Normal Interpolation on AX=Y in CSL-algebra AlgL

Abstract

Let L be a commutative subspace lattice on a Hilbert space H and X and Y be operators on H. Let M X =.n.i=1EiXfi :n 2 . ; fi 2 H and Ei 2 L.and M Y =.n.i=1EiY fi :n 2 . ; fi 2 H and Ei 2 L.: Then the following are equivalent. (i) There is an operator A in AlgL such that AX = Y, Ag = 0 for all g in M X?,AA = AA and every E in L reduces A. (ii) supfK (E;f) :n 2 . ;fi 2 H and Ei 2 Lg < 1 ,M Y M X and there is an operator T acting on H such that hEXf;Tgi = hEY f;Xgi and hETf;Tgi = hEY f;Y gi forall f; gin H and E in L, where K (E;f) = k. ni=1 EiY fik=k. ni=1 EiXfik.