Abstract We characterize the discrete maximal regularity for second order difference equations by means of spectral and R-boundedness properties of the resolvent set. Keywords: Discrete maximal regularitySecond order difference equations R-boundednessUMD spacesKeywords: Primary 39A12Secondary 42A45, 47D09 Acknowledgements This note was originated in a discussion between both authors, while the second author was visiting the Universidade Federal de Pernambuco and after the first author was visiting the Universidad de Santiago de Chile. Both authors are grateful for the hospitality of the respectives institutions. Notes ¶ The first author is partially supported by CNPQ/Brazil Grant# 300068/2005-0. § The second author is partially supported by Proyecto Anillo ACT-13. Email: clizama@usach.cl Additional informationNotes on contributorsClaudio Cuevas ¶ ¶ The first author is partially supported by CNPQ/Brazil Grant# 300068/2005-0. Carlos Lizama § § The second author is partially supported by Proyecto Anillo ACT-13. Email: clizama@usach.cl