## Banach Journal of Mathematical Analysis

### Interpolation of $S$-numbers and entropy numbers of operators

#### Abstract

We introduce the notion of $\vec{s}$ -numbers for operators in Banach couples. We investigate variants of the approximation, Gelfand, and Kolmogorov numbers. In particular, we derive upper estimates of these numbers for operators between spaces generated by interpolation functors on Banach couples satisfying interpolation variants of approximation properties. We also study two-sided interpolation of entropy numbers.

#### Article information

Source
Banach J. Math. Anal., Volume 13, Number 2 (2019), 427-448.

Dates
Received: 19 August 2018
Accepted: 5 December 2018
First available in Project Euclid: 27 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1551258125

Digital Object Identifier
doi:10.1215/17358787-2018-0046

Mathematical Reviews number (MathSciNet)
MR3927881

Zentralblatt MATH identifier
07045466

#### Citation

Mastyło, Mieczysław; Szwedek, Radosław. Interpolation of $S$ -numbers and entropy numbers of operators. Banach J. Math. Anal. 13 (2019), no. 2, 427--448. doi:10.1215/17358787-2018-0046. https://projecteuclid.org/euclid.bjma/1551258125

#### References

• [1] J. Bergh and J. Löfström, Interpolation Spaces: An Introduction, Grundlehren Math. Wiss. 223, Springer, Berlin, 1976.
• [2] B. Carl, On $s$-numbers, quasi $s$-numbers, $s$-moduli and Weyl inequalities of operators in Banach spaces, Rev. Mat. Complut. 23 (2010), no. 2, 467–487.
• [3] B. Carl and I. Stephani, Entropy, Compactness and the Approximation of Operators, Cambridge Tracts in Math. 98, Cambridge Univ. Press, Cambridge, 1990.
• [4] B. Carl and H. Triebel, Inequalities between eigenvalues, entropy numbers, and related quantities of compact operators in Banach spaces, Math. Ann. 251 (1980), no. 2, 129–133.
• [5] F. Cobos and J. Peetre, Interpolation of compactness using Aronszajn–Gagliardo functors, Israel J. Math. 68 (1989), no. 2, 220–240.
• [6] D. E. Edmunds and Y. Netrusov, Entropy numbers and interpolation, Math. Ann. 351 (2011), no. 4, 963–977.
• [7] Y. Gordon, H. König, and C. Schütt, Geometric and probabilistic estimates for entropy and approximation numbers of operators, J. Approx. Theory 49 (1987), no. 3, 219–239.
• [8] S. Janson, Interpolation of subcouples and quotient couples, Ark. Mat. 31 (1993), no. 2, 307–338.
• [9] B. S. Kašin, Kolmogorov diameters of octahedra (in Russian), Dokl. Akad. Nauk SSSR 214 (1974), 1024–1026; English translation in Dokl. Math. 15 (1974), 304–307.
• [10] H. König, Eigenvalue Distribution of Compact Operators, Oper. Theory Adv. Appl. 16, Birkhäuser, Basel 1986.
• [11] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Lecture Notes in Math. 338, Springer, Berlin, 1973.
• [12] J. L. Lions, Sur les espaces d’interpolation: Dualité, Math. Scand. 9 (1961), 147–177.
• [13] M. Mastyło, Interpolation estimates for entropy numbers with applications to non-convex bodies, J. Approx. Theory 162 (2010), no. 1, 10–23.
• [14] M. Mastyło and R. Szwedek, Eigenvalues and entropy moduli of operators in interpolation spaces, J. Geom. Anal. 27 (2017), no. 2, 1131–1177.
• [15] V. I. Ovchinnikov, The method of orbits in interpolation theory, Math. Rep. 1 (1984), no. 2, 349–515.
• [16] A. Pełczyński and H. P. Rosenthal, Localization techniques in $L^{p}$ spaces, Studia Math. 52 (1974/75), 263–289.
• [17] A. Persson, Compact linear mappings between interpolation spaces, Ark. Mat. 5 (1964), 215–219.
• [18] A. Pietsch, $s$-numbers of operators in Banach spaces, Studia Math. 51 (1974), 201–223.
• [19] A. Pietsch, Operator Ideals, North-Holland Math. Lib. 20, North-Holland, Amsterdam 1980.
• [20] A. Pietsch, Eigenvalues and $s$-numbers, Cambridge Stud. Adv. Math. 13, Cambridge Univ. Press, Cambridge, 1987.
• [21] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math. 94, Cambridge Univ. Press, Cambridge, 1989.
• [22] A. Szankowski, On the uniform approximation property in Banach spaces, Israel J. Math. 49 (1984), no. 4, 343–359.
• [23] R. Szwedek, Geometric interpolation of entropy numbers, Q. J. Math. 69 (2018), no. 2, 377–389.
• [24] M. F. Teixeira and D. E. Edmunds, Interpolation theory and measures of noncompactness, Math. Nachr. 104 (1981), 129–135.