Abstract
In [1] we gave a fairly short proof that there is an operator on the space ℓ1, without nontrivial invariant subspaces, and we conjectured that the same might be true of any space ℓ1 ⊕ W where W is a separable Banach space. This conjecture turns out to be true, and by proving it here we give the first example of a reasonably large class of Banach spaces for which the solution to the invariant subspace problem is known. This continues the sequence of counter-examples which began on an unknown Banach space (Enflo [2], Read [4], Beauzamy [3], (simplification of [2])), proceeded to the space ℓ1 (Read [5,1]) and here continues with the case of any separable Banach space containing ℓ1 as a complemented subspace. No counter-example is known to the author for a Banach space which does not contain ℓ1.
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References
C.J. Read, A short proof concerning the invariant subspace problem, J. Lond. Math. Soc., (2) 33 (1986).
P. Enflo, Acta Math, to appear.
B. Beauzamy, Operators without invariant subspaces, simplification of a result of P. Enflo, J. Integral Equations and Operator Theory (1985).
C.J. Read, A solution to the invariant subspace problem, Bull. London. Math. Soc., 16 (1984) 337–401.
C.J. Read, A solution to the invariant subspace problem on the space ℓ1, Bull. Lond. Math. Soc., 17 (1985), 305–317.
R. Ovsepian and A. Pelcynski, The existence in every separable Banach space of a fundamental and total bounded biorthogonal sequence and related constructions of unifromly bounded orthonormal systems in L 2. Studia Math, 54 (1975) 149–159.
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© 1988 Springer-Verlag
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Read, C.J. (1988). The invariant subspace problem on a class of nonreflexive Banach spaces, 1. In: Lindenstrauss, J., Milman, V.D. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 1317. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0081733
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DOI: https://doi.org/10.1007/BFb0081733
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