# Extensions of linear operators from hyperplanes of ${l}_{\infty }^{\left(n\right)}$

• Volume: 36, Issue: 3, page 443-458
• ISSN: 0010-2628

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## Abstract

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Let $Y\subset {l}_{\infty }^{\left(n\right)}$ be a hyperplane and let $A\in ℒ\left(Y\right)$ be given. Denote $\begin{array}{ccc}\hfill 𝒜=& \left\{L\in ℒ\left({l}_{\infty }^{\left(n\right)},Y\right):L\mid Y=A\right\}\phantom{\rule{4.0pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}\hfill & \hfill {\lambda }_{A}=inf\left\{\parallel L\parallel :L\in 𝒜\right\}.\end{array}$ In this paper the problem of calculating of the constant ${\lambda }_{A}$ is studied. We present a complete characterization of those $A\in ℒ\left(Y\right)$ for which ${\lambda }_{A}=\parallel A\parallel$. Next we consider the case ${\lambda }_{A}>\parallel A\parallel$. Finally some computer examples will be presented.

## How to cite

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Baronti, Marco, Fragnelli, Vito, and Lewicki, Grzegorz. "Extensions of linear operators from hyperplanes of $l^{(n)}_\infty$." Commentationes Mathematicae Universitatis Carolinae 36.3 (1995): 443-458. <http://eudml.org/doc/247744>.

@article{Baronti1995,
abstract = {Let $Y \subset l^\{(n)\}_\{\infty \}$ be a hyperplane and let $A \in \{\mathcal \{L\}\}(Y)$ be given. Denote \begin\{@align\}\{1\}\{-1\}\{\mathcal \{A\}\} = & \lbrace L\in \{\mathcal \{L\}\}(l^\{(n)\}\_\{\infty \},Y):L\mid Y = A\rbrace \text\{ and\} \ & \lambda \_\{A\} = \inf \lbrace \parallel L \parallel : L\in \{\mathcal \{A\}\}\rbrace . \end\{@align\} In this paper the problem of calculating of the constant $\lambda _\{A\}$ is studied. We present a complete characterization of those $A \in \{\mathcal \{L\}\}(Y)$ for which $\lambda _\{A\} = \parallel A \parallel$. Next we consider the case $\lambda _\{A\} > \parallel A \parallel$. Finally some computer examples will be presented.},
author = {Baronti, Marco, Fragnelli, Vito, Lewicki, Grzegorz},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {linear operator; extension of minimal norm; element of best approximation; strongly unique best approximation},
language = {eng},
number = {3},
pages = {443-458},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Extensions of linear operators from hyperplanes of $l^\{(n)\}_\infty$},
url = {http://eudml.org/doc/247744},
volume = {36},
year = {1995},
}

TY - JOUR
AU - Baronti, Marco
AU - Fragnelli, Vito
AU - Lewicki, Grzegorz
TI - Extensions of linear operators from hyperplanes of $l^{(n)}_\infty$
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1995
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 36
IS - 3
SP - 443
EP - 458
AB - Let $Y \subset l^{(n)}_{\infty }$ be a hyperplane and let $A \in {\mathcal {L}}(Y)$ be given. Denote \begin{@align}{1}{-1}{\mathcal {A}} = & \lbrace L\in {\mathcal {L}}(l^{(n)}_{\infty },Y):L\mid Y = A\rbrace \text{ and} \ & \lambda _{A} = \inf \lbrace \parallel L \parallel : L\in {\mathcal {A}}\rbrace . \end{@align} In this paper the problem of calculating of the constant $\lambda _{A}$ is studied. We present a complete characterization of those $A \in {\mathcal {L}}(Y)$ for which $\lambda _{A} = \parallel A \parallel$. Next we consider the case $\lambda _{A} > \parallel A \parallel$. Finally some computer examples will be presented.
LA - eng
KW - linear operator; extension of minimal norm; element of best approximation; strongly unique best approximation
UR - http://eudml.org/doc/247744
ER -

## References

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1. Baronti M., Papini P.L., Norm one projections onto subspaces of ${l}^{p}$, Ann. Mat. Pura Appl. IV (1988), 53-61. (1988) MR0980971
2. Blatter J., Cheney E.W., Minimal projections onto hyperplanes in sequence spaces, Ann. Mat. Pura Appl. 101 (1974), 215-227. (1974) MR0358179
3. Collins H.S., Ruess W., Weak compactness in spaces of compact operators and vector valued functions, Pacific J. Math. 106 (1983), 45-71. (1983) MR0694671
4. Odyniec Wl., Lewicki G., Minimal Projections in Banach Spaces, Lecture Notes in Math. 1449, Springer-Verlag. Zbl1062.46500MR1079547
5. Singer I., On the extension of continuous linear functionals..., Math. Ann. 159 (1965), 344-355. (1965) Zbl0141.12002MR0188758
6. Singer I., Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces, Springer-Verlag, Berlin, Heidelberg, New York, 1970. Zbl0197.38601MR0270044
7. Sudolski J., Wojcik A., Some remarks on strong uniqueness of best approximation, Approximation Theory and its Applications 6 (1990), 44-78. (1990) Zbl0704.41016MR1078687

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