An elementary spectral regularization (Q2722498)

The article provides an alternative simpler proof to a less exact form of an inequality of \textit{M. Lifshits} and the author [C. R. Acad. Sci., Paris, Sér. I 324, No. 1, 99-103 (1997; Zbl 0892.46080)]. Let \(U:H\to H\) be a contraction in a Hilbert space \((H,\|\;\|)\). Fix \(f\in H\), put NEWLINE\[NEWLINE A_n(f)= A^U_n(f)= {1\over n} \sum^{n- 1}_{j=0}U^j(f),\quad n\geq 1. NEWLINE\]NEWLINE Using the spectral lemma, the paper demonstrates that NEWLINE\[NEWLINE \|A_nf-A_mf\|^2\leq 4\mu\{[\tfrac{1}{m}, \tfrac{1}{n}[\}+M^{(\mu)}\{[\tfrac{1}{m}, \tfrac{1}{n}[\}. NEWLINE\]NEWLINE Here \(\mu\) is the spectral measure of \(f\) relative to \(U\) and \(M^{(\mu)}\) is a regularization of \(\mu\).NEWLINENEWLINEFor the entire collection see [Zbl 0958.00034].