Elements of a normed algebra whose 2βΏπ‘β powers lie close to the identity
Paul Robert Chernoff Β· Proceedings of the American Mathematical Society Β· 1969
In [1], R. H. Cox stated that if x is a square matrix all of whose positive powers lie within a distance a <1 of the identity matrix 1, then x =1. Nakamura and Yoshida [3] extended this result to bounded operators on Hilbert space; their argument used the mean ergodic theorem. Hirschfeld [2] recently presented a proof, based on spectral theory, showing that the result is valid for elements of any normed algebra. Actually this had been established earlier by Wallen [4], who gave a concise and elementary argument using a significantly weakened hypothesis: he required only that IXn-1lI = o(n) and that lim inf n-1(||X_JJ+J|x22-JJJ+ * * +|Jxn_J-1)<1. (Wils [5] also deals with this subject matter; I thank the referee for calling this reference to my attention.) In this note we examine a different weakening of Cox's hypotheses: we impose conditions only on the 2nth powers of x.