Orthonormal sets and isoabelian operators on Banach spaces

Puttamadaiah, C. and Huche Gowda, (1987) Orthonormal sets and isoabelian operators on Banach spaces. Tamkang J. Math., 18 (4). pp. 13-18.

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Abstract

This paper is concerned with several Hilbert space concepts which can be extended to a Banach space setting through use of a semi-inner product [⋅,⋅] as in a paper by G. Lumer [Trans. Amer. Math. Soc. 100 (1961), 29–43; MR0133024 (24 #A2860)]. A system (ei) in a Banach space is said to be orthonormal (with respect to a compatible semi-inner product [⋅,⋅] if [ei,ej]=[ej,ei]=δij. If T is a bounded operator on X, T is said to be adjoint abelian if there is a semi-inner product for which [Tx,y]=[x,Ty] for all x,y, and T is said to be isoabelian if T is invertible and [Tx,y]=[x,T−1y] for all x,y. The last two definitions are due to J. G. Stampfli [Canad. J. Math. 21 (1969), 505–512; MR0239450 (39 #807)]. In fact, an operator is isoabelian if and only if it is an invertible isometry [ D. Koehler and P. Rosenthal , Studia Math. 36 (1970), 213–216; MR0275209 (43 #966)]. Several results are given concerning orthonormal sets, adjoint abelian operators, and isoabelian operators. Most have short proofs relying on the interplay among the various definitions.

Item Type: Article
Subjects: Physical Sciences > Mathematics
Divisions: PG Campuses > Manasagangotri, Mysore > Mathematics
Depositing User: Kodandarama
Date Deposited: 21 May 2013 07:55
Last Modified: 31 Aug 2013 06:22
URI: http://eprints.uni-mysore.ac.in/id/eprint/9812

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