# Article

 Title: Extensions of linear operators from hyperplanes of $l^{(n)}_\infty$ (English) Author: Baronti, Marco Author: Fragnelli, Vito Author: Lewicki, Grzegorz Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 36 Issue: 3 Year: 1995 Pages: 443-458 . Category: math . Summary: Let $Y \subset l^{(n)}_{\infty }$ be a hyperplane and let $A \in {\Cal L}(Y)$ be given. Denote \align {\Cal A} = & \{L\in {\Cal L}(l^{(n)}_{\infty },Y):L\mid Y = A\} \text{ and} \ & \lambda_{A} = \inf \{\parallel L \parallel : L\in {\Cal A}\}. \endalign In this paper the problem of calculating of the constant $\lambda_{A}$ is studied. We present a complete characterization of those $A \in {\Cal L}(Y)$ for which $\lambda_{A} = \parallel A \parallel$. Next we consider the case $\lambda_{A} > \parallel A \parallel$. Finally some computer examples will be presented. (English) Keyword: linear operator Keyword: extension of minimal norm Keyword: element of best approximation Keyword: strongly unique best approximation MSC: 41A35 MSC: 41A52 MSC: 41A55 MSC: 41A65 MSC: 46A22 MSC: 47A20 idZBL: Zbl 0831.41014 idMR: MR1364484 . Date available: 2009-01-08T18:19:11Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/118772 . Reference: [1] Baronti M., Papini P.L.: Norm one projections onto subspaces of $l^p$.Ann. Mat. Pura Appl. IV (1988), 53-61. MR 0980971 Reference: [2] Blatter J., Cheney E.W.: Minimal projections onto hyperplanes in sequence spaces.Ann. Mat. Pura Appl. 101 (1974), 215-227. MR 0358179 Reference: [3] Collins H.S., Ruess W.: Weak compactness in spaces of compact operators and vector valued functions.Pacific J. Math. 106 (1983), 45-71. MR 0694671 Reference: [4] Odyniec Wl., Lewicki G.: Minimal Projections in Banach Spaces.Lecture Notes in Math. 1449, Springer-Verlag. Zbl 1062.46500, MR 1079547 Reference: [5] Singer I.: On the extension of continuous linear functionals....Math. Ann. 159 (1965), 344-355. Zbl 0141.12002, MR 0188758 Reference: [6] Singer I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces.Springer-Verlag, Berlin, Heidelberg, New York, 1970. Zbl 0197.38601, MR 0270044 Reference: [7] Sudolski J., Wojcik A.: Some remarks on strong uniqueness of best approximation.Approximation Theory and its Applications 6 (1990), 44-78. Zbl 0704.41016, MR 1078687 .

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