ö MASAMICHI TAKESAKI Thus we get 2(M - Reco)||Ä/(/z)S#||2 = 2(|ù | - Re ù)(Á/(Á)Á/(Á)| # ||*) = 2(|ù | - Rew)((*7(W(*)l)*lf#) = 2(|ù | - âåù)(Äî|Á/(£)*/ /(*)£ ) 2|ù | ||Á:/(£)Ä||| ||*/(*)£|| - 2 Re ù(*/(£)Äî|Á:/(Á:)î ) ||Á/(£)Äî|| 2 + M2||fc/(fc)!l|2 - 2 Re ù(*/(Á:)Äî|Á/(£)î ) = ||Á/(*)(Ä - ù)||| 2 = ||fc/(Ä:)77l|2 = ÉÉ/(^)^'î 0 ÉÉ 2 ÉÉ^'ÉÉ 2 ÉÉ/(Á:)Á:î 0 ||2 = llx'||||«*/(Ä:)fc|0||2 = ||÷'|ÉÉÉ/(Á)"*Ë|ÉÉ202 = ÉÉ*'ÉÉ 2 ÉÉ/(Ë)îÉÉ·2# Hence we have the inequality (21) \\hf(h)k*W2 Ô(ù) 2 ||÷'||2||/(Á)|#||2, / e K(0, ~) . Now let A = /~ë^(ë ) be the spectral decomposition of h. Inequality (21) means that / 0 " É/(ë)|2ë 2 d\\e(\)f\\* c 2 j · - |/(ë)| 2 d|k(X)£#||2, / e K(0, °°), with c = ã(ù)||÷'||. This inequality means that the measure i/||e(X)£#||2 on (0, °°) is sup- ported by [0, c]. Hence we have e(c)|* = î* . Now, for every y' G M' we have e(c)a*y'%0 = e(c)/£# = y'e(c)f = y'f = a*y'i0 so that e(c)a* = a* and l l e * / M = \\c)a*y%\\ = lle(c)Att V^oll c||/* 0 ||. Hence a* is bounded and so is a. Thus £ is left bounded and ÉÉð/î)! ! c. Q.E.D. LEMMA 1.9. For euch ç e SI', s 0, sei ^(A + sT'v. Ihm for each ^ , £2 e ^(Ä1/ 2 ) ç ^ Ä - 1 ' 2 ) , (22) (ð,(ç)£r If2) = (/ð,á ) "/Ä"1 ' 2 f ÷ ÉÄ% 1 ) + sf/n^JA1^ ÉÄ 1 /2f2).

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