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Two geometric constants for operators acting on a separable Banach space

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Martín Peinador, Elena and Indurain, E. and Plans, A. and Rodés-Usán, Álvaro (1988) Two geometric constants for operators acting on a separable Banach space. Revista matemática de la Universidad Complutense de Madrid, 1 (1-3). pp. 23-30. ISSN 0214-3577

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Official URL: http://revistas.ucm.es/index.php/REMA/article/view/REMA8888110023A/17319


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Abstract

Let A be an operator from a separable Banach space X into another Banach space Y. For every M-basis (an) of X the authors define two numbers: hA = hA,(an) = infn||A|[an,...]||and HA = HA,(an) = supnm(A|[an,...]), where [ ] stands for closed linear span and m stands for minimum modulus, i.e. m(A)=inf||x||=1||Ax||. First they prove that reflexivity of X can be characterized by the stability of HA,(an) under changes of the M-basis. In the case X is a separable reflexive Banach space these constants are related with s-numbers. The authors show that HA is the infimum of the Gelʹfand numbers of A and hA is a lower bound of the Bernstein numbers of A defined by J. Zemánek [Studia Math. 80 (1984), no. 3, 219–234]. They prove that a separable Banach space X is reflexive if and only if the infimum of the Gelʹfand numbers of every operator A from X into a Banach space Y can be computed in terms of one sequence of closed, nested, finite codimensional subspaces with null intersection. Several relationships between these numbers and the spectral theory are discussed. Finally, in the framework of a separable Hilbert space X and a selfadjoint operator A on X, it is shown that HA and hA are respectively the maximum and the minimum of the limit points of the spectrum of A. If the operator is not selfadjoint, HA and hA are exactly the maximum and minimum of the limit points of the spectrum of (A*A)1/2.


Item Type:Article
Subjects:Sciences > Mathematics > Topology
ID Code:21869
Deposited On:14 Jun 2013 18:01
Last Modified:12 Dec 2018 15:13

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