Vol. 103, No. 2, 1982

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Shadow and inverse-shadow inner products for a class of linear transformations

George Golightly

Vol. 103 (1982), No. 2, 389–399

Suppose {H,(,)} is a complete inner product space and H1 is a dense subspace of H. In case T is a linear transformation from H1 to H1 (perhaps not bounded), a necessary and sufficient condition is obtained in Theorem 1 for the existence of an inner product (,)1 for H1 such that (i) the identity is continuous from {H1,(,)1} to {H,(,)} and (ii) T is bounded in {H1,(,)1}. When this condition holds, the inverse-shadow inner product is defined on H1, for sufficiently large positive numbers β, by (x,y)β,T = Σp=0((T∕β)px,(T∕β)py). An extension of Theorem 1 provides a necessary and sufficient condition for the existence of an inner product (,)1 for H1 such that {H1,(,)1} 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 {H1,(,)1}, with H1 dense in H, for which (i) holds. According to this description, given in Theorem 4, each such inner product (,)1 is a scalar-multiple of an inverse-shadow inner product (,)δ,c, where C is a bounded operator on H mapping H1 to H1 and δ = 1.

Mathematical Subject Classification 2000
Primary: 46C05
Secondary: 47A05
Published: 1 December 1982
George Golightly