Isomorphic Schauder decompositions in certain Banach spaces

Vitalii Marchenko

Open Mathematics (2014)

  • Volume: 12, Issue: 11, page 1714-1732
  • ISSN: 2391-5455

Abstract

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We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use ℓψ-Hilbertian and ∞-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition to the case of Banach spaces and introduce the class of Banach spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type, cotype, infratype and M-cotype of a Banach space and study the properties of unconditional Schauder decompositions in Banach spaces possessing certain geometric structure.

How to cite

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Vitalii Marchenko. "Isomorphic Schauder decompositions in certain Banach spaces." Open Mathematics 12.11 (2014): 1714-1732. <http://eudml.org/doc/269687>.

@article{VitaliiMarchenko2014,
abstract = {We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use ℓψ-Hilbertian and ∞-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition to the case of Banach spaces and introduce the class of Banach spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type, cotype, infratype and M-cotype of a Banach space and study the properties of unconditional Schauder decompositions in Banach spaces possessing certain geometric structure.},
author = {Vitalii Marchenko},
journal = {Open Mathematics},
keywords = {Isomorphic Schauder decompositions; Unconditional Schauder decompositions; Projections; Orlicz spaces; isomorphic Schauder decompositions; unconditional Schauder decompositions; projections; Khintchine inequality},
language = {eng},
number = {11},
pages = {1714-1732},
title = {Isomorphic Schauder decompositions in certain Banach spaces},
url = {http://eudml.org/doc/269687},
volume = {12},
year = {2014},
}

TY - JOUR
AU - Vitalii Marchenko
TI - Isomorphic Schauder decompositions in certain Banach spaces
JO - Open Mathematics
PY - 2014
VL - 12
IS - 11
SP - 1714
EP - 1732
AB - We extend a theorem of Kato on similarity for sequences of projections in Hilbert spaces to the case of isomorphic Schauder decompositions in certain Banach spaces. To this end we use ℓψ-Hilbertian and ∞-Hilbertian Schauder decompositions instead of orthogonal Schauder decompositions, generalize the concept of an orthogonal Schauder decomposition to the case of Banach spaces and introduce the class of Banach spaces with Schauder-Orlicz decompositions. Furthermore, we generalize the notions of type, cotype, infratype and M-cotype of a Banach space and study the properties of unconditional Schauder decompositions in Banach spaces possessing certain geometric structure.
LA - eng
KW - Isomorphic Schauder decompositions; Unconditional Schauder decompositions; Projections; Orlicz spaces; isomorphic Schauder decompositions; unconditional Schauder decompositions; projections; Khintchine inequality
UR - http://eudml.org/doc/269687
ER -

References

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  1. [1] Adduci J., Mityagin B., Eigensystem of an L 2-perturbed harmonic oscillator is an unconditional basis, Cent. Eur. J. Math., 2012, 10(2), 569–589 http://dx.doi.org/10.2478/s11533-011-0139-3 Zbl1259.47059
  2. [2] Adduci J., Mityagin B., Root system of a perturbation of a selfadjoint operator with discrete spectrum, Integral Equations Operator Theory, 2012, 73(2), 153–175 http://dx.doi.org/10.1007/s00020-012-1967-7 Zbl1294.47022
  3. [3] Ahmad K., A note on equivalence of sequences of subspaces in Banach spaces, An. Stiint. Univ. ”Ovidius” Constanta Ser. Mat., 1989, 27, 9–12 
  4. [4] Allexandrov G., Kutzarova D., Plichko A., A separable space with no Schauder decomposition, Proc. Amer. Math. Soc., 1999, 127(9), 2805–2806 http://dx.doi.org/10.1090/S0002-9939-99-05370-8 Zbl0921.46007
  5. [5] Bari N.K., Biorthogonal systems and bases in Hilbert space, Moskov. Gos. Univ. Učenye Zapiski. Matematika, 1951, 148(4), 69–107 (in Russian) 
  6. [6] Bilalov B.T., Veliev S.G., Some Questions of Bases, Elm, Baku, 2010 (in Russian) Zbl1149.34352
  7. [7] Bonet J., Ricker W.J., Schauder decompositions and the Grothendieck and Dunford-Pettis properties in Köthe echelon spaces of infinite order, Positivity, 2007, 11(1), 77–93 http://dx.doi.org/10.1007/s11117-006-2014-1 Zbl1131.46005
  8. [8] Chadwick J.J.M., Cross R.W., Schauder decompositions in non-separable Banach spaces, Bull. Aust. Math. Soc., 1972, 6(1), 133–144 http://dx.doi.org/10.1017/S0004972700044336 Zbl0221.46018
  9. [9] Clark C., On relatively bounded perturbations of ordinary differential operators, Pacific J. Math., 1968, 25(1), 59–70 http://dx.doi.org/10.2140/pjm.1968.25.59 Zbl0185.23101
  10. [10] Clement P., De Pagter B., Sukochev F.A., Witvliet H., Schauder decompositions and multiplier theorems, Studia Math., 2000, 138(2), 135–163 Zbl0955.46004
  11. [11] Curtain R.F., Zwart H.J., An Introduction to Infinite-Dimensional Linear Systems Theory, Texts in Applied Mathematics, Volume 21, Springer-Verlag, New-York, 1995 http://dx.doi.org/10.1007/978-1-4612-4224-6 Zbl0839.93001
  12. [12] Davis W.J., Schauder decompositions in Banach spaces, Bull. Amer. Math. Soc., 1968, 74(6), 1083–1085 http://dx.doi.org/10.1090/S0002-9904-1968-12054-3 Zbl0167.12803
  13. [13] De la Rosa M., Frerick L., Grivaux S., Peris A., Frequent hypercyclicity, chaos, and unconditional Schauder decompositions, Israel J. Math., 2012, 190(1), 389–399 http://dx.doi.org/10.1007/s11856-011-0210-6 Zbl1258.47012
  14. [14] De Pagter B., Ricker W.J., Products of commuting Boolean algebras of projections and Banach space geometry, Proc. Lond. Math. Soc. (3), 2005, 91(3), 483–508 http://dx.doi.org/10.1112/S0024611505015303 Zbl1093.46010
  15. [15] Djakov P., Mityagin B., Criteria for existence of Riesz bases consisting of root functions of Hill and 1D Dirac operators, J. Funct. Anal., 2012, 263(8), 2300–2332 http://dx.doi.org/10.1016/j.jfa.2012.07.003 Zbl1263.34121
  16. [16] Fage M.K., Idempotent operators and their rectification, Dokl. Akad. Nauk, 1950, 73, 895–897 (in Russian) 
  17. [17] Fage M.K., The rectification of bases in Hilbert space, Dokl. Akad. Nauk, 1950, 74, 1053–1056 (in Russian) 
  18. [18] Gelfand I.M., A remark on N.K. Bari’s paper “Biorthogonal systems and bases in Hilbert space”, Moskov. Gos. Univ. Učenye Zapiski. Matematika, 1951, 148(4), 224–225 (in Russian) 
  19. [19] Gohberg I.C., Krein M.G., Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Transl. Math. Monogr., 18, American Mathematical Society, Providence, Rhode Island, 1969 Zbl0181.13504
  20. [20] Grinblyum M.M., On the representation of a space of type B in the form of a direct sum of subspaces, Dokl. Akad. Nauk, 1950, 70, 749–752 (in Russian) 
  21. [21] Gurarii V.I., Gurarii N.I., Bases in uniformly convex and uniformly smooth Banach spaces, Izv. Ross. Akad. Nauk Ser. Mat., 1971, 35(1), 210–215 (in Russian) Zbl0199.43902
  22. [22] Haagerup U., The best constants in the Khintchine inequality, Studia Math., 1982, 70, 231–283 Zbl0501.46015
  23. [23] Haase M., A decomposition theorem for generators of strongly continuous groups on Hilbert spaces, J. Operator Theory, 2004, 52, 21–37. Zbl1104.47042
  24. [24] Haase M., The Functional Calculus for Sectorial Operators, Oper. Theory Adv. Appl., vol. 169, Birkhäuser, Basel, 2006. http://dx.doi.org/10.1007/3-7643-7698-8 
  25. [25] Hughes E., Perturbation theorems for relative spectral problems, Canad. J. Math., 1972, 24(1), 72–81 http://dx.doi.org/10.4153/CJM-1972-009-8 Zbl0255.47023
  26. [26] Jain P.K., Ahmad K., Maskey S.M., Domination and equivalence of sequences of subspaces in dual spaces, Czechoslovak Math. J., 1986, 36(3), 351–357 http://dx.doi.org/10.1007/BF01597835 Zbl0627.46012
  27. [27] Jain P.K., Ahmad K., Schauder decompositions and best approximations in Banach spaces, Port. Math., 1987, 44(1), 25–39 Zbl0636.41016
  28. [28] Jain P.K., Ahmad K., Unconditional Schauder decompositions and best approximations in Banach spaces, Indian J. Pure Appl. Math., 1981, 12(12), 1456–1467 Zbl0501.46014
  29. [29] Johnson W.B., Finite-dimensional Schauder decompositions in π λ and dual π λ spaces, Illinois J. Math., 1970, 14(4), 642–647 
  30. [30] Johnson W.B., Lindenstrauss J., Handbook of the Geometry of Banach Spaces, Volume 1, Elsevier, 2001 
  31. [31] Johnson W.B., Lindenstrauss J., Handbook of the Geometry of Banach Spaces, Volume 2, Elsevier, 2003 
  32. [32] Kadets M.I, Kadets V.M., Series in Banach Spaces, Conditional and Unconditional Convergence, Birkhäuser, Berlin, 1997 Zbl0876.46009
  33. [33] Kato T., Perturbation Theory for Linear Operators, 2nd ed. (reprint), Classics Math., Springer, Berlin, 1995 
  34. [34] Kato T., Similarity for sequences of projections, Bull. Amer. Math. Soc., 1967, 73(6), 904–905 http://dx.doi.org/10.1090/S0002-9904-1967-11836-6 Zbl0156.38103
  35. [35] Köthe G., Toeplitz O., Linear Räume mit unendlich vielen Koordinaten und Ringe unendlicher Matrizen, J. Reine Angew. Math., 1934, 171, 193–226 Zbl0009.25704
  36. [36] Krein M., Milman D., Rutman M., On a property of a basis in a Banach space, Comm. Inst. Sci. Math. Mec. Univ. Kharkoff [Zapiski Inst. Mat. Mech.], 1940, 16(4), 106–110 (in Russian, with English summary) Zbl0023.13105
  37. [37] Lindenstrauss J., Tzafriri L., Classical Banach Spaces I and II, Reprint of the 1977, 1979 ed., Springer-Verlag, Berlin, 1996 Zbl0403.46022
  38. [38] Lorch E.R., Bicontinuous linear transformations in certain vector spaces, Bull. Amer. Math. Soc., 1939, 45, 564–569 http://dx.doi.org/10.1090/S0002-9904-1939-07035-3 Zbl0022.05302
  39. [39] Marcus A.S., A basis of root vectors of a dissipative operator, Dokl. Akad. Nauk, 1960, 132(3), 524–527 (in Russian) 
  40. [40] Marcus A.S., Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Monogr., 71, American Mathematical Society, Providence, Rhode Island, 1988 
  41. [41] Mityagin B., Siegl P., Root system of singular perturbations of the harmonic oscillator type operators, preprint available at http://arxiv.org/abs/1307.6245 Zbl1330.47023
  42. [42] Orlicz W., Über die Divergenz von allgemeinen Orthogonalreihen & Über unbedingte Convergenz in Funktionraümen, Studia Math., 1933, 4, 27–37 
  43. [43] Rabah R., Sklyar G.M., Rezounenko A.V., Generalized Riesz basis property in the analysis of neutral type systems, C. R. Math. Acad. Sci. Paris, Ser. I, 2003, 337, 19–24 http://dx.doi.org/10.1016/S1631-073X(03)00251-6 Zbl1035.34092
  44. [44] Rabah R., Sklyar G.M., Rezounenko A.V., Stability analysis of neutral type systems in Hilbert space, J. Differential Equations, 2005, 214, 391–428 http://dx.doi.org/10.1016/j.jde.2004.08.001 Zbl1083.34058
  45. [45] Rabah R., Sklyar G.M., The analysis of exact controllability of neutral-type systems by the moment problem approach, SIAM J. Control Optim., 2007, 46(6), 2148–2181 http://dx.doi.org/10.1137/060650246 Zbl1149.93011
  46. [46] Retherford J.R., Basic sequences and the Paley-Wiener criterion, Pacific J. Math., 1964, 14, 1019–1027 http://dx.doi.org/10.2140/pjm.1964.14.1019 Zbl0182.16502
  47. [47] Retherford J.R., Some remarks on Schauder bases of subspaces, Rev. Roumaine Math. Pures Appl., 1966, 11, 787–792 Zbl0149.08902
  48. [48] Sanders B.L., Decompositions and reflexivity in Banach spaces, Proc. Amer. Math. Soc., 1965, 16(2), 204–208 http://dx.doi.org/10.1090/S0002-9939-1965-0172092-8 Zbl0144.16802
  49. [49] Sanders B.L., On the existence of [Schauder] decompositions in Banach spaces, Proc. Amer. Math. Soc., 1965, 16(5), 987–990 Zbl0134.11002
  50. [50] Singer I., Bases in Banach Spaces I, Springer-Verlag, Berlin, 1970 http://dx.doi.org/10.1007/978-3-642-51633-7 
  51. [51] Singer I., Bases in Banach Spaces II, Springer-Verlag, Berlin, 1981 http://dx.doi.org/10.1007/978-3-642-67844-8 Zbl0467.46020
  52. [52] Singer I., On Banach spaces with symmetric bases, Rev. Roumaine Math. Pures Appl., 1961, 6, 159–166 (in Russian) Zbl0107.32602
  53. [53] Vizitei V.N., Marcus A.S., Convergence of multiple decompositions in a system of eigenelements and adjoint vectors of an operator pencil, Mat. Sb. (N.S.), 1965, 66(108):2, 287–320 (in Russian) 
  54. [54] Vizitei V.N., On the stability of bases of subspaces in a Banach space, In: Studies on Algebra and Mathematical Analysis, Moldov. Acad. Sci., Kartja Moldovenjaska, Chişinău, 1965, 32–44 (in Russian) 
  55. [55] Wermer J., Commuting spectral measures on Hilbert space, Pacific J. Math., 1954, 4, 355–361 http://dx.doi.org/10.2140/pjm.1954.4.355 Zbl0056.34701
  56. [56] Wyss C., Riesz bases for p-subordinate perturbations of normal operators, J. Funct. Anal., 2010, 258(1), 208–240 http://dx.doi.org/10.1016/j.jfa.2009.09.001 Zbl1185.47014
  57. [57] Zwart H., Riesz basis for strongly continuous groups, J. Differential Equations, 2010, 249, 2397–2408 http://dx.doi.org/10.1016/j.jde.2010.07.020 Zbl1203.47020

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