POLYNOMIALLY BOUNDED OPERATORS AND BOUNDED PRODUCTS

Abstract

Two commuting operators S and T are polynomially bounded if there exists a constant C such that p(S,T)≤Cp∞ for every polynomial p in two variables, where the sup norm is computed over the product of the spectrums of the two operators. Two operator valued measures E and F defined on the Borel sets of the plane satisfy a boundedness condition if their product has a bounded extension to the algebra generated by the measurable rectangles. In this note we point out a connection between these two properties for scalar operators S and T and their spectral resolutions E and F, respectively.