Invertible Interpolation on AX = Y in AlgL

Abstract

Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for n-operators satises the equationAX i = Yi, for i = 1, 2, ..., n. In this article, we showed the following : Let L be a subspace lattice on a Hilbert space H and let X and Y be operators in B(H). Then the following are equivalent:(1)sup.kE? Y fkkE? Xf k:f 2H ;E2L.< 1 , sup.kXf kkY fk:f 2 H.< 1andrange X = H =range Y .(2)There exists an invertible operatorA in AlgL such that AX = Y.