PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
2014 | 12 | 11 | 1714-1732
Tytuł artykułu

Isomorphic Schauder decompositions in certain Banach spaces

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Twórcy
Bibliografia
  • [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
  • [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
  • [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] 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
  • [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] Bilalov B.T., Veliev S.G., Some Questions of Bases, Elm, Baku, 2010 (in Russian)
  • [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
  • [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
  • [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
  • [10] Clement P., De Pagter B., Sukochev F.A., Witvliet H., Schauder decompositions and multiplier theorems, Studia Math., 2000, 138(2), 135–163
  • [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
  • [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
  • [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
  • [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
  • [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
  • [16] Fage M.K., Idempotent operators and their rectification, Dokl. Akad. Nauk, 1950, 73, 895–897 (in Russian)
  • [17] Fage M.K., The rectification of bases in Hilbert space, Dokl. Akad. Nauk, 1950, 74, 1053–1056 (in Russian)
  • [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] 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
  • [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] 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)
  • [22] Haagerup U., The best constants in the Khintchine inequality, Studia Math., 1982, 70, 231–283
  • [23] Haase M., A decomposition theorem for generators of strongly continuous groups on Hilbert spaces, J. Operator Theory, 2004, 52, 21–37.
  • [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] 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
  • [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
  • [27] Jain P.K., Ahmad K., Schauder decompositions and best approximations in Banach spaces, Port. Math., 1987, 44(1), 25–39
  • [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
  • [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] Johnson W.B., Lindenstrauss J., Handbook of the Geometry of Banach Spaces, Volume 1, Elsevier, 2001
  • [31] Johnson W.B., Lindenstrauss J., Handbook of the Geometry of Banach Spaces, Volume 2, Elsevier, 2003
  • [32] Kadets M.I, Kadets V.M., Series in Banach Spaces, Conditional and Unconditional Convergence, Birkhäuser, Berlin, 1997
  • [33] Kato T., Perturbation Theory for Linear Operators, 2nd ed. (reprint), Classics Math., Springer, Berlin, 1995
  • [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
  • [35] Köthe G., Toeplitz O., Linear Räume mit unendlich vielen Koordinaten und Ringe unendlicher Matrizen, J. Reine Angew. Math., 1934, 171, 193–226
  • [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)
  • [37] Lindenstrauss J., Tzafriri L., Classical Banach Spaces I and II, Reprint of the 1977, 1979 ed., Springer-Verlag, Berlin, 1996
  • [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
  • [39] Marcus A.S., A basis of root vectors of a dissipative operator, Dokl. Akad. Nauk, 1960, 132(3), 524–527 (in Russian)
  • [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] Mityagin B., Siegl P., Root system of singular perturbations of the harmonic oscillator type operators, preprint available at http://arxiv.org/abs/1307.6245
  • [42] Orlicz W., Über die Divergenz von allgemeinen Orthogonalreihen & Über unbedingte Convergenz in Funktionraümen, Studia Math., 1933, 4, 27–37
  • [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
  • [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
  • [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
  • [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
  • [47] Retherford J.R., Some remarks on Schauder bases of subspaces, Rev. Roumaine Math. Pures Appl., 1966, 11, 787–792
  • [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
  • [49] Sanders B.L., On the existence of [Schauder] decompositions in Banach spaces, Proc. Amer. Math. Soc., 1965, 16(5), 987–990
  • [50] Singer I., Bases in Banach Spaces I, Springer-Verlag, Berlin, 1970 http://dx.doi.org/10.1007/978-3-642-51633-7
  • [51] Singer I., Bases in Banach Spaces II, Springer-Verlag, Berlin, 1981 http://dx.doi.org/10.1007/978-3-642-67844-8
  • [52] Singer I., On Banach spaces with symmetric bases, Rev. Roumaine Math. Pures Appl., 1961, 6, 159–166 (in Russian)
  • [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] 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] 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
  • [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
  • [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
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-014-0441-y
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.