Banach Journal of Mathematical Analysis

$(X_{d}, X_{d}^{*})$-Bessel multipliers in Banach spaces

Mohammad Hasan Faroughi , Elnaz Osgooei , and Asghar Rahimi

Full-text: Open access

Abstract

Multipliers have recently been introduced as operators for Bessel sequences and frames in Hilbert spaces. In this paper, we define the concept of $(X_{d}, X_{d}^{*})$ and $(l^{\infty}, X_{d}, X_{d}^{*})$-Bessel multipliers in Banach spaces and investigate the compactness of these multipliers. Also, we study the possibility of invertibility of $(l^{\infty}, X_{d}, X_{d}^{*})$-Bessel multiplier depending on the properties of its corresponding sequences and its symbol. Furthermore, we prove that every $(X_{d}, X_{d}^{*})$-Bessel multiplier is a $\lambda$-nuclear operator.

Article information

Source
Banach J. Math. Anal., Volume 7, Number 2 (2013), 146 -161 .

Dates
First available in Project Euclid: 20 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1363784228

Digital Object Identifier
doi:10.15352/bjma/1363784228

Mathematical Reviews number (MathSciNet)
MR3039944

Zentralblatt MATH identifier
1266.42083

Subjects
Primary: 42C40
Secondary: 47B99 42C15 41A58

Keywords
$X_{d}$-Bessel sequence ($X_{d}, X_{d}^{*}$)-Bessel multiplier $\lambda$-nuclear operator

Citation

Faroughi , Mohammad Hasan; Osgooei , Elnaz; Rahimi , Asghar. $(X_{d}, X_{d}^{*})$-Bessel multipliers in Banach spaces. Banach J. Math. Anal. 7 (2013), no. 2, 146 --161. doi:10.15352/bjma/1363784228. https://projecteuclid.org/euclid.bjma/1363784228


Export citation

References

  • P. Balazs, J.P. Antoine and A. Gryboś, Weighted and controlled frames, Int. J. Wavelets Multiresolut. Inf. Process. 8 (2010), no. 1, 109–132.
  • P. Balazs, Basic definition and properties of Bessel multipliers, J. Math. Anal. Appl. 325 (2007), no. 1, 571–585.
  • P. Balazs, Hilbert-Schmidt operators and frames-classification, best approximation by multipliers and algorithms, Int. J. Wavelets Multiresolut. Inf. Process. 6 (2008), no. 2, 315–330.
  • P. Balazs, Matrix representation of operators using frames, Sampl. Theory Signal Image Process. 7 (2008), no. 1, 39–54.
  • P.G. Casazza, O. Christensen and D.T. Stoeva, Frame expansions in separable Banach spaces, J. Math. Anal. Appl. 307 (2005), no. 2, 710–723.
  • P.G. Casazza, D. Han and D.R. Larson, Frames for Banach spaces, Contemp. Math. 247 (1999), 149–182.
  • O. Christensen, Atomic decomposition via projective group representations, Rocky Mountain J. Math. 26 (1996), no. 4, 1289–1312.
  • O. Christensen and C. Heil, Perturbations of Banach frames and atomic decompositions, Math. Nachr. 185 (1997), no. 1, 33–47.
  • E. Dubinsky and M.S. Ramanujan, On $\lambda$-nuclearity, Mem. Amer. Math. Soc. no. 128, 1972.
  • M. Fabian, P. Habala, P. Hájek, V.M. Santalucía, J. Pelant and V. Zizler, Functional Analysis and Infinite-Dimensional Geometry, Springer, New York, 2001.
  • K. Gröchenig, Describing functions: atomic decompositions versus frames, Monatsh. Math. 112 (1991), no. 1, 1–42.
  • S.K. Kaushik, A generalization of frames in Banach spaces, J. Contemp. Math. Anal. 44 (2009), no. 4, 212–218.
  • G. Köthe, Topological Vector Spaces I, Springer, New York, 1969.
  • B. Liu and R. Liu, Upper Beurling density of systems formed by translates of finite sets of elements in $L^{p}(\mathbb{R}^{d})$, Banach J. Math. Anal. 6 (2012), no. 2, 86–97.
  • E. Malkowsky and V. Rakočević, An introduction into the theory of sequence spaces and measures of noncompactness, Zb. Rad. 9 (2000), no. 17, 143–234.
  • A. Rahimi and P. Balazs, Multipliers for p-Bessel sequences in Banach spaces, Integral Equations Operator Theory 68 (2010), no. 2, 193–205.
  • A. Rahimi, Multipliers of generalized frames in Hilbert spaces, Bull. Iran. Math. Soc. 37 (2011), no. 1, 63–80.
  • M.S. Ramanujan, Generalized nuclear maps in normed linear spaces, J. reine angew. Math. 1970 (1970), no. 244, 190–197.
  • A.F. Ruston, Direct products of Banach spaces and linear functional equations, Proc. London Math. Soc. (3)1 (1951), no. 1, 327–384.
  • A.F. Ruston, On the Fredholm theory of integral equations for operators belonging to the trace class of a general Banach space, Proc. London. Math. Soc. (2) 53 (1951), no. 1, 109–124.
  • R. Schatten, Norm Ideals of Completely Continuous Operators, Springer, Berlin, 1960.
  • D.T. Stoeva and P. Balazs, Detailed characterization of unconditional convergence and invertibility of multipliers, arXiv:1007.0673v1.
  • D.T. Stoeva and P. Balazs, Invertibility of multipliers, Appl. Comput. Harmon. Anal. 33 (2012), no. 2, 292–299.
  • D.T. Stoeva and P. Balazs, Representation of the inverse of a multiplier as a multiplier, arXiv:1108.6286v3.
  • D.T. Stoeva and P. Balazs, Unconditional convergence and invertibility of multipliers, arXiv:0911.2783v3.
  • D.T. Stoeva, Characterization of atomic decompositions, Banach frames, $X_{d}$-frames, duals and synthesis-pseudo-duals, with application to Hilbert frame theorey, arXiv:1108.6282v2.
  • D.T. Stoeva, $X_d$-frames in Banach spaces and their duals, Int. J. Pure Appl. Math. 52 (2009), no. 1, 1–14.
  • D.T. Stoeva, $X_d$-Riesz bases in separable Banach spaces, “Collection of papers, ded. to the 60th Anniv. of M. Konstantinov", BAS Publ. House, 2008.
  • S. Suantai and W. Sanhan, On $\beta$-dual of vector-valued sequence spaces of Maddox, Int. J. Math. Sci. 30 (2002), no. 7, 383–392.
  • H. Zhang and J. Zhang, Frames, Riesz bases, and sampling expansions in Banach spaces via semi-inner products, Appl. Comput. Harmon. Anal. 31 (2011), no. 1, 1–25.