Publication: The weak summability dominion of a sequence S of the Hilbert space in relation with the set of linear bounded operators
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1981
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Universidad Nacional Autónoma de México
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Abstract
Let H be a separable, real Hilbert space, L(H) the Banach space of all bounded linear operators on H. For a given sequence (xn)n∈N⊆H with xn≠0 for all n∈N let C(xn):={T∈L(H):∑n∈NTxn<∞} and M(xn):={x∈H:∑ n∈N|(xn,x)|<∞}. The author studies injective (i.e. one-to-one, not necessarily invertible) operators, finite rank operators, and completely continuous operators in C(xn). The following results are shown: (1) C(xn) contains an injective operator if and only if M (xn)=H. (2) C(xn) is contained in the set of all finite rank operators on H if and only if the linear subspace M (xn)⊆H is of finite dimension. (3) C(xn) contains operators which are not completely continuous if and only if M(xn) contains an infinite-dimensional closed linear subspace of H. Finally it is proved that whenever all operators in C(xn) are completely continuous, they must necessarily be Hilbert-Schmidt operators.
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Fillmore, P. A., Williams, J. P. “On Operator Ranges”. Advances in Math., v. 7, nº. 3 (1971), 254-281
Julia, G. “Introduction Mathématiques aux Théories Quantiques”. Ed. Gauthier-Villars (1955)
Martín Peinador, E. “On the set of bounded linear operators transforming a certain sequence of a Hilbert space into an absolutely summable one”. Acts of the Colloquium on Topology (Budapest 1978)
Plans, A. “Zerlegung von Folgen in Hilbertraum in Heterogonalsystem”. Archiv der Mathematik, v. X (1959)
Schatten, R. “Norm ideals of completely continuous operators”. Springer-Verlag (1970)