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.
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. MR2654396 1192.42016 10.1142/S0219691310003377 P. Balazs, J.P. Antoine and A. Gryboś, Weighted and controlled frames, Int. J. Wavelets Multiresolut. Inf. Process. 8 (2010), no. 1, 109–132. MR2654396 1192.42016 10.1142/S0219691310003377
P. Balazs, Basic definition and properties of Bessel multipliers, J. Math. Anal. Appl. 325 (2007), no. 1, 571–585. MR2273547 1105.42023 10.1016/j.jmaa.2006.02.012 P. Balazs, Basic definition and properties of Bessel multipliers, J. Math. Anal. Appl. 325 (2007), no. 1, 571–585. MR2273547 1105.42023 10.1016/j.jmaa.2006.02.012
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. MR2450414 05315903 10.1142/S0219691308002379 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. MR2450414 05315903 10.1142/S0219691308002379
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. MR2142455 1091.46007 10.1016/j.jmaa.2005.02.015 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. MR2142455 1091.46007 10.1016/j.jmaa.2005.02.015
O. Christensen, Atomic decomposition via projective group representations, Rocky Mountain J. Math. 26 (1996), no. 4, 1289–1312. MR1447588 10.1216/rmjm/1181071989 euclid.rmjm/1181071989
0891.22006 O. Christensen, Atomic decomposition via projective group representations, Rocky Mountain J. Math. 26 (1996), no. 4, 1289–1312. MR1447588 10.1216/rmjm/1181071989 euclid.rmjm/1181071989
0891.22006
O. Christensen and C. Heil, Perturbations of Banach frames and atomic decompositions, Math. Nachr. 185 (1997), no. 1, 33–47. MR1452474 0868.42013 10.1002/mana.3211850104 O. Christensen and C. Heil, Perturbations of Banach frames and atomic decompositions, Math. Nachr. 185 (1997), no. 1, 33–47. MR1452474 0868.42013 10.1002/mana.3211850104
E. Dubinsky and M.S. Ramanujan, On $\lambda$-nuclearity, Mem. Amer. Math. Soc. no. 128, 1972. MR420215 0244.47015 E. Dubinsky and M.S. Ramanujan, On $\lambda$-nuclearity, Mem. Amer. Math. Soc. no. 128, 1972. MR420215 0244.47015
M. Fabian, P. Habala, P. Hájek, V.M. Santalucía, J. Pelant and V. Zizler, Functional Analysis and Infinite-Dimensional Geometry, Springer, New York, 2001. MR1831176 0981.46001 M. Fabian, P. Habala, P. Hájek, V.M. Santalucía, J. Pelant and V. Zizler, Functional Analysis and Infinite-Dimensional Geometry, Springer, New York, 2001. MR1831176 0981.46001
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. MR2945990 1243.42044 10.1001/jama.2011.1959 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. MR2945990 1243.42044 10.1001/jama.2011.1959
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. MR1780493 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. MR1780493
A. Rahimi and P. Balazs, Multipliers for p-Bessel sequences in Banach spaces, Integral Equations Operator Theory 68 (2010), no. 2, 193–205. MR2721082 1198.42030 10.1007/s00020-010-1814-7 A. Rahimi and P. Balazs, Multipliers for p-Bessel sequences in Banach spaces, Integral Equations Operator Theory 68 (2010), no. 2, 193–205. MR2721082 1198.42030 10.1007/s00020-010-1814-7
A.F. Ruston, Direct products of Banach spaces and linear functional equations, Proc. London Math. Soc. (3)1 (1951), no. 1, 327–384. MR44734 0043.11003 10.1112/plms/s3-1.1.327 A.F. Ruston, Direct products of Banach spaces and linear functional equations, Proc. London Math. Soc. (3)1 (1951), no. 1, 327–384. MR44734 0043.11003 10.1112/plms/s3-1.1.327
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. MR42612 0054.04906 10.1112/plms/s2-53.2.109 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. MR42612 0054.04906 10.1112/plms/s2-53.2.109
D.T. Stoeva and P. Balazs, Detailed characterization of unconditional convergence and invertibility of multipliers, arXiv:1007.0673v1. 1007.0673v1 MR2927462 10.1016/j.acha.2011.11.001 1246.42017 D.T. Stoeva and P. Balazs, Detailed characterization of unconditional convergence and invertibility of multipliers, arXiv:1007.0673v1. 1007.0673v1 MR2927462 10.1016/j.acha.2011.11.001 1246.42017
D.T. Stoeva and P. Balazs, Invertibility of multipliers, Appl. Comput. Harmon. Anal. 33 (2012), no. 2, 292–299. MR2927462 10.1016/j.acha.2011.11.001 1246.42017 D.T. Stoeva and P. Balazs, Invertibility of multipliers, Appl. Comput. Harmon. Anal. 33 (2012), no. 2, 292–299. MR2927462 10.1016/j.acha.2011.11.001 1246.42017
D.T. Stoeva and P. Balazs, Representation of the inverse of a multiplier as a multiplier, arXiv:1108.6286v3. 1108.6286v3 D.T. Stoeva and P. Balazs, Representation of the inverse of a multiplier as a multiplier, arXiv:1108.6286v3. 1108.6286v3
D.T. Stoeva and P. Balazs, Unconditional convergence and invertibility of multipliers, arXiv:0911.2783v3. 0911.2783v3 MR2927462 10.1016/j.acha.2011.11.001 1246.42017 D.T. Stoeva and P. Balazs, Unconditional convergence and invertibility of multipliers, arXiv:0911.2783v3. 0911.2783v3 MR2927462 10.1016/j.acha.2011.11.001 1246.42017
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. 1108.6282v2 1108.6282v2 1108.6282v2 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. 1108.6282v2 1108.6282v2 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. MR2515188 1173.42322 D.T. Stoeva, $X_d$-frames in Banach spaces and their duals, Int. J. Pure Appl. Math. 52 (2009), no. 1, 1–14. MR2515188 1173.42322
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. MR1908346 1026.46003 10.1155/S0161171202012772 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. MR1908346 1026.46003 10.1155/S0161171202012772
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. MR2795872 10.1016/j.acha.2010.09.007 1221.42067 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. MR2795872 10.1016/j.acha.2010.09.007 1221.42067
