Positive Interpolation on Aχ = 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 article, we obtained the following : Let H be a Hilbert space and L be a commutative subspace lattice on H. Let x and y be vectors in H. Then the following statements are equivalent.(1) There exists an operator A in AlgL such that Ax = y,A is positive and every E in L reduces A.(2) sup.k. ni=1 iEiykk. ni=1 iEixk:n 2 . ;i 2 . and Ei 2 L.< 1 and. . mk=1 kEky;. mk=1 kEkx.0, for all m 2 . , all k 2 . and all Ek 2 L.