28 в our gain and Tzafriri, On a problem of Kadison and Singer

Theorem 4.1. If the symbol (peL^(T) satisfies the condition

i : i^ ( « ) pi«r<cx ) ,

neZ

for some т > 0, then the corresponding Laurent operator L^ has the paving property.

Remark . The condition imposed above on the symbol ф is clearly equivalent to the requirement that cp belongs to the intersection of L^(F) with the Besov space й^2,2(^)5 mentioned in the Introduction. As we have pointed out there, the class of Laurent operators for which Theorem 4.1 applies forms a subalgebra of the algebra of all Laurent operators acting on Ь2(Т).

Corollary 4.2. There exists an open neighborhood V of the rationals on the circle T such that д V has positive measure and the corresponding Laurent operator L^^ has the paving property.

Proof of Corollary 4.2. First, notice that if/is an interval of length 2ô on the circle, for some Ô > 0, then the Fourier coefficients of the characteristic function Xi of / satisfy, as is easily verified, the condition

\Un ) \u2miniôM\n\ ) , for all 0 Ф « 6 Z.

Let now {t^}^= 1 be an enumeration of the rationals on (0,2n) and, for each r, let /^ be an

interval centered at /^, contained in (0, 2n) and of length 2ô^, where e. g. 5^ < 2 "''" ^. In order

00

to show that V = \J Ir satisfies the assertion of Corollary 4.2, it suffices, in view of Theorem

r=l

4 . 1 , to verify that x^e й^2,2Дога110 < т < 1. To this end, for fc = 1, 2,... and/e L2 ( F), put

and notice that

\\Saau4min ( 2'^'4 , , 2 - " ' ) , for all к and r. Hence,

WSavWu i \\saiM+ t \\Sa,M

r=l r=k+l

^4 ( â : 2 - '^ / 2^2'^ / 2 f оЛйЦк2-^'^-\-2-^^^)й^к2-^'\