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Pelczar-Barwacz, Anna

Strictly singular operators in asymptotic \(\ell_p\) Banach spaces. (English) Zbl 1300.46009

Ill. J. Math. 56, No. 3, 861-883 (2012).
Author’s abstract: We present a condition on higher-order asymptotic behavior of basic sequences in a Banach space ensuring the existence of bounded noncompact strictly singular operators on a subspace. Applications concern asymptotic \(\ell_p\) spaces, \(1 \leq p < \infty\), in particular, convexified mixed Tsirelson spaces and related asymptotic HI spaces.
Reviewer: Daniel Li (Lens)

MSC:

46B06 Asymptotic theory of Banach spaces
52A23 Asymptotic theory of convex bodies
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces

Keywords:

asymptotic \(\ell_p\) space; convexified mixed Tsirelson space; HI space; Schreier families; strictly singular operator
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References:

[1] \beginbarticle \bauthor\binitsD. \bsnmAlspach and \bauthor\binitsS. A. \bsnmArgyros, \batitleComplexity of weakly null sequences, \bjtitleDiss. Math. \bvolume321 (\byear1992), page 1-\blpage44. \endbarticle \endbibitem
[2] \beginbarticle \bauthor\binitsG. \bsnmAndroulakis and \bauthor\binitsK. \bsnmBeanland, \batitleA hereditarily indecomposable asymptotic \(\ell_2\) Banach space, \bjtitleGlasg. Math. J. \bvolume48 (\byear2006), page 503-\blpage532. \endbarticle \endbibitem · Zbl 1121.46007 · doi:10.1017/S0017089506003193
[3] \beginbarticle \bauthor\binitsG. \bsnmAndroulakis, \bauthor\binitsE. \bsnmOdell, \bauthor\binitsT. \bsnmSchlumprecht and \bauthor\binitsN. \bsnmTomczak-Jaegermann, \batitleOn the structure of the spreading models of a Banach space, \bjtitleCanad. J. Math. \bvolume57 (\byear2005), no. \bissue4, page 673-\blpage707. \endbarticle \endbibitem · Zbl 1090.46004 · doi:10.4153/CJM-2005-027-9
[4] \beginbarticle \bauthor\binitsG. \bsnmAndroulakis and \bauthor\binitsF. \bsnmSanacory, \batitleAn extension of Schreier unconditionality, \bjtitlePositivity \bvolume12 (\byear2008), no. \bissue2, page 313-\blpage340. \endbarticle \endbibitem · Zbl 1156.46017 · doi:10.1007/s11117-007-2126-2
[5] \beginbarticle \bauthor\binitsG. \bsnmAndroulakis and \bauthor\binitsT. \bsnmSchlumprecht, \batitleStrictly singular, non-compact operators exist on the space of Gowers and Maurey, \bjtitleJ. London Math. Soc. (2) \bvolume64 (\byear2001), page 655-\blpage674. \endbarticle \endbibitem · Zbl 1015.46007 · doi:10.1112/S0024610701002769
[6] \beginbarticle \bauthor\binitsS. A. \bsnmArgyros and \bauthor\binitsI. \bsnmDeliyanni, \batitleExamples of asymptotic \(\ell_1\) Banach spaces, \bjtitleTrans. Amer. Math. Soc. \bvolume349 (\byear1997), page 973-\blpage995. \endbarticle \endbibitem · Zbl 0869.46002 · doi:10.1090/S0002-9947-97-01774-1
[7] \beginbarticle \bauthor\binitsS. A. \bsnmArgyros, \bauthor\binitsI. \bsnmDeliyanni and \bauthor\binitsA. \bsnmManoussakis, \batitleDistortion and spreading models in modified mixed Tsirelson spaces, \bjtitleStudia Math. \bvolume157 (\byear2003), no. \bissue3, page 199-\blpage236. \endbarticle \endbibitem · Zbl 1028.46019 · doi:10.4064/sm157-3-1
[8] \beginbarticle \bauthor\binitsS. A. \bsnmArgyros, \bauthor\binitsI. \bsnmDeliyanni, \bauthor\binitsD. \bsnmKutzarova and \bauthor\binitsA. \bsnmManoussakis, \batitleModified mixed Tsirelson spaces, \bjtitleJ. Funct. Anal. \bvolume159 (\byear1998), page 43-\blpage109. \endbarticle \endbibitem · Zbl 0931.46017 · doi:10.1006/jfan.1998.3310
[9] \beginbarticle \bauthor\binitsS. A. \bsnmArgyros and \bauthor\binitsR. \bsnmHaydon, \batitleA hereditarily indecomposable \(\mc{L}_\infty\)-space that solves the scalar-plus-compact problem, \bjtitleActa Math. \bvolume206 (\byear2011), no. \bissue1, page 1-\blpage54. \endbarticle \endbibitem · Zbl 1223.46007 · doi:10.1007/s11511-011-0058-y
[10] \beginbarticle \bauthor\binitsS. A. \bsnmArgyros, \bauthor\binitsS. \bsnmMercourakis and \bauthor\binitsA. \bsnmTsarpalias, \batitleConvex unconditionality and summability of weakly null sequences, \bjtitleIsrael J. Math. \bvolume107 (\byear1998), page 157-\blpage193. \endbarticle \endbibitem · Zbl 0942.46007 · doi:10.1007/BF02764008
[11] \beginbarticle \bauthor\binitsK. \bsnmBeanland, \batitleOperators on asymptotic \(\ell_p\) spaces which are not compact perturbations of a multiple of the identity, \bjtitleIllinois J. Math. \bvolume52 (\byear2008), no. \bissue2, page 515-\blpage532. \endbarticle \endbibitem
[12] \beginbarticle \bauthor\binitsA. \bsnmBrunel and \bauthor\binitsL. \bsnmSucheston, \batitleOn B-convex Banach spaces, \bjtitleMath. Systems Theory \bvolume7 (\byear1974), no. \bissue4, page 294-\blpage299. \endbarticle \endbibitem · Zbl 0323.46018 · doi:10.1007/BF01795947
[13] \beginbarticle \bauthor\binitsI. \bsnmDeliyanni and \bauthor\binitsA. \bsnmManoussakis, \batitleAsymptotic \(\ell_p\) hereditarily indecomposable Banach spaces, \bjtitleIllinois J. Math. \bvolume51 (\byear2007), page 767-\blpage803. \endbarticle \endbibitem
[14] \beginbarticle \bauthor\binitsS. \bsnmDilworth, \bauthor\binitsE. \bsnmOdell, \bauthor\binitsT. \bsnmSchlumprecht and \bauthor\binitsA. , \batitlePartial unconditionality, \bjtitleHouston J. Math. \bvolume35 (\byear2009), no. \bissue4, page 1251-\blpage1311. \endbarticle \endbibitem
[15] \beginbarticle \bauthor\binitsI. \bsnmGasparis, \batitleA dichotomy theorem for subsets of the power set of the natural numbers, \bjtitleProc. Amer. Math. Soc. \bvolume129 (\byear2001), no. \bissue3, page 759-\blpage764. \endbarticle \endbibitem · Zbl 0962.46006 · doi:10.1090/S0002-9939-00-05594-5
[16] \beginbarticle \bauthor\binitsI. \bsnmGasparis, \batitleStrictly singular non-compact operators on hereditarily indecomposable Banach spaces, \bjtitleProc. Amer. Math. Soc. \bvolume131 (\byear2002), page 1181-\blpage1189. \endbarticle \endbibitem · Zbl 1019.46006 · doi:10.1090/S0002-9939-02-06657-1
[17] \beginbarticle \bauthor\binitsW. T. \bsnmGowers and \bauthor\binitsB. \bsnmMaurey, \batitleThe unconditional basic sequence problem, \bjtitleJ. Amer. Math. Soc. \bvolume6 (\byear1993), page 851-\blpage874. \endbarticle \endbibitem · Zbl 0827.46008 · doi:10.2307/2152743
[18] \beginbchapter \bauthor\binitsW. T. \bsnmGowers, \bctitleA remark about the scalar-plus-compact problem, \bbtitleConvex geometric analysis (\bconflocationBerkeley, CA, \bconfdate1996), \bsertitleMath. Sci. Res. Inst. Publ., vol. \bseriesno34, \bpublisherCambridge Univ. Press, \blocationCambridge, \byear1999, pp. page 111-\blpage115. \endbchapter \endbibitem
[19] \beginbarticle \bauthor\binitsD. \bsnmKutzarova, \bauthor\binitsA. \bsnmManoussakis and \bauthor\binitsA. \bsnmPelczar-Barwacz, \batitleIsomorphisms and strictly singular operators in mixed Tsirelson spaces, \bjtitleJ. Math. Anal. Appl. \bvolume388 (\byear2012), page 1040-\blpage1060. \endbarticle \endbibitem · Zbl 1242.46014 · doi:10.1016/j.jmaa.2011.10.053
[20] \beginbarticle \bauthor\binitsH. \bsnmLemberg, \batitleSur un théorème de J.-L. Krivine sur la finie représentation de \(\ell_p\) dans un espace de Banach, \bjtitleC. R. Acad. Sci. Paris Sér. I Math. \bvolume292 (\byear1981), no. \bissue14, page 669-\blpage670. \endbarticle \endbibitem
[21] \beginbarticle \bauthor\binitsA. \bsnmManoussakis, \batitleA note on certain equivalent norms on Tsirelson’s space, \bjtitleGlasg. Math. J. \bvolume46 (\byear2004), page 379-\blpage390. \endbarticle \endbibitem · Zbl 1060.46006 · doi:10.1017/S0017089504001867
[22] \beginbchapter \bauthor\binitsE. \bsnmOdell, \bctitleOn Schreier unconditional sequences, \bbtitleBanach spaces (Mérida, 1992), \bsertitleContemp. Math., vol. \bseriesno144, \bpublisherAmer. Math. Soc., \blocationProvidence, RI, \byear1993, pp. page 197-\blpage201. \endbchapter \endbibitem · doi:10.1090/conm/144/1209461
[23] \beginbarticle \bauthor\binitsE. \bsnmOdell, \bauthor\binitsN. \bsnmTomczak-Jaegermann and \bauthor\binitsR. \bsnmWagner, \batitleProximity to \(\ell_1\) and distortion in asymptotic \(\ell_1\) spaces, \bjtitleJ. Funct. Anal. \bvolume150 (\byear1997), page 101-\blpage145. \endbarticle \endbibitem · Zbl 0890.46015 · doi:10.1006/jfan.1997.3106
[24] \beginbchapter \bauthor\binitsT. \bsnmSchlumprecht, \bctitleHow many operators exist on a Banach space?, \bbtitleTrends in Banach spaces and operator theory (Memphis, TN, 2001), \bsertitleContemp. Math., vol. \bseriesno321, \bpublisherAmer. Math. Soc., \blocationProvidence, RI, \byear2003, pp. page 295-\blpage333. \endbchapter \endbibitem · doi:10.1090/conm/321/05651
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