A remark on the spectrum of fractional difference operator

Abstract

This work computes the spectrum of the backward fractional difference operator ${\mathrm{\Delta }}^{(\alpha )}$ $(0<\alpha <1)$ over various Banach spaces ${\ell }_1,c$ and $c_0$. Note that recently, the spectrum of the fractional difference operator ${\mathrm{\Delta }}_{\nu }^{(\alpha )}$, ($\nu =({\nu }_k)$ being either a constant or strictly decreasing sequence of positive real numbers satisfying some conditions) on the sequence space ${\ell }_1$ has been computed by Baliarsingh P. [On the spectrum of fractional difference operator. Linear Multilinear Algebra. 2021;1–13] and it has been noticed that the results obtained are unified and more general, but lack of accuracy for the case $0<\alpha <1$. In this work, we improve the results by providing the sharper estimations and some illustrative examples. Also, individual sets describing the point, the residual and the continuous spectrum of the operator ${\mathrm{\Delta }}^{(\alpha )}$ on the Banach space c and $c_0$ are computed.

Keywords:

• Sequence spaces
• fractional difference operator
• regular value
• resolvent set
• spectrum of an operator

2010 Mathematics Subject Classifications:

• 47A10
• 40A05
• 46A45

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is partially supported by National Board for Higher Mathematics (NBHM), DAE, Mumbai [grant number 02011/7/2020/NBHM(RP)/R&D/6289].