LIZ-FACTORIZATION OF OPERATORS ON lx

Necessary and sufficent conditions are obtained for ¿(/-factorization of operators on /,. In particular it is shown that uniform invertibility of the compressions of the operator is not sufficient to insure an LU-factorization of the operator, thus answering a question of de Boor, Jia, and Pinkus. The question of when a bounded linear operator on lp, 1 < p <, oo, has an Li/-factorization has been much studied recently. Barkar and Gohberg [2] have shown that if A is an operator on lp which has an Lt/-factorization, then A and its compressions A„ = P„APn are uniformly invertible, i.e. sup„ {H^i"1!!, M"1!!} < oo. In the other direction, various classes of operators such as invertible, diagonally dominant operators on /\ [7] and invertible, totally positive operators [3, 1] on I have been shown to have L [/-factorizations. For these kinds of operators it is known [1] that their compressions satisfy a stronger condition than uniform invertibility; namely, that the inverses of the compressions are order bounded, i.e. HsupJ-d;;1! II < oo. Left open, then, is the possibility (first raised in [3] with a negative expectation) that uniform invertibility might be sufficient for a matrix operator on lx to have an LU-factorization. In this paper an example is given that shows that uniform invertibility is not sufficient for factoring an operator on lx (or /,). However, we also show that uniform invertibility of the compressions is sufficient to ensure an LU-factorization when the operator has an inverse whose columns decay at a certain rate away from the diagonal. Among the operators with this property are the banded operators. We wish to express thanks to the referee for several helpful suggestions. We now fix some terminology and notation. If x = (x,) is an element of /x we denote its usual projection onto the span of the first n basis vectors by P„x. A bounded linear operator A on /, is said to be upper (respectively lower) triangular if P„AP„ = APn (respectively P„A) for all n. We say that A is unit upper (lower) triangular if it is upper (lower) triangular and its diagonal entries in the matrix representation for A relative to the usual basis e¡ of l1 are all ones. An operator A is said to have an L[/-factorization (relative to the usual basis e¡ of /,) if there exist invertible operators L and U so that A = LU and the operators L, L'1 are unit lower triangular while U, U'1 are upper triangular. An operator A is said to be Received by the editors June 10, 1985. 1980 Mathematics Subject Classification. Primary 47A68; Secondary 46E40.