Complete Multipliers

We investigate whether the completeness of a complete orthonormal sequence for L2(IT, 77) is preserved if the sequence is perturbed by multiplying a portion of it by a fixed function. For the particular sequence i(277)-1/2emxi we show that given any xb £ L°°(-77, 77), except lb = 0 a.e., there is a nontrivial portion of Í (2rr)-^ einx\ which will maintain completeness under this perturbation. We investigate whether the completeness of a complete orthonormal sequence (CONS) for L (n, rt) (complex-valued) is preserved if the sequence is perturbed by multiplying a portion of it by a fixed function. To be more precise, given a function if, £ L°°(n, tt) and a CONS $ = \ for n d S, is also complete in L . If all subsets of Z are 2,/, .-sets we call the pair (if,, $) a complete pair (for L ), and if ( 0 a.e. and either Im aib > 0 a.e. or Im ays < 0 a.e. on the zero set of Re aib. We now consider in more detail the case when ifi (which we assume is not the function which is 0 a.e.) is not a complete multiplier. In particular we ask if there always must exist a nontrivial 2 . ^,-set (where S is trivial if S is empty or S = Z). As a partial answer to this question we first determine when a finite 2 , -set i/>,* Received by the editors February 9, 1971 and, in revised form, April 29, 1971. AMS (MOS) subject classifications (1969). Primary 4217; Secondary 3067. ' ation, complete orthonorn Copyright © 1973. American Mathematical Society 399 nmj U ) Oj sifications 9). ary ; ndary .