Normal Interpolation on Ax=y in CSL-Algebra ALG L

Abstract

Given vectors x and y in a Hilbert space, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equation Txi = yi, for i = 1 ;2; ;n. In this paper the following is proved : Let H be a Hilbert space and L be a commutative subspace lattice on H. Let x and y be vectors in H. Let M x =.n.i=1iEix :n 2 N;i 2 . and Ei 2 L.and M y =.n.i=1iEiy :n 2 N;i 2 . and Ei 2 L.: Then the following are equivalent.(1)There exists an operator A in AlgL such that Ax = y,Af = 0 for all f in M x?,AA = AAand AE = EA for all E 2 L.(2)sup.k. ni=1 iEiykk. ni=1 iEiyk:n 2 N;i 2 . and Ei 2 L.< 1 ,M y M x and there is a vector h in M x such that hEy;xi = hEx;hi and hEy;yi = hEh;hi for all E in L.