Spectral measures which fail to be equicontinuous
Spectral measures, and their equivalent formulation in terms of \(\sigma\)- complete Boolean algebras of projections, play a fundamental role in the spectral theory of normal operators in Hilbert spaces. An essential ingredient in this theory is the uniform boundedness of the spectral measure. It was soon realized that to treat certain operators in Banach spaces such as \(L^ \infty\), the \(\sigma\)-additivity has to be relaxed. This was achieved by using the weak* topology rather than the norm topology of \(L^ \infty\). Here the natural analogue of the uniform boundedness of the projections is the requirement that they form an equicontinuous family of operators. So, granted that there exist non- equicontinuous spectral measures (of interest) the question is, how often they do occur. The aim of this note is to show that they occur frequently in a systematic way.
\(\sigma\)-complete Boolean algebras of projections
uniform boundedness of the spectral measure
scientific article; zbMATH DE number 598225