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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A note on some operator theory in certain semi-inner-product spaces.


Author: D. O. Koehler
Journal: Proc. Amer. Math. Soc. 30 (1971), 363-366
MSC: Primary 47.10
MathSciNet review: 0281024
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Abstract: Let X be a smooth uniformly convex Banach space and let $ [\cdot ,\cdot ]$ be the unique semi-inner-product generating the norm of X. If A is a bounded linear operator on X, $ {A^\dag }$ mapping X to X is called the generalized adjoint of A if and only if $ [A(x),y] = [x,{A^\dag }(y)]$ for all x and y in X. In this setting adjoint abelian (iso abelian) operators [5] are characterized as those operators A for which $ {A^\dag } = A({A^\dag } = {A^{ - 1}}$, i.e. the invertible isometries). It is also shown that the compression spectrum of an operator is contained in its numerical range.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0281024-6
Keywords: Adjoint abelian operators, iso abelian operators, generalized adjoint, isometries, compression spectrum
Article copyright: © Copyright 1971 American Mathematical Society