Journl of Applied Mthemtics nd Physics, 03,, 65-70 Pulished Online Novemer 03 (http://wwwscirporg/journl/jmp) http://dxdoiorg/0436/jmp03500 The Modified Heinz s Inequlity Tkshi Yoshino Mthemticl Institute, Tohoku University, Sendi, Jpn Emil: yoshino@mthtohokucjp Received August 3, 03; revised Septemer 4, 03; ccepted Septemer 8, 03 Copyright 03 Tkshi Yoshino This is n open ccess rticle distriuted under the Cretive Commons Attriution License, which permits unrestricted use, distriution, nd reproduction in ny medium, provided the originl work is properly cited ABSTRACT In my pper [], we imed to determine the est possile rnge of such tht the modified Heinz s inequlity ( AAA) ( ABA) holds for ny ounded liner opertors A nd B on Hilert spce such s A B I some 0 nd for ny given nd such s 0 nd 0 But the counter-exmples prepred in [] nd lso in [] were not sufficient nd, in this pper, we shll constitute the sufficient counter-exmples which will stisfy ll the lcking prts Keywords: Heinz s Inequlity By the sme wy s in the proof of Theorem ([]), we hve the following Lemm For ny x,,, nd such s 0,0 x,0 nd 0 O where O ( ) is function of such tht O is 0 ounded, let x 0 A nd B 0 If AAA ABA for ny rel numers, nd such s 0 nd 0, then we hve the following inequlity () Theorem ([]) The region of such tht the opertor inequlity AAA ABA holds for ny ounded liner opertors A nd B on Hilert spce such s A B I some 0 nd for ny given nd such s 0 nd 0 is s follows: ) 0, 0, ) 0, mx, min 0, 3) 0,, 0 4) 0, mx, min, nd 5),0, mx 0, x x ( x ) x 0 x x ()
66 T YOSHINO Proof Since the sufficiency of our rnge for the modified Heinz s inequlity is lredy proved in [], we hve only to constitute counter-exmples of A nd B in the outside of our rnges For ny x,,, nd such s in Lemm, let x 0 A nd B 0 Let x y,0 nd Then we hve A B I ecuse 0, x y, x nd ecuse x xx xx x y 0 Also if AAA ABA 0,, then, y multiply- ing x to the oth sides of the inequlity () in Lemm, we hve y y y = y y y y y y y y y y y y Let x, 0 nd Then we hve A B I ecuse 0, x nd ecuse x xx 0 AAA x Also if ABA () (3) (4) (5), then, y multiply- ing to the oth sides of the inequlity () in Lemm, we hve the following inequlity (6) And, y multiplying to the oth sides of the ove inequlity, we hve Let x nd 0 Then we hve (7) A B I
T YOSHINO 67 ecuse 0 x nd ecuse x x 0 Also if AAA ABA plying to the oth sides of the inequlity () in Lemm, we hve the following inequlity, then, y multi- For 0, let 0 Then ecuse And let 0, x y y 0 nd (8) 0 Then we hve A B xi ecuse x 0, x 0 nd ecuse x x0 Also if AAA ABA we hve the following inequlity, then, y Lemm, y y y y y y Since y y, y y y y, nd since, y multiplying inequlity, we hve, for, to the oth sides of the ove, 0 nd
68 T YOSHINO y y y y y Cse Let 0,,0 Then ecuse y y y y y nd y y y This contrdicts (3) Cse Let,0, 0 Then ecuse If y y y y 0, then we hve y 0 y y (9) nd, if 0, then we hve lso y y y 0, 0, 0 0, 0 nd hence, y (4) nd (5), we hve 0 nd this contrdicts the fct tht 0 for ll Cse 3 Let 0,, 0 Then nd y y y y y y ecuse y By (3), we hve 0 nd this contrdicts the fct tht 0 for ll Cse 4 Let,0,0 mx0, Then 0 nd nd hence 0 Therefore we hve 0 0 nd 0 0 This contrdicts (6)
T YOSHINO 69 Cse 5 Let 0,, mx, In this cse mx, 0 nd mx, 0 nd hence 0 ecuse Then, in the cse where 0, we hve nd hence 0 Therefore we hve nd This contrdicts (6) And, in the cse where, we hve nd hence 0 nd Therefore we hve 0 0 0 0 This contrdicts (7) Cse 6 Let 0,, Then 0 nd 0 And, y (8), we hve 0 0 0 0 nd this is contrdiction Cse 7 Let 0,, 0 Then 0 nd 0 ecuse Therefore we hve 0 0 nd 0 0 This contrdicts (6) Cse 8 Let 0,, 0 Then 0 Since y nd since 0 in the cse where, 0, 0, ecuse
70 T YOSHINO we hve y y y 0 y nd y y This contrdicts (9) Therefore we completed the proof of the est possiility of the rnges in our theorem REFERENCES [] T Yoshino, A Modified Heinz s Inequlity, Liner Alger nd its Applictions, Vol 40, No -3, 007, pp 686-699 http://dxdoiorg/006/jl0060803 [] T Yoshino, The Best Possile Rnge of Modified Heinz s Inequlity, Interntionl Journl of Funct Anl Oper Theory Appl, Vol 3, No, 0, pp -7