Spectra and dynamics of generalized Cesàro operators in (LF) and (PLB) sequence spaces. (English) Zbl 07873839
The main object of the paper is the so-called generalized Cesàro operator \(C_t,\,t\in[0,1]\) defined as
\[
C_t(x):=\Big(\frac{t^nx_0+t^{n-1}x_1+\ldots+x_n}{n+1}\Big)_{n\in\mathbb{N}_0}
\]
where \(x=(x_n)_{n\in\mathbb{N}_0}\) 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. The paper consists of 5 sections. After Introduction (Section 1) the authors present their preliminary results (Section 2). The next section delivers the definition of the (LF)-spaces \(L(p-),\,C(p-),\,D(p-)\) as well as 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, the study power boundedness, mean ergodicity and hypercyclicity of \(C_t\).
Reviewer: Krzysztof Piszczek (Poznań)
MSC:
| 46A45 | Sequence spaces (including Köthe sequence spaces) |
| 47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |
| 46A13 | Spaces defined by inductive or projective limits (LB, LF, etc.) |
| 47A10 | Spectrum, resolvent |
| 47A35 | Ergodic theory of linear operators |
| 47A16 | Cyclic vectors, hypercyclic and chaotic operators |
Keywords:
generalized Cesàro operator; spectrum; power boundedness; uniform mean ergodicity; supercyclicity; sequence spaces; (LF)-space; (PLB)-spaceReferences:
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