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, sav, R 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, if E = C0(R) or C(b)(R), and if T:E --> L1(R) 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(b)(R) and a non-zero continuous linear operator T:E --> L1(R) such that T commutes with each shift S(a). It is well-known that there is a discontinuous linear operator T:L1(R+) --> L1(R+) 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.