Interpolation Problems 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 forn vectors satisfies the equation Txi = yi for all 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 =(nXi=1iEix :n 2 N;i 2 C and Ei 2 L) and M y =(nXi=1iEiy :n 2 N;i 2 C 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?,AE = EA for all E 2 L and A is invertible. (2)supkP ni=1 iEiykkP ni=1 iEixk:n 2 N;i 2 C and Ei 2 L< 1 ,supkP ni=1 iEixkkP ni=1 iEiyk:n 2 N;i 2 C and Ei 2 L< 1 and M x and M y are densei n H.