On isometric equivalence of certain Volterra operators

Abstract

otherwise, as in V, the complex valued function G(x, y) is continuously differentiable. The only difference from V is the presence of the constant a which affects the proof of Theorem 2 of V. A version of that theorem in the more general case where a is an arbitrary constant of absolute value 1 will be published elsewhere [4], All other theorems and proofs of V remain valid. The class D of functions with which we are principally concerned may be described as follows: the functions F are of the general form (1) where, in addition, G and m satisfy any one of the following: (A) G is analytic in a suitable region and m is an arbitrary positive integer (see Lemma 4 of V); (B) G(x, y)=G(y — x) where G (0)^0 and GGG2 in a neighborhood of y — x and otherwise G(i)G?i[0, l] and m is an arbitrary positive integer; (C) GGG2 and m=l. One very important property of the operators Tf for FG-D is the fact (see Theorem 3 of V) that their only reducing manifolds are the subspaces Lp[0, c] oi Lp[0, l] for all cG [0, 1 ] (see also [2; 5 and 6]). This property is crucial for the establishment of unitary invariants (in the case p — 2) of the operators Tf in §4 of V. As is usual, we define q by l/p + l/q= 1. Two continuous linear transformations Pi and T2 mapping Lp [0, 1 ] into itself are called isometrically equivalent if there exists an isometry 77of Lp[0, l] onto itself such that Ti= UT2U~X (regarding isometries for p7?2, see, e.g., [l, p. 178]; the considerations of the present paper are valid without this restriction). Two preliminary lemmas are needed in order to extend some results on Hilbert spaces and spectral theory to general p. We shall use the following notation: Ma is the operator multiplication by the characteristic function ca(x) oi the