Some Applications of the Duhamel Product

Abstract

The Duhamel product of functions f and g is defined by the formula \((f \circledast g)(x) = \frac{d}{{dx}}\int\limits_0^x {f(x - t)g(t)dt.}\). In the present paper, the Duhamel product is used in the study of spectral multiplicity for direct sums of operators and in the description of cyclic vectors of the restriction of the integration operator \(f(x,y) \mapsto \int\limits_0^x {\int\limits_0^y {f(t,\tau )d\tau \;dt} }\) in two variables to its invariant subspace consisting of functions that depend only on the product xy. Bibliography: 13 titles.