Biinvariants subspaces for some weighted spaces

We study the biinvariant subspaces for the usual shift on the weighted spac es L-omega(2) = {f is an element of L-2 (T): parallel to f parallel to omega = (Sigma(n is an element of Z) \f(n)omega(2)(n)(1/2) < +infinity}, where omega(n) = (1+n)p, n greater than or equal to 0 and omega(n)/((1+\n\) )(p) -->(n-->-infinity) +infinity for some integer p greater than or equal to 1. We show that the analytic part of all biinvariant subspaces is spectr al if Sigma(n greater than or equal to 2) 1/(nlog omega(-n)) diverges, but that this does not hold when Sigma(n greater than or equal to 2) 1/(n log o mega(-n)) converges.