Shadow and inverse-shadow inner products for a class of linear transformations
George O. Golightly · Pacific Journal of Mathematics · 1982
Suppose {H, ( , •)} is a complete inner product space and Hi is a dense subspace of H.In case T is a linear transformation from Hi to H ± (perhaps not bounded), a necessary and sufficient condition is obtained in Theorem 1 for the existence of an inner product ( , )i for H λ such that (i) the identity is continuous from {flΊ,( , )J to {#,(•, •)} and (ii) T is bounded in {-Hi, ( , )J When this condition holds, the inverse-shadow inner product is defined on H l9 for sufficiently large positive numbers β, by (x,y) β , τ = Σ£=o ((T/β)*x, (T/β)»y).An extension of Theorem 1 provides a necessary and sufficient condition for the existence of an inner product ( , )i for H t such that {H lf ( , )J is complete and (i) and (ii) hold.This latter condition, stated in Theorem 5 in terms of a pair of inverse-shadow inner products, depends on a description of those complete inner product spaces {H lf ( , )i}> with H ± dense in H, for which (i) holds.According to this description, given in Theorem 4, each such inner product ( , )i is a scalarmultiple of an inverse-shadow inner product ( , )s,c> where C is a bounded operator on H mapping H± to H ι and δ -1.