Vol. 49, No. 1, 1973

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On numerical ranges of elements of locally m-convex algebras

John R. Giles and Donald Otto Koehler

Vol. 49 (1973), No. 1, 79–91

The concept of numerical range is extended from normed algebras to locally m-convex algebras. It is shown that the approximating relations between the numerical range and the spectrum of an element are preserved in the generalization. The set of elements with bounded numerical range is characterized and the relation between boundedness of the spectrum and of the numerical range is discussed. The Vidav-Palmer theory is generalized to give a characterization of b-algebras by numerical range.

Mathematical Subject Classification 2000
Primary: 46H05
Received: 21 August 1972
Revised: 28 September 1972
Published: 1 November 1973
John R. Giles
Donald Otto Koehler