Theorem 1 contains an
abstract characterization and unitary invariants of operators T which are
finite direct sums of n Volterra operators (αjV f)(x) = αj∫0xf(y)dy with
real nonzero αj defined on a Hilbert space ℋ which is a direct sum of nℒ2(I1) spaces on the unit interval I1. This is done by demanding that the
dimension of (T + T∗)ℋ be n; that the subspaces ℋj of ℋ generated by T and
the eigenvectors ej of T + T∗ be orthogonal to all ek for k≠j; and that the
spectrum of T be 0. Theorem 2 contains an abstract characterization and
unitary invariants of finite commuting sets {Wj}1n of Volterra operators
which are real nonzero multiples of integration in the various coordinate axis
directions on a Hilbert space ℋ which is the ℒ2 space on the unit cube in n real
dimensions. The characterization is given by demanding that the Wj commute
with all Wk and Wk∗ for k≠j; that ∏(Wj+ WJ∗)ℋ = ℰ have dimension 1;
that ℋ be spanned by the Wj’s and ℰ; and that the Wj’s have spectrum
0.
