Spectra and dynamics of generalized Cesàro operators in (LF) and (PLB) sequence spaces (Q6564726)

The main object of the paper is the so-called \textit{generalized Cesàro operator} \(C_t\), \(t\in[0,1]\), defined as\N\[\NC_t(x):=\Big(\frac{t^nx_0+t^{n-1}x_1+\ldots+x_n}{n+1}\Big)_{n\in\mathbb{N}_0}\N\]\Nwhere \(x=(x_n)_{n\in\mathbb{N}_0}\) is any sequence. For \(t=1\) the authors recover the classical Cesàro operator \(C_1\). The framework for the study of this operator are specific (LF)- and (PLB)-spaces. These are sequence spaces arising from \(\ell_p\)-spaces, \(p\in[1,\infty]\), by implementing a projective/inductive limit procedure. \N\NThe paper consists of 5 sections. After an \textit{Introduction} (Section~1) the authors present their preliminary results (Section~2). The next section contains the definition of the (LF)-spaces \(L(p-),\,C(p-),\,D(p-)\) as well as of the (PLB)-spaces \(L(p+),\,C(p+),\,D(p+)\) together with their basic properties. The main part of the paper are Sections~4 and~5. In the first of them the authors study spectra of \(C_t\) and in the last one they deal with linear dynamics of this operator. Specifically, they study power boundedness, mean ergodicity and hypercyclicity of \(C_t\).