ABSTRACT
O'el>(L) +O'e6(M) ~ [O'ew(L) +O'(M») U [O'(L) +O'ew(M»). It follows then that ~ E [O'ew(L) +O'(M)] U [O'(L) +O'ew(M»). Now, since Ue :J O'e6(L) and ~ :J O'e6(M) are open sets, we know that their complements to O'(L) and O'(M), respectively, are finite, say
As in the proof of (14.34) we have
(14.39)
O'ew(K) = UUO'ew(KIE, ® Fj) ,=1 j=1
where E, is the nullspace of (L - Q,I),,(L,ClI), Fj is the nu1lspace of (M - (JjI),,(i:I,{Jj) (i =1, ... ,Pi; =1, ... , q), and
In fact, let Pt denote the projector of U to Ei' Qj the projector of V to Fj, P =PI EB ... EB PI" and Q =Ql EB ... EB Qj. Since P and Q are
projectors of finite rank, P and Q are obviously regular operators. Consequently, the operators 1-P and 1-Q are also regular. Thus the operators P®Q, P®(I - Q), (I - P)®Q, and (I - P)®(I - Q) are continuous in [U 4-V]. Let A be one of these operators. Since A2 = A, A is a continuous projector. Consequently,
(14.41)
JI I R(P®Q) = LLEBEi ® F;,
i=l ;=1 JI
;=1
(14.43)
Taking into account that (I - P)U = Rlt (I - Q)V = R2' the set R«I - P)®(I - Q» is closed in [U 4-V], and Rl ® R2 is dense in R«I - P)®(I - Q», we get (14.42) R«I - P)®(I - Q» = Rl®R2. From the equalities (14.40) - (14.42) it follows that
JI I [U 4-V] = LLEBEi®F;
i=1 ;=1 EB Ef=1 EBEi ® R2 EB E1=1 EBR1 ® F; EB R1®R2.
