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Zbl 1073.47039
Yildirim, M.
On the spectrum and fine spectrum of the compact Rhaly operators.
(English)
[J] Indian J. Pure Appl. Math. 34, No. 10, 1443-1452 (2003). ISSN 0019-5588; ISSN 0975-7465/e

Given a sequence $a= \{a_n\}$ of scalars, the Rhaly matrix $R_a$ is the lower triangular matrix with constant row-segments, $$R_a= \left[\matrix a_0 & 0 & 0 & \cdots\\ a_1 & a_1 & 0 & \cdots\\ a_2 & a_2 & a_2 & \cdots\\ \vdots & \vdots & \vdots & \vdots\endmatrix\right].$$ Let $c_0$, $bv$ and $bv_0$ denote, respectively, the space of null sequences, sequences such that $\sum^\infty_{k=0} |x_{k+1}- x_k|< \infty$, and $bv_0= bv\cap c_0$.\par In [Bull. Lond. Math. Soc. 21, No. 4, 399--406 (1989; Zbl 0695.47024)], {\it H.~C.~Rhaly} determined the spectrum of the Rhaly operator $R_a$ regarded as an operator on the Hilbert space $\ell_2$, normed by $\Vert x\Vert= (\sum_n|x_n|^2)^{1/2}$. The purpose of the present paper is to characterize the spectrum and fine spectrum of Rhaly operators acting on $bv_0$ and $bv$.
[Bahman Yousefi (Shiraz)]
MSC 2000:
*47B37 Operators on sequence spaces, etc.
46B45 Banach sequence spaces
47A10 Spectrum and resolvent of linear operators

Keywords: Rhaly operator; Cesàro operator; spectrum; point spectrum

Citations: Zbl 0695.47024

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