Publication: Invariant subspaces for positive operators on Banach spaces with unconditional basis
Full text at PDC
Advisors (or tutors)
American Mathematical Society
We prove that every lattice homomorphism acting on a Banach space X with the lattice structure given by an unconditional basis has a non-trivial closed invariant subspace. In fact, it has a non-trivial closed invariant ideal, which is no longer true for every positive operator on such a space. Motivated by these examples, we characterize tridiagonal positive operators without non-trivial closed invariant ideals on X extending to this context a result of Grivaux on the existence of non-trivial closed invariant subspaces for tridiagonal operators.
 Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory, Graduate Studies in Mathematics, vol. 50, American Mathematical Society, Providence, RI, 2002, DOI 10.1090/gsm/050. MR1921782  Y. A. Abramovich and C. D. Aliprantis, Problems in operator theory, Graduate Studies in Mathematics, vol. 51, American Mathematical Society, Providence, RI, 2002, DOI 10.1090/gsm/051. MR1921783  Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw, Invariant subspaces of operators on lp-spaces, J. Funct. Anal. 115 (1993), no. 2, 418–424, DOI 10.1006/jfan.1993.1097. MR1234398  Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw, Invariant subspace theorems for positive operators, J. Funct. Anal. 124 (1994), no. 1, 95–111, DOI 10.1006/jfan.1994.1099. MR1284604  Y. A. Abramovich, C. D. Aliprantis, and O. Burkinshaw, Invariant subspaces for positive operators acting on a Banach space with basis, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1773–1777, DOI 10.2307/2160990. MR1242069  C. D. Aliprantis and O. Burkinshaw, Positive operators, Pure and Applied Mathematics, vol. 119, Academic Press, Inc., Orlando, FL, 1985. MR809372  F. Chamizo, E. A. Gallardo-Gutiérrez, M. Monsalve-López, and A. Ubis, Invariant subspaces for Bishop operators and beyond, Adv. Math. 375 (2020), 107365, 25, DOI 10.1016/j.aim.2020.107365. MR4136602  A. M. Davie, Invariant subspaces for Bishop’s operators, Bull. London Math. Soc. 6 (1974), 343–348, DOI 10.1112/blms/6.3.343. MR353015  E. A. Gallardo-Gutiérrez and M. Monsalve-López, A closer look at Bishop operators, Operator theory, functional analysis and applications, Oper. Theory Adv. Appl., vol. 282, Birkhäuser/Springer, Cham,  c 2021, pp. 255–281, DOI 10.1007/978-3-030-51945-2 13. MR4248021  S. Grivaux, Invariant subspaces for tridiagonal operators (English, with English and French summaries), Bull. Sci. Math. 126 (2002), no. 8, 681–691, DOI 10.1016/S0007-4497(02)01137-5. MR1944393  D. W. Hadwin, E. A. Nordgren, H. Radjavi, and P. Rosenthal, An operator not satisfying Lomonosov’s hypothesis, J. Functional Analysis 38 (1980), no. 3, 410–415, DOI 10.1016/0022-1236(80)90073-7. MR593088  A. K. Kitover and A. W. Wickstead, Invariant sublattices for positive operators, Indag. Math. (N.S.) 18 (2007), no. 1, 39–60, DOI 10.1016/S0019-3577(07)80005-X. MR2330731  K. B. Laursen and M. M. Neumann, An introduction to local spectral theory, London Mathematical Society Monographs. New Series, vol. 20, The Clarendon Press, Oxford University Press, New York, 2000. MR1747914  V. I. Lomonosov, Invariant subspaces of the family of operators that commute with a completely continuous operator (Russian), Funkcional. Anal. i Priloˇzen. 7 (1973), no. 3, 55–56. MR0420305  H. Radjavi and V. G. Troitsky, Invariant sublattices, Illinois J. Math. 52 (2008), no. 2, 437–462. MR2524645  A. L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. Math. Surveys, No. 13. MR0361899  V. G. Troitsky, A remark on invariant subspaces of positive operators, Proc. Amer. Math. Soc. 141 (2013), no. 12, 4345–4348, DOI 10.1090/S0002-9939-2013-11709-0. MR3105876