Translation-invariant linear operators

The theory of translation-invariant operators on various spaces of functions (or measures or distributions) is a well-trodden field. The problem is to decide, first, whether or not a linear operator between two function spaces on, say, IR or R + which commutes with one or many translations on the two spaces is necessarily continuous, and, second, to give a canonical form for all such continuous operators. In some cases each such operator is zero. The second problem is essentially the 'multiplier problem', and it has been extensively discussed; see [7], for example. In this paper, we shall give some further results about these two problems. In Section 1, we shall introduce the subject and recall some of the known results. In Section 2, we shall show that, HE = C 0 (U) or C(U), and if T-.E-^L^U) is a linear operator which commutes with a single non-trivial shift S a , then necessarily T = 0, but that, on the other hand, there is a closed linear subspace E of C(U) and a non-zero continuous linear operator T:E^-L l (W) such that T commutes with each shift S a. It is well-known that there is a discontinuous linear operator T:L 1 (M +)-+L l (U +) such that T commutes with a single left shift L a. In Section 3, we shall show that there are discontinuous linear operators which commute with all left shift operators. 1. Introduction Let E and F be linear spaces. Then SC(E,F) is the space of all linear maps from E into F. We write £?(E) for <£{E,E)\ the identity in S£(E) is I E , and the set of invertible operators in the algebra <£{E) is Invj£?(jf?). In the case where E and JF are Banach spaces, 38(E,F) denotes the Banach space of all bounded maps in £P(E,F), and we write 0S(E) for @(E,E). The spectrum of Te0S(E) is a[T). Let E and F be linear spaces, and let Re£C(E) and SeJ£(F). A linear map T.E-+F intertwines the pair (R, S) if TR = ST. HE and F are Banach spaces and if Re38{E) and Se@l(F), then we ask whether or not a map TeJ£(E,F) which intertwines (R,S) is automatically continuous. More generally, we consider the automatic continuity of a linear map T which intertwines a family of pairs of operators on E and F. We also consider the canonical form of continuous operators …