NORM IDEALS OF OPERATORS ON HILBERT SPACE
Kazö Tsuji · Institutional Repositories DataBase (IRDB) · 1974
Norm ldeals of Operators on Hilbert Space 53Then it is obvious that {sj} is the sequence of singular numbeTs ofa compact opeTatoT.We shall then clenote by AvB the compact operatoT.The following lemmas will justify the notation in Definition 1,3.LEMMA 1.4.Let A, BEV, ana let {s;} be the seqttenee deLfined by ,1 (1.3) s,=2[sj(A)+s,(B)], (i=1, 2,...).If a compact operatoT C has {s5•} as the sequence of singulaT 7zh2Lmbers, then AvB te C for eveTy di E .if.PRooF.Let sj be defined by (1,2).Then we have s; s. s, .sg2s;• (i=1, 2,...).k le le SinceZs;•s.ZsjÅq=2.Zs;-(k=1, 2,...), we have J---1 J'=1 J=1 Åë(O.[s(C)]) S. Åë(O.[s(AV B)]) S. 2Åë (a.[s(C)]) , n == 1, 2,• .•, and hence AvB2 C. LEMMA 1.5.I7ToT every A,B E 9, ÅëÅë (1,4) AÅqAvB ancl BÅqÅqAvB.PRooF.Trivial.Åë LEMMA 1.6.B te AvB if and only if AÅq(B.PRooF.ByLemma1.5wehaveBÅqg{AvB.If A-2(B, then there exists a positive number a such that (1.5) Åë(O.[s(A)])S.aÅë(d.[s(B)])(n =1, 2,,..).Let C be the compact operator which has l-År--[sj(A)+sj(B)]l as the sequence of singular numbers.Then by (1.5) we have Åë(dn[S(C)]) = Åë (-line (On [S(A)] + Cn [S(B)])l S -li-{Åë(dn[S(A)]) + Åë(dn[S (B)])} g-li-(1+a)Åë(6.[s(B)])(n= 1, 2,...).Hence C:l( B. From Lemma 1.4, AvB Åql{ B. Therefore we have B •e AvB.Conversely, if B2AvB, then A-ll(B because we have by Lemma 1.5 e AÅqAvB.CoRoLLARy 1.7.A •e (A vB) Ab B if and only if A .e B. LEMMA 1.8.If A2( C and B •!k C for A, B, CE g, then AvB -!{ C. ee PRooF.If A-ÅqC and B-ÅqC, then there exists a positive number a such that Åë(a.[s(A)])S.aÅë(o.[s(C)])and Åë(O.[s(B)])S.aÅë(o.[s(C)])(n =1, 2,...).Hence we have ÅëIrli-(On[s(A)]+On[s(B)]) l S S {Åë(On[s(A)]) +Åë(On[s(B)])} SaÅë(O.[s(C)])(n=1, 2,...).Therefore by Lemma 1.4 we have AvB •l2( c.LEMMA 1.9.If Ai 2 A2 and Bi •e B2, then AivBiÅíA2VB2.