Edelstein’s Astonishing Affine Isometry

Abstract

In 1964, Michael Edelstein presented an amazing affine isometry acting on the space of square-summable sequences. This operator has no fixed points, but a suborbit that converges to 0 while another escapes in norm to infinity! We revisit, extend, and sharpen his construction. Moreover, we sketch a connection to modern optimization and monotone operator theory.

Keywords:

• MSC: Primary 47H09
• Secondary 47H05
• 90C25

Acknowledgments

The authors thank the referees and the editors for their careful reading and constructive comments. HHB was partially supported by the Natural Sciences and Engineering Research Council of Canada. WMM was partially supported by a Natural Sciences and Engineering Research Council of Canada Postdoctoral Fellowship.

Additional information

Notes on contributors

Heinz H. Bauschke

HEINZ BAUSCHKE is a former doctoral student of Jonathan Borwein and currently a Professor of Mathematics at the University of British Columbia (Okanagan campus in Kelowna, B.C., Canada). His main areas of interest are in convex analysis, optimization, and monotone operator theory. He has published more than 100 papers and the book Convex Analysis and Monotone Operator Theory in Hilbert Spaces.

Department of Mathematics, UBC Okanagan, Kelowna, B.C., Canada V1V 1V7

[email protected]

Sylvain Gretchko

SYLVAIN GRETCHKO is a Software Engineer and a graduate student in mathematics at the University of Göttingen. He is particularly interested in experimental mathematics.

Faculty of Mathematics and Computer Science, University of Göttingen, 37073 Göttingen, Germany

[email protected]

Walaa M. Moursi

WALAA MOURSI is an Assistant Professor in the Department of Combinatorics and Optimization at the University of Waterloo. Her research interests are convex analysis, monotone operator theory and continuous optimization.

Department of Combinatorics and Optimization, Faculty of Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

[email protected]

Matthew Saurette

MATTHEW SAURETTE is an undergraduate student in mathematics at the University of British Columbia (Okanagan campus in Kelowna, B.C., Canada). His interests are in optimization and mathematical biology.

Department of Mathematics, UBC Okanagan, Kelowna, B.C., Canada V1V 1V7

[email protected]