A \\to A, with A being a commutative Banach algebra with unity, and X the spectrum of A. We ensure the existence of a unique partial dynamical system \\varphi: \\Delta \\subseteq X \\to X associated with the endomorphism. By adding the condition that A is a uniform algebra, we obtain an expression for the spectral radius of the weighted endomorphisms a\\alpha, a \\in A, which depends on the invariant or ergodic probabilities measures of the partial dynamical system (X, \\varphi). Moreover, this spectral radius will coincide with the spectral radius of the weighted shift operators aT, where \\alpha(a) = TaS for some operator S. In the second part of the project, for the shift operator \\sigma: \\Delta \\to X_A, where X_A is a generalized Markov shift space, we identify a *-endomorphism \\alpha: \\mathcal_A \\to \\mathcal_A associated with (X_A, \\sigma), where \\mathcal_A is the algebra whose spectrum is X_A. We give examples under conditions that guarantee the continuous extension of the shift operator and other examples where we show the impossibility of extending the shift operator continuously over the entire space. For the particular case of the renewal shift, we characterize every element of the algebra and provide an explicit formula to calculate r(a\\alpha) for all a \\in \\mathcal_A.