The Chalmers-Metcalf operator and minimal extensions. (English) Zbl 1467.46011
Summary: Let \(X\) be a real or complex Banach space and let \(Y \subset X\) be a finite-dimensional subspace. Fix \(A \in \mathcal{L}(Y)\). Then we define
\[
\mathcal{P}_A(X, Y) = \{P \in \mathcal{L}(X, Y) : P |_Y = A \} .
\]
An operator \(P_o \in \mathcal{P}_A(X, Y)\) is called a minimal extension (a minimal projection if \(A = \mathrm{Id}_Y\)) if
\[
\| P_o \| = \inf \{\| P \| : P \in \mathcal{P}_A(X, Y) \} .
\]
The aim of this paper is to present a variety of theorems characterizing minimal extensions, which generalize previously obtained results (in the real case) for minimal projections. We include several new applications, in which these theorems are utilized to determine minimal projections. Moreover, these characterizations employ so-called Chalmers-Metcalf operators (which are defined within the context of Theorem 1) and the form of these operators (when properly restricted) is also considered here. Indeed, we show that under certain assumptions, this form becomes quite simple – essentially the identity map – and this is of benefit in determining minimal extensions. We note that it has been conjectured that the assumptions we put in place to guarantee this simple form can be significantly weakened and we address this question.
References:
| [1] | Aksoy, A. G.; Lewicki, G., Minimal projections with respect to various norms, Stud. Math., 210, 1, 1-16 (2012) · Zbl 1252.41019 |
| [2] | Aksoy, A.; Leviatan, D.; Lewicki, G., In memoriam: Bruce L. Chalmers (1938-2015), J. Approx. Theory, 208, 1-8 (2016) · Zbl 1338.46004 |
| [3] | Basso, G., Computation of maximal projection constants, J. Funct. Anal., 277, 10, 3560-3585 (2019) · Zbl 1443.46004 |
| [4] | Blatter, J.; Cheney, E. W., Minimal projections onto hyperplanes in sequence spaces, Ann. Mat. Pura Appl., 101, 215-227 (1974) · Zbl 0303.46017 |
| [5] | Castejón, A.; Lewicki, G., Subspaces of codimension two with large projection constants, J. Math. Anal. Appl., 420, 1391-1407 (2014) · Zbl 1320.46011 |
| [6] | Castejón, A.; Lewicki, G., Some results on absolute projection constants, Numer. Funct. Anal. Optim., 40, 1, 34-51 (2019) · Zbl 1423.46013 |
| [7] | Chalmers, B. L.; Lewicki, G., Minimal projections onto some subspaces of \(l_1^{( n )}\), dedicated to Julian Musielak, Funct. Approx. Comment. Math., 26, 85-92 (1998) · Zbl 0922.46008 |
| [8] | Chalmers, B. L.; Lewicki, G., Symmetric subspaces of \(l_1\) with large projection constants, Stud. Math., 134, 119-133 (1999) · Zbl 0926.41018 |
| [9] | Chalmers, B. L.; Lewicki, G., Symmetric spaces with maximal projection constants, J. Funct. Anal., 200, 1-22 (2003) · Zbl 1032.46014 |
| [10] | Chalmers, B. L.; Lewicki, G., Three-dimensional subspace of \(l_\infty^{( 5 )}\) with maximal projection constant, J. Funct. Anal., 257, 553-592 (2009) · Zbl 1201.46011 |
| [11] | Chalmers, B. L.; Lewicki, G., A proof of the Grünbaum conjecture, Stud. Math., 201, 1, 103-129 (2010) · Zbl 1255.46005 |
| [12] | Chalmers, B. L.; Metcalf, F. T., The determination of minimal projections and extensions in \(L^1\), Trans. Am. Math. Soc., 329, 289-305 (1992) · Zbl 0753.41018 |
| [13] | Chalmers, B. L.; Metcalf, F. T., A characterization and equations for minimal projections and extensions, J. Oper. Theory, 32, 31-46 (1994) · Zbl 0827.41016 |
| [14] | Cheney, E. W.; Franchetti, C., Minimal projections in \(L_1\)-space, Duke Math. J., 43, 501-510 (1976) · Zbl 0335.41021 |
| [15] | Cheney, E. W.; Hobby, C. R.; Morris, P. D.; Schurer, F.; Wulbert, D. E., On the minimal property of the Fourier projection, Trans. Am. Math. Soc., 143, 249-258 (1969) · Zbl 0187.05603 |
| [16] | Cheney, E. W.; Light, W. A., Approximation Theory in Tensor Product Spaces, Lecture Notes in Math. (1985), Springer Verlag: Springer Verlag Berlin · Zbl 0575.41001 |
| [17] | Cheney, E. W.; Morris, P. D., On the existence and characterization of minimal projections, J. Reine Angew. Math., 270, 61-76 (1974) · Zbl 0303.41037 |
| [18] | Ciesielski, M.; Lewicki, G., On certain class of norms in semimodular spaces and their monotonicity properties, J. Math. Anal. Appl., 475, 490-518 (2019) · Zbl 1417.46005 |
| [19] | Deregowska, B.; Lewandowska, B., Minimal projections onto hyperplanes in vector-valued sequence spaces, J. Approx. Theory, 194, 1-13 (2015) · Zbl 1328.47033 |
| [20] | Deregowska, B.; Lewandowska, B., On the minimal property of de la Vallée Poussin’s operator, Bull. Aust. Math. Soc., 91, 1, 129-133 (2015) · Zbl 1317.42002 |
| [21] | Fisher, S. D.; Morris, P. D.; Wulbert, D. E., Unique minimality of Fourier projections, Trans. Am. Math. Soc., 265, 235-246 (1981) · Zbl 0545.42001 |
| [22] | Foucart, S.; Skrzypek, L., On maximal relative projection constants, J. Math. Anal. Appl., 447, 1, 309-328 (2017) · Zbl 1371.46010 |
| [23] | Grünbaum, B., Projection constants, Trans. Am. Math. Soc., 95, 451-465 (1960) · Zbl 0095.09002 |
| [24] | Kozdȩba, M., Minimal projection onto certain subspace of \(L_p(X \times Y \times Z))\), Numer. Funct. Anal. Optim., 39, 13, 1407-1422 (2018) · Zbl 1414.43002 |
| [25] | König, H., Spaces with large projection constants, Isr. J. Math., 50, 181-188 (1985) · Zbl 0582.46012 |
| [26] | König, H.; Lewis, D. R.; Lin, P. K., Finite-dimensional projection constants, Stud. Math., 75, 341-358 (1983) · Zbl 0529.46013 |
| [27] | König, H.; Tomczak-Jaegermann, N., Bounds for projection constants and 1-summing norms, Trans. Am. Math. Soc., 320, 799-823 (1990) · Zbl 0713.46012 |
| [28] | König, H.; Schuett, C.; Tomczak-Jaegermann, N., Projection constants of symmetric spaces and variants of Khinchine’s inequality, J. Reine Angew. Math., 511, 1-42 (1999) · Zbl 0926.46008 |
| [29] | Lambert, P. V., Minimum norm property of the Fourier projection in spaces of continuous functions, Bull. Soc. Math. Belg., 21, 359-369 (1969) · Zbl 0207.12601 |
| [30] | Lewicki, G., Minimal extensions in tensor product spaces, J. Approx. Theory, 97, 366-383 (1999) · Zbl 0927.46015 |
| [31] | Lewicki, G., Asymptotic estimate of absolute projection constants, Univ. Iagel. Acta Math., 51, 51-60 (2013) · Zbl 1325.46013 |
| [32] | Lewicki, G.; Mastyło, M., Asymptotic behavior of factorization and projection constants, J. Geom. Anal., 27, 1, 335-365 (2017) · Zbl 1372.46010 |
| [33] | Lewicki, G.; Prophet, M., Minimal shape-preserving projections onto \(\operatorname{\Pi} i_n\): generalizations and extensions, Numer. Funct. Anal. Optim., 27, 7-8, 847-873 (2006) · Zbl 1129.46014 |
| [34] | Lewicki, G.; Prophet, M., Minimal multi-convex projections, Stud. Math., 178, 2, 71-91 (2007) · Zbl 1120.46010 |
| [35] | Lewicki, G.; Prophet, M., Minimal shape-preserving projections in tensor product spaces, J. Approx. Theory, 162, 5, 931-951 (2010) · Zbl 1198.46013 |
| [36] | Lewicki, G.; Skrzypek, L., On properties of Chalmers-Metcalf operators, (Banach Spaces and Their Applications in Analysis (2007), Walter de Gruyter: Walter de Gruyter Berlin), 375-389 · Zbl 1134.41015 |
| [37] | Lewicki, G.; Skrzypek, L., Chalmers-Metcalf operator and uniqueness of minimal projections, J. Approx. Theory, 148, 71-91 (2007) · Zbl 1125.41017 |
| [38] | Lewicki, G.; Skrzypek, L., Minimal projections onto hyperplanes in \(l_p^n\), J. Approx. Theory, 202, 42-63 (2016) · Zbl 1339.46012 |
| [39] | Mielczarek, D., The unique minimality of an averaging projection, Monatshefte Math., 154, 157-171 (2008) · Zbl 1168.47019 |
| [40] | Musielak, J., Orlicz Spaces and Modular Spaces, Lecture Notes in Math., vol. 1034 (1983), Springer Verlag · Zbl 0557.46020 |
| [41] | Odyniec, W.; Lewicki, G., Minimal Projections in Banach Spaces, Lecture Notes in Mathematics, vol. 1449 (1990), Springer-Verlag · Zbl 1062.46500 |
| [42] | Rudin, W., Functional Analysis (1974), McGraw Hill · Zbl 0303.46001 |
| [43] | Rudin, W., Real and Complex Analysis (1974), McGraw Hill Inc. · Zbl 0278.26001 |
| [44] | Skrzypek, L., The uniqueness of minimal projections in smooth matrix spaces, J. Approx. Theory, 107, 315-336 (2000) · Zbl 0969.41021 |
| [45] | Skrzypek, L., Minimal projections in spaces of functions of N variables, J. Approx. Theory, 123, 214-231 (2003) · Zbl 1029.41018 |
| [46] | Shekhtman, B.; Skrzypek, L., Uniqueness of minimal projections onto two-dimensional subspaces, Stud. Math., 168, 273-284 (2005) · Zbl 1073.41029 |
| [47] | Sokołowski, F., Minimal projections onto subspaces with codimension 2, Numer. Funct. Anal. Optim., 38, 8, 1045-1059 (2017) · Zbl 1385.46005 |
| [48] | Wojtaszczyk, P., Banach Spaces for Analysts (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0724.46012 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
