Approximation numbers of matrix transformations and inclusion maps

Main Article Content

M. Gupta
L. R. Acharya

Abstract

In this paper we establish relationships of the approximation numbers of matrix transformations acting between the vector-valued sequence spaces spaces of the type $\lambda(X)$ defined corresponding to a scalar-valued sequence space $\lambda$ and a Banach space $(X,\|.\|)$ as $$\lambda(X)=\{\overline x=\{x_i\}: x_i\in X, \forall~i\in \mathbb{N},~\{\|x_i\|_X\}\in \lambda\};$$ with those of their component operators. This study leads to a characterization of a diagonal operator to be approximable. Further, we compute the approximation numbers of inclusion maps acting between $\ell^p(X)$ spaces for $1\leq p\leq \infty$.

Article Details

How to Cite
Gupta, M., & Acharya, L. R. (2011). Approximation numbers of matrix transformations and inclusion maps. Tamkang Journal of Mathematics, 42(2), 193–203. https://doi.org/10.5556/j.tkjm.42.2011.924
Section
Papers

References

L. R. Acharya and M. Gupta, On Kolmogorov numbers of matrix transformations, Banach spaces and their applications in analysis, de Gruyter Proceedings in Mathematics, Berlin, New York (2007), 219–228.

B. Carl and I. Stephani, Entropy, Compactness and the Approximation of Operators, Cambridge Univ. Press, Cambridge, 1990.

P. Enflo, A counter example to the approximation problem in Banach spaces, Acta. Math., 130 (1973), 309-317.

N. De Grande - De Kimpe, Generalized sequence spaces, Bull. Soc.Math. Belgique 23 (1971), 123–166.

M. Gupta and J. Patterson, The generalized $l_p$ spaces, Tamkang J.Math., 13 (1982), 161–179.

——-,Matrix transformations on generalized sequence spaces, J.Math. Anal. Appl., 106 (1985), 54–68.

A. Hinrichs, Approximation numbers of identity operators between symmetric sequence spaces, J. Approx. Theory 118 (2002), 305–315.

A.Hinrichs and C.Michels, Approximation numbers of inclusions between Schatten classes, Rend. Circ.Mat. Palermo (2) Suppl. No. 76, (2005), 395–411.

C. V. Hutton, J. S.Morrell and J. R. Retherford,Diagonal operators, approximation numbers and Kolmogoroff diameters, J. Approx. Theory 16 (1976), 48–80.

J. A. SampaioMartins,On the approximation numbers of embeddings of sequence spaces,Mathematical studies in honor of Professor Luis de Albuquerque (Portuguese), Univ. Coimbra, Coimbra, 1994.

J. Patterson, Generalized sequence spaces and matrix transformations, Dissertation, I. I. T. Kanpur, India, 1980.

A. Pietsch, Einige neue Klassen von kompakten linearen Abbildungen, Rev. Roumaine Math. Pures Appl. 8 (1963), 427–447.

——-, Nuclear Locally Convex Spaces, Springer-Verlag, Berlin, Heidelberg, New York, 1972.

——-, s− numbers of operators in Banach spaces, StudiaMath., 51 (1974), 201–223.

——-, Operator Ideals, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.

——-, Eigenvalues and s-numbers, Cambridge University Press, New York, NY, USA, 1986.

L. Skrzypczak, On approximation numbers of Sobolov embeddings of weighted sequence spaces, J. Approx. Theory 136 (2005), 91–107.