For given operators X and Y acting on a Hilbert space Η, an interpolating operator is a bounded operator A Such that AX=Y. Let X and Y be bounded operators acting on a Hilbert space Η. Assume that the ranges of X anc Y are dense Η. Then the followi...
For given operators X and Y acting on a Hilbert space Η, an interpolating operator is a bounded operator A Such that AX=Y. Let X and Y be bounded operators acting on a Hilbert space Η. Assume that the ranges of X anc Y are dense Η. Then the following are equivalent:
(1) There is a unitary operator A in B(Η) such that AX=Y.
(2) rangeY*=rangeX* and <Xf,Xg>=<Xf,Yg> for all f, g in Η