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On the spectrum and fine spectrum of the compact Rhaly operators. (English) Zbl 1073.47039

Given a sequence $a=\left\{{a}_{n}\right\}$ of scalars, the Rhaly matrix ${R}_{a}$ is the lower triangular matrix with constant row-segments,

${R}_{a}=\left[\begin{array}{cccc}{a}_{0}& 0& 0& \cdots \\ {a}_{1}& {a}_{1}& 0& \cdots \\ {a}_{2}& {a}_{2}& {a}_{2}& \cdots \\ ⋮& ⋮& ⋮& ⋮\end{array}\right]·$

Let ${c}_{0}$, $bv$ and $b{v}_{0}$ denote, respectively, the space of null sequences, sequences such that ${\sum }_{k=0}^{\infty }|{x}_{k+1}-{x}_{k}|<\infty$, and $b{v}_{0}=bv\cap {c}_{0}$.

In [Bull. Lond. Math. Soc. 21, No. 4, 399–406 (1989; Zbl 0695.47024)], H. C. Rhaly determined the spectrum of the Rhaly operator ${R}_{a}$ regarded as an operator on the Hilbert space ${\ell }_{2}$, normed by $\parallel x\parallel =\left({\sum }_{n}|{x}_{n}{{|}^{2}\right)}^{1/2}$. The purpose of the present paper is to characterize the spectrum and fine spectrum of Rhaly operators acting on $b{v}_{0}$ and $bv$.

##### MSC:
 47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 46B45 Banach sequence spaces 47A10 Spectrum and resolvent of linear operators
##### Keywords:
Rhaly operator; Cesàro operator; spectrum; point spectrum