Vol. 18, No. 3, 1966

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Characterizations of direct sums and commuting sets of Volterra operators

G. K. Kalisch

Vol. 18 (1966), No. 3, 545–552
Abstract

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 which is a direct sum of n 2(I1) spaces on the unit interval I1. This is done by demanding that the dimension of (T + T)be n; that the subspaces j of generated by T and the eigenvectors ej of T + T be orthogonal to all ek for kj; 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 which is the 2 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 kj; that (Wj + WJ)= have dimension 1; that be spanned by the Wj’s and ; and that the Wj’s have spectrum 0.

Mathematical Subject Classification
Primary: 47.25
Secondary: 47.40
Milestones
Received: 5 March 1964
Revised: 31 December 1964
Published: 1 September 1966
Authors
G. K. Kalisch