Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

THERE ARE NO NUMERICAL RADIUS PEAK n-LINEAR MAPPINGS ON c0

  • Sung Guen Kim (Department of Mathematics Kyungpook National University)
  • Received : 2022.05.11
  • Accepted : 2022.09.20
  • Published : 2023.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

For n ≥ 2 and a real Banach space E, 𝓛(nE : E) denotes the space of all continuous n-linear mappings from E to itself. Let Π (E) = {[x*, (x1, . . . , xn)] : x*(xj) = ||x*|| = ||xj|| = 1 for j = 1, . . . , n }. An element [x*, (x1, . . . , xn)] ∈ Π(E) is called a numerical radius point of T ∈ 𝓛(nE : E) if |x*(T(x1, . . . , xn))| = v(T), where the numerical radius v(T) = sup[y*,y1,...,yn]∈Π(E)|y*(T(y1, . . . , yn))|. For T ∈ 𝓛(nE : E), we define Nradius(T) = {[x*, (x1, . . . , xn)] ∈ Π(E) : [x*, (x1, . . . , xn)] is a numerical radius point of T}. T is called a numerical radius peak n-linear mapping if there is a unique [x*, (x1, . . . , xn)] ∈ Π(E) such that Nradius(T) = {±[x*, (x1, . . . , xn)]}. In this paper we present explicit formulae for the numerical radius of T for every T ∈ 𝓛(nE : E) for E = c0 or l∞. Using these formulae we show that there are no numerical radius peak mappings of 𝓛(nc0 : c0).

Keywords

Acknowledgement

The author is thankful to the referee for the careful reading and considered suggestions leading to a better-presented paper.

References

  1. R. M. Aron, C. Finet, and E. Werner, Some remarks on norm-attaining n-linear forms, in Function spaces (Edwardsville, IL, 1994), 19-28, Lecture Notes in Pure and Appl. Math., 172, Dekker, New York, 1995.
  2. E. Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 97-98. https://doi.org/10.1090/S0002-9904-1961-10514-4
  3. Y. S. Choi, D. Garcia, S. G. Kim, and M. Maestre, Norm or numerical radius attaining polynomials on C(K), J. Math. Anal. Appl. 295 (2004), no. 1, 80-96. https://doi.org/10.1016/j.jmaa.2004.03.005
  4. Y. S. Choi and S. G. Kim, Norm or numerical radius attaining multilinear mappings and polynomials, J. London Math. Soc. (2) 54 (1996), no. 1, 135-147. https://doi.org/10.1112/jlms/54.1.135
  5. S. Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London, 1999. https://doi.org/10.1007/978-1-4471-0869-6
  6. S. G. Kim, The norming set of a polynomial in P(2l2∞), Honam Math. J. 42 (2020), no. 3, 569-576. https://doi.org/10.5831/HMJ.2020.42.3.569
  7. S. G. Kim, The norming set of a bilinear form on l2∞, Comment. Math. 60 (2020), no. 1-2, 37-63.
  8. S. G. Kim, The norming set of a symmetric bilinear form on the plane with the supremum norm, Mat. Stud. 55 (2021), no. 2, 171-180. https://doi.org/10.30970/ms.55.2.171-180
  9. S. G. Kim, The norming set of a symmetric 3-linear form on the plane with the l1-norm, New Zealand J. Math. 51 (2021), 95-108. https://doi.org/10.53733/177
  10. S. G. Kim, NA(𝓛(nl1 : l1)) = NRA(𝓛(nl1 : l1)), Acta Sci. Math. (Szeged) 88 (2022), no. 3-4, 769-775. https://doi.org/10.1007/s44146-022-00048-5
  11. S. G. Kim, Three kinds of numerical indices of lp-spaces, Glas. Mat. Ser. III 57(77) (2022), no. 1, 49-61. https://doi.org/10.3336/gm.57.1.04
  12. S. G. Kim, Numerical radius points of a bilinear mapping in 𝓛(2l21: l21), Preprint.
  13. M. J. Sevilla and R. Paya, Norm attaining multilinear forms and polynomials on preduals of Lorentz sequence spaces, Studia Math. 127 (1998), no. 2, 99-112. https://doi.org/10.4064/sm-127-2-99-112