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

  • VITALII MARCHENKO (a1)

Abstract

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.

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[1]Adduci, J. and Mityagin, B., ‘Eigensystem of an L 2 -perturbed harmonic oscillator is an unconditional basis’, Cent. Eur. J. Math. 10(2) (2012), 569589.
[2]Adduci, J. and Mityagin, B., ‘Root system of a perturbation of a selfadjoint operator with discrete spectrum’, Integral Equations Operator Theory 73(2) (2012), 153175.
[3]Allexandrov, G., Kutzarova, D. and Plichko, A., ‘A separable space with no Schauder decomposition’, Proc. Amer. Math. Soc. 127(9) (1999), 28052806.
[4]Bilalov, B. T. and Veliev, S. G., Some Questions of Bases (Elm, Baku, 2010) (in Russian).
[5]Cazassa, P. G. and Christensen, O., ‘Perturbation of operators and applications to frame theory’, J. Fourier Anal. Appl. 3(5) (1997), 543557.
[6]Chadwick, J. J. M. and Cross, R. W., ‘Schauder decompositions in non-separable Banach spaces’, Bull. Aust. Math. Soc. 6(1) (1972), 133144.
[7]Clark, C., ‘On relatively bounded perturbations of ordinary differential operators’, Pacific J. Math. 25(1) (1968), 5970.
[8]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), 5980.
[9]Djakov, P. and Mityagin, B., ‘Bari-Markus property for Riesz projections of 1D periodic Dirac operators’, Math. Nachr. 283(3) (2010), 443462.
[10]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).
[11]Haagerup, U., ‘The best constants in the Khintchine inequality’, Studia Math. 70 (1982), 231283.
[12]Hughes, E., ‘Perturbation theorems for relative spectral problems’, Canad. J. Math. 24(1) (1972), 7281.
[13]Johnson, W. B. and Lindenstrauss, J., Handbook of the Geometry of Banach Spaces, Vol. 1 (Elsevier, Amsterdam, 2001).
[14]Johnson, W. B. and Lindenstrauss, J., Handbook of the Geometry of Banach Spaces, Vol. 2 (Elsevier, Amsterdam, 2003).
[15]Kadets, M. I. and Kadets, V. M., Series in Banach Spaces, Conditional and Unconditional Convergence (Birkhäuser, Berlin, 1997).
[16]Kato, T., ‘Similarity for sequences of projections’, Bull. Amer. Math. Soc. 73(6) (1967), 904905.
[17]Kato, T., Perturbation Theory for Linear Operators, 2nd edn, Classics in Mathematics (Springer, Berlin, 1995), reprint.
[18]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), 106110; (in Russian, with English summary).
[19]Lindenstrauss, J. and Tzafriri, L., Classical Banach Spaces I and II (Springer, Berlin, 1996), (reprint of the 1977, 1979 editions).
[20]Marchenko, V., ‘Isomorphic Schauder decompositions in certain Banach spaces’, Cent. Eur. J. Math. 12(11) (2014), 17141732.
[21]Marcus, A. S., ‘A basis of root vectors of a dissipative operator’, Dokl. Akad. Nauk 132(3) (1960), 524527; (in Russian).
[22]Marcus, A. S., Introduction to the Spectral Theory of Polynomial Operator Pencils, Translations of Mathematical Monographs, 71 (American Mathematical Society, Providence, RI, 1988).
[23]Sanders, B. L., ‘On the existence of [Schauder] decompositions in Banach spaces’, Proc. Amer. Math. Soc. 16(5) (1965), 987990.
[24]Singer, I., Bases in Banach Spaces I (Springer, Berlin, 1970).
[25]Singer, I., Bases in Banach Spaces II (Springer, Berlin, 1981).
[26]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), 3244; (in Russian).
[27]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), 287320; (in Russian).
[28]Wermer, J., ‘Commuting spectral measures on Hilbert space’, Pacific J. Math. 4 (1954), 355361.
[29]Wyss, C., ‘Riesz bases for p-subordinate perturbations of normal operators’, J. Funct. Anal. 258(1) (2010), 208240.
[30]Zwart, H., ‘Riesz basis for strongly continuous groups’, J. Differential Equations 249 (2010), 23972408.
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