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On the spectrum and fine spectrum of the compact Rhaly operators. (English) Zbl 1073.47039
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 $bv_0= bv\cap c_0$. 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 $bv$.

47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46B45Banach sequence spaces
47A10Spectrum and resolvent of linear operators