Algebraic Geometry

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By Fulton W.

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In case L we need only take is a bounded and positive bounded below operator, Y2=l , x0=0 ~ and A=I to obtain Kolom~'s Theorem (21) from Theorem (25). In the article [121], Petryshyn gives applications of his two methods to both ordinary and partial differential equations. III. 3 84 SECTION 4 THE E X T ~ S I O N OF SOME ALGEBRAIC M~I'HODS Motivated by thoughts of advancing from solving in n n equations unknowns to solving an infinite number of equations in an infinite number of unknowns, it seems reasonable to attempt to carry certain methods of linear algebra over into the theory of operator equations.

Au,Kv> = (Ku,Av> is a real Hilbert space this property In this section we use the inner product (u,v>K = (Au,Kv> , u, v ¢ D(A) , and the corresponding norm ll'II K . The space as discussed in Chapter I. i. 7). L Now assume that we are given an operator defined in ~ . We choose a Kpd operator and such that for constants ~l > 0 and Re(Lu,Ku> • ~I(AU,Ku) , A which is densely so that D(A) = D(L) I~>0, U ¢ D(L) , (4) and (5) I(Lu,Kv>I 2 ~ ~(Au,Ku>(Av,Kv> for all u, v ¢ D(L) . Define an operator F(A,K) , W is bounded.

If we i=l,2,... ,n , then = Lw n - 7 1 t n L Z n = L(Xn_X* ) _ 71tnLZn = L(Xn+l-X*) = L(Xn-~ltnZn ) - Lx* = LWn+ 1 • by induction, Az n = Lw n = Lxn-Y' Fr~ as usual, = L ( x 0 - x * ) = Lw 0 , and Az I = A z 0 - T l t 0 L z 0 = L ( x 0 - T l t 0 Z o ) assume e. l Then Wn+ 1 = w n - YltnZn Note by , n=O,1,2, . . (16) we o b t a i n III. 3 (17) 47 (AWn+l,KWn+ I) = ;lWn+lll~ (18) Now (17) may be written in the form ~w n = zn n=0,1,2, .... or Wn= ~'Iz n , Thus, using (14), we may write (18) as 2 ~ {_ ½ 71 llWn+lllK = IlWn11~ - 2~ 1 (Re(~Zn,Zn>K)2 + Re(~Zn,Zn> K Re(zn,~-iZn> K 2 } " (Zn,Zn> K Using the inequalities (7) and (i0), we can express (19) as (Zn,Zn>~ 11Wn+llI~~ llWnll~ - Fl(2k~l-Yl ) (Re(~Zn,Zn>K)2 because Re(~Zn,Zn) K Re(zn,~'iZn> K (Zn,Zn)~ Inequality (12) shows that (llWnllK} ~ k~ I • 2k~l " YI > 0 .

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