The Index of Invariant Subspaces of Bounded below Operators on Banach Spaces

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Author(s)

George Chailos

For an operator on a Banach space , let be the collection of all its invariant subspaces. We consider the index function on and we show, amongst others, that if is a bounded below operator and if , , then If in addition are index 1 invariant subspaces of , with nonzero intersection, we show that . Furthermore, using the index function, we provide an example where for some , holds .

KEYWORDS

Index; Invariant Subspaces; Bounded below Operators; Banach Spaces

Cite this paper

G. Chailos, "The Index of Invariant Subspaces of Bounded below Operators on Banach Spaces," Advances in Pure Mathematics, Vol. 2 No. 2, 2012, pp. 124-127. doi: 10.4236/apm.2012.22018.

 [1] G. Chailos, “On Reproducing Kernels and Invariant Subspaces of the Bergman Shift,” Journal of Operator Theory, Vol. 51, No. 1, 2004, pp. 181-200. [2] G. Chailos, “Algebraic Properties of the Index of Invariant Subspaces of Operators on Banach Spaces,” Bulletin of the Irish Mathematical Society, Vol. 61, 2008, pp. 9- 13. [3] J. B. Conway, “A Course in Functionl Analysis,” 2nd Edition, Springer-Verlag, New York, 1990. [4] H. Hedenmalm, B. Korenblum and K. Zhu, “Theory of Bergman Spaces,” Springer-Verlag, New York, 2000. doi:10.1007/978-1-4612-0497-8 [5] T. Hungerford, “Algebra,” Springer-Verlag, New York, 1996. [6] S. Richter, “Invariant Subspaces in Banach Spaces of Analytic Functions,” Transactions of the American Mathematical Society, Vol. 304, No. 2, 1987, pp. 585-616. doi:10.1090/S0002-9947-1987-0911086-8