Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-07T03:13:29.285Z Has data issue: false hasContentIssue false
  • English
  • Français

STABILITY OF UNCONDITIONAL SCHAUDER DECOMPOSITIONS IN $\ell _{p}$ SPACES

Part of: Special classes of linear operators General theory of linear operators Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  03 August 2015

VITALII MARCHENKO
VITALII MARCHENKO*
Affiliation:
Mathematical Division, Institute for Low Temperature Physics and Engineering of NAS of Ukraine, 47, Lenin Ave, 61103, Kharkiv, Ukraine email [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We use the best constants in the Khintchine inequality to generalise a theorem of Kato [‘Similarity for sequences of projections’, Bull. Amer. Math. Soc.73(6) (1967), 904–905] on similarity for sequences of projections in Hilbert spaces to the case of unconditional Schauder decompositions in $\ell _{p}$ spaces. We also sharpen a stability theorem of Vizitei [‘On the stability of bases of subspaces in a Banach space’, in: Studies on Algebra and Mathematical Analysis, Moldova Academy of Sciences (Kartja Moldovenjaska, Chişinău, 1965), 32–44; (in Russian)] in the case of unconditional Schauder decompositions in any Banach space.

Keywords

Khintchine inequalityunconditional Schauder decompositionprojectiontypecotypeinfratypeisomorphismSchauder–Orlicz decompositionsymmetric basis

MSC classification

Primary: 46B15: Summability and bases
Secondary: 47A46: Chains (nests) of projections or of invariant subspaces, integrals along chains, etc. 47B40: Spectral operators, decomposable operators, well-bounded operators, etc.
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 92 , Issue 3 , December 2015 , pp. 444 - 456
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

References

Adduci, J. and Mityagin, B., ‘Eigensystem of an L 2 -perturbed harmonic oscillator is an unconditional basis’, Cent. Eur. J. Math. 10(2) (2012), 569–589.CrossRefGoogle Scholar
Adduci, J. and Mityagin, B., ‘Root system of a perturbation of a selfadjoint operator with discrete spectrum’, Integral Equations Operator Theory 73(2) (2012), 153–175.CrossRefGoogle Scholar
Allexandrov, G., Kutzarova, D. and Plichko, A., ‘A separable space with no Schauder decomposition’, Proc. Amer. Math. Soc. 127(9) (1999), 2805–2806.CrossRefGoogle Scholar
Bilalov, B. T. and Veliev, S. G., Some Questions of Bases (Elm, Baku, 2010) (in Russian).Google Scholar
Cazassa, P. G. and Christensen, O., ‘Perturbation of operators and applications to frame theory’, J. Fourier Anal. Appl. 3(5) (1997), 543–557.CrossRefGoogle Scholar
Chadwick, J. J. M. and Cross, R. W., ‘Schauder decompositions in non-separable Banach spaces’, Bull. Aust. Math. Soc. 6(1) (1972), 133–144.CrossRefGoogle Scholar
Clark, C., ‘On relatively bounded perturbations of ordinary differential operators’, Pacific J. Math. 25(1) (1968), 59–70.CrossRefGoogle Scholar
Djakov, P. and Mityagin, B., ‘Bari-Markus property for Riesz projections of Hill operators with singular potentials’, in: Functional Analysis and Complex Analysis, Contemporary Mathematics, 481 (American Mathematical Society, Providence, RI, 2009), 59–80.CrossRefGoogle Scholar
Djakov, P. and Mityagin, B., ‘Bari-Markus property for Riesz projections of 1D periodic Dirac operators’, Math. Nachr. 283(3) (2010), 443–462.CrossRefGoogle Scholar
Gohberg, I. C. and Krein, M. G., Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Translations of Mathematical Monographs, 18 (American Mathematical Society, Providence, RI, 1969).CrossRefGoogle Scholar
Haagerup, U., ‘The best constants in the Khintchine inequality’, Studia Math. 70 (1982), 231–283.CrossRefGoogle Scholar
Hughes, E., ‘Perturbation theorems for relative spectral problems’, Canad. J. Math. 24(1) (1972), 72–81.CrossRefGoogle Scholar
Johnson, W. B. and Lindenstrauss, J., Handbook of the Geometry of Banach Spaces, Vol. 1 (Elsevier, Amsterdam, 2001).Google Scholar
Johnson, W. B. and Lindenstrauss, J., Handbook of the Geometry of Banach Spaces, Vol. 2 (Elsevier, Amsterdam, 2003).Google Scholar
Kadets, M. I. and Kadets, V. M., Series in Banach Spaces, Conditional and Unconditional Convergence (Birkhäuser, Berlin, 1997).Google Scholar
Kato, T., ‘Similarity for sequences of projections’, Bull. Amer. Math. Soc. 73(6) (1967), 904–905.CrossRefGoogle Scholar
Kato, T., Perturbation Theory for Linear Operators, 2nd edn, Classics in Mathematics (Springer, Berlin, 1995), reprint.CrossRefGoogle Scholar
Krein, M., Milman, D. and Rutman, M., ‘On a property of a basis in a Banach space’, Comm. Inst. Sci. Math. Mec. Univ. Kharkoff [Zapiski Inst. Mat. Mech.] 16(4) (1940), 106–110; (in Russian, with English summary).Google Scholar
Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces I and II (Springer, Berlin, 1996), (reprint of the 1977, 1979 editions).CrossRefGoogle Scholar
Marchenko, V., ‘Isomorphic Schauder decompositions in certain Banach spaces’, Cent. Eur. J. Math. 12(11) (2014), 1714–1732.Google Scholar
Marcus, A. S., ‘A basis of root vectors of a dissipative operator’, Dokl. Akad. Nauk 132(3) (1960), 524–527; (in Russian).Google Scholar
Marcus, A. S., Introduction to the Spectral Theory of Polynomial Operator Pencils, Translations of Mathematical Monographs, 71 (American Mathematical Society, Providence, RI, 1988).Google Scholar
Sanders, B. L., ‘On the existence of [Schauder] decompositions in Banach spaces’, Proc. Amer. Math. Soc. 16(5) (1965), 987–990.CrossRefGoogle Scholar
Singer, I., Bases in Banach Spaces I (Springer, Berlin, 1970).CrossRefGoogle Scholar
Singer, I., Bases in Banach Spaces II (Springer, Berlin, 1981).CrossRefGoogle Scholar
Vizitei, V. N., ‘On the stability of bases of subspaces in a Banach space’, in: Studies on Algebra and Mathematical Analysis, Moldova Academy of Sciences (Kartja Moldovenjaska, Chişinău, 1965), 32–44; (in Russian).Google Scholar
Vizitei, V. N. and Marcus, A. S., ‘Convergence of multiple decompositions in a system of eigenelements and adjoint vectors of an operator pencil’, Mat. Sb. 66(108)(2) (1965), 287–320; (in Russian).Google Scholar
Wermer, J., ‘Commuting spectral measures on Hilbert space’, Pacific J. Math. 4 (1954), 355–361.CrossRefGoogle Scholar
Wyss, C., ‘Riesz bases for p-subordinate perturbations of normal operators’, J. Funct. Anal. 258(1) (2010), 208–240.CrossRefGoogle Scholar
Zwart, H., ‘Riesz basis for strongly continuous groups’, J. Differential Equations 249 (2010), 2397–2408.CrossRefGoogle Scholar