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Algebraic Geometry

By Shafarevich I.R.

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The next method equation A*Ax = A*y is equivalent is suggestive which~ of applying Theorem as we have shown in Lemma to (i) when (i) is solvable. III. 2 (6) to the (I. 8), $9 (n) Theorem. Let A such that be a bounded 0 < (~ < 2 operator on ~ . 1~ infllAx-yll Further, Ix n} x~ converges if and only if (13) Ax = PM(A*)y is solvable. If (13) is solvable, is the minimal solution then x n ~ PN(A)x0 + ~ , where of (13). Proof. Noting that A*PN(A*)y=O , we apply AXn+ 1 : Ax n + C~A*[PM(A*)y Setting Vn+ 1 = AXn+ I- PM(A*)y E M(A*) A to (12) and obtain - Ax n] , we can write this in the form Vn+ I = v n - C~A*v n The operator Also AA* is positive I{AA*II = IIAI;2 .

For every y E ~ , III. 2 34 inf llAx-yll = IIPN(A*)yll . x¢~ Every solution x' ~ ~" of AX=PM(A*)y (3) satisfies (4) inf llAx-Yll = flax'-YlI" x¢~" Proof. Since •hus ~ ~-~(A*)y = M(A*) ~ M(A*) . , we have ~iti~ Ax¢ M(A*) for any x ~ ~ . y : FM(A*)y + PN(A*)y , ~e obtain llAx_yll2 = IIAx-PM(A*)Yll2 + IIP~(A*)Yll2 from the m t h a g o r e a n theorem. inf IIAx-PM(A*)Yll xc~ But = if we take the infimum of the above expansion over all obtain (3). If Ax' = PM(A*)y , then o . ~us, x e~ , we Ax'-y = [PM(A*)-I]y = -PN(A*)y and we obtain (4).

Here for B . = We note that B and , and :k B , is the spectral resolution are bounded and self-adJoint. If. 3 (12) 28 Theorem. (13) Let let A be a bounded linear operator with B=A*A . Assume that sequence of polynomials g(B) c [m,M] [l-Pn(k)k~ where 0 ~ m ~ M . If the [m,M] [m,M]\[0] , then the sequence defined by (9) converges to a solution y ~ R(A) closed, and is uniformly bounded on and converges to zero at each point of [Xn] R(A) x* of (i) provided . Proof. If m > 0 , then (12) and the Lebesgue dominated conver- gence theorem show that llXn+l-X*ll ~ 0 .

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