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Hermitian operators and isometries on sums of Banach spaces

Published online by Cambridge University Press:  20 January 2009

R. J. Fleming  and
J. E. Jamison
R. J. Fleming
Affiliation:
Department of Mathematics, Central Michigan University, Mount Pleasant, Michigan, U.S.A.
J. E. Jamison
Affiliation:
Department of Mathematics, Memphis State University, Memphis, TN. 38152, U.S.A.
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Let E be a Banach sequence space with the property that if (αi) ∈ E and |βi|≦|αi| for all i then (βi) ∈ E and ‖(βi)‖E≦‖(αi)‖E. For example E could be co, lp or some Orlicz sequence space. If (Xn) is a sequence of real or complex Banach spaces, then E can be used to construct a vector sequence space which we will call the E sum of the Xn's and symbolize by ⊕EXn. Specifically, ⊕EXn = {(xn)|(xn)∈Xn and (‖xn‖)∈E}. The E sum is a Banach space with norm defined by: ‖(xn)‖ = ‖(‖xn‖)‖E. This type of space has long been the source of examples and counter-examples in the geometric theory of Banach spaces. For instance, Day [7] used E=lp and Xk=lqk, with appropriate choice of qk, to give an example of a reflexive Banach space not isomorphic to any uniformly conves Banach space. Recently VanDulst and Devalk [33] have considered Orlicz sums of Banach spaces in their studies of Kadec-Klee property.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 32 , Issue 2 , June 1989 , pp. 169 - 191
Copyright
Copyright © Edinburgh Mathematical Society 1989

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