## Taiwanese Journal of Mathematics

### ASYMPTOTIC REGULARITY OF LINEAR POWER BOUNDED OPERATORS

#### Abstract

Let $T$ be a linear power bounded operator on a Banach space $X$ and let $S_{\lambda} = (1−\lambda)I + \lambda T$ be the averaged map of $T$, where $\lambda \in (0,1)$. It is shown that $S_{\lambda}$ is asymptotically regular on $X$; that is, $\lim_{n \to \infty} \| S_{\lambda}^{n}x - S_{\lambda}^{n+1}x \| = 0$ for every $x \in X$. Hence the sequence $\{S_{\lambda}^{n}x\}$ converges strongly provided it has a weak cluster point.

#### Article information

Source
Taiwanese J. Math., Volume 10, Number 2 (2006), 417-429.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500403834

Digital Object Identifier
doi:10.11650/twjm/1500403834

Mathematical Reviews number (MathSciNet)
MR2208276

Zentralblatt MATH identifier
1106.47032

#### Citation

Xu, Hong-Kun; Yamada, Isao. ASYMPTOTIC REGULARITY OF LINEAR POWER BOUNDED OPERATORS. Taiwanese J. Math. 10 (2006), no. 2, 417--429. doi:10.11650/twjm/1500403834. https://projecteuclid.org/euclid.twjm/1500403834

#### References

• [1.] J. B. Baillon, Un theoreme de type ergodique pour les contractions non lineaires dans un espace de Hilbert, C. R. Acad. Sci. Paris Ser. I, Math. 280 (1975), 1511-1514.
• [2.] F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings, J. Math. Anal. Appl., 20 (1967), 197-228.
• [3.] R. E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math., 32 (1979), 107-116.
• [4.] C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems, 20 (2004), 103-120.
• [5.] W. G. Jr. Dotson, On the Mann iterative process, Trans. Amer. Math. Soc., 149 (1970), 65-73.
• [6.] W. F. Eberlein, Abstract ergodic theorems, Proc. Natl. Acad. Sci., 34 (1948), 43-47.
• [7.] W. F. Eberlein, Abstract ergodic theorems and almost periodic functions, Trans. Amer. Math. Soc., 67 (1949), 217-240.
• [8.] W. F. Eberlein, On retrogression in mean ergodic theory, J. Approx. Theory, 42 (1984), 293-298.
• [9.] X. L. Ge, Lecture Notes in Nonlinear Analysis-Selected Topics, Zhejiang University, 1984. (Unpublished)
• [10.] S. Ishikawa, Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59 (1976), 65-71.
• [11.] A. Levi and H. Stark, Restoration from phase and magnitude by generalized projections, in Image Recovery Theory and Applications, (Ed. by H. Stark), Academic Press, Orlando, 1987, pp. 277-320.
• [12.] C. I. Podilchuk and R. J. Mammone, Image recovery by convex projections using a least-squares constraint, J. Opt. Soc. Am. A, 7 (1990), 517-521.
• [13.] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 272-276.
• [14.] M. I. Sezan and H. Stark, Applications of convex projection theory to image recovery in tomography and related areas, in Image Recovery Theory and Applications, (Ed. by H. Stark), Academic Press, Orlando, 1987, pp. 415-462.
• [15.] K. Yosida, Functional Analysis, 3rd Edition, Springer-Verlag, Berlin/Heidelburg, 1971.
• [16.] D. Youla, Mathematical theory of image restoration by the method of convex projections, in Image Recovery Theory and Applications, (Ed. by H. Stark), Academic Press, Orlando, 1987, pp. 29-77.
• [17.] D. Youla, On deterministic convergence of iterations of relaxed projection operators, J. Visual Comm. Image Representation, 1 (1990), 12-20.