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1、?41o?1?xx?Vol.41, No. 1 1998?1?ACTA MATHEMATICA SINICAJan., 1998XBUPamDf?jWq?u?x(?d?y?y?L?310027)gcr?y?s?t?v?zw?GJA?s? MR(1991)nYFM47D25kZFMO177Finite Rank Operators in Some Ideals of Nest AlgebrasLu FangyanLu Shijie(Department of Mathematics, Zhejiang University, Hangzhou 310027, China)Abstract In
2、this paper, we investigate finite rank operators in ideals of nest algebras.And then we use properties of finite rank operators to characterize such ideals.Keywords Next algebras, Ideals, Finite rank operators1991 MR Subject Classification 47D25Chinese Library Classification O1771Tbug?J?zl?r?v?kp?zl
3、?B?hJ?RingroseG?J?JacobsonC?rS1.?R?Larson2?D?A?Cn(?C)ug?fBqZM?zl?Ga?3?Erdoszl?JacobsonCk?ug?4?Hopenwasser?zl?i?JacobsonCM?C?ug?ErdosMPower5?ugkp?Z?ug?ug?T?ltn?ug?Q?6Y?AnousisMKatsoulis?N?ytJ?T (N)?BE?q?B?f?e?ug(?JN)?BE?W?Y?c?zl?f?e?ugJN?c?E?ZMLarsonug?G?H?Hilberts?H?JN?H?B?K(0)MH?s?eM?E?N N,?N= N?:
4、N? N,N? 0,?N?SE?E(T)? 0,?N?c?SE?E(T)? i+p.k?q?q?y1t?i+pt=iq?eq?G?Wy1i x2i T (N)MJN?T (N)?ug?i+p?t=iq?x2t,x2 i?y1i y1t=?y1i x2i?F 1 JN.?y =i+p?t=iq?x2i,x2t?y1t,?y Ni? Ni,?y1i y?T (N)?f?JN?f?e?G?y1i y = 0.?x2i?2?= 0,?y ?= 0.?y1i= 0.?x2i y1i= 0.?Wi?G?F2= 0.?JN?n?E?Fq?JN?cB?E?M?F =?wi zi,?wi Mi, zi Mi,
5、Mi N.?7?M16.11?u?x?Q2.5 RN,RNMJN?fP?c?E?3RN,RNHJNCLI116?ww?41odN3.17,9T (N)?c?E?T (N)?S?M?S?SY3.2 RN,RNMJN?fP?M?hR?F?T (N)?c?E?V?w limF= I.?T JN?FT?JN?c?E?w limFT = T.?m?AJN?c?E?T?JN?c?E?R?2.5 RN,RNMJN?fP?tP?EN3.3?1) N?2) RN?S?3) RN?S?4) RN?S?5) RN?S?6) JN?S?7) JN?S?8) JN?S?hR1)5).?A?N?YP?O?T RN,?AT
6、A= 0, .?JT RN?w limT= T,?ATAx?2= (ATAx,ATAx) = lim (ATAx,ATAx) = 0.?Ox HM?x?ATA= 0?O?x?T RN.a?5)?K4), 5)?K2), 2)?K3).?u1.1M?M3.2,?4)?K6), 5)?K7)Z2)?K8).?u2.2M?u2.1,?JN?yt?RNMJN?KT (N)?G?c?E?k?u3.1?2)8)?x?JN?A?N?YP?P = I ?P(A) ?= 0.?M = PN : N N?P?BytJ?PRNP R MMPJNP JM.?PRNPMPJNP?KT (M)?G?c?E?k?2)8)?
7、x?EN3.4?1) N?cJ?2) RN= RN; 3) RN?S?4) RN?S?5) RN?S?6) RN?S?hR1)?2)?b?1)M2)?x?u3)6)?x?M3.2,?J3)6)?O?B?x?2),k?1)?x?1?K?vh?d?F?117SY3.5 (1)?DN?J?T (N)?f?DN+ RN= D + R : D DN,R RN,?Lat(DN+ RN) = N.(2)?NMM?H?BJ?JJN= JM?M = N.L?JM?N?J?RM?K?RN?hR(1)?M Lat(DN+ RN), N,M?sY?N N, P(M)(I P(N)MP(N)(I P(M)?BW?wP(
8、N)?N?Q?WDN?G?P(M)?D?N= N?k?M?BP(N)eT?j?xMyE?T = P(N)(I P(M)(x y)P(M)(I P(N).?RN.?TP(M) = P(M)TP(M) = 0.?T = TP(M),?T = 0.?xMy?q?P(M)(I P(N) = 0UP(N)(I P(M) = 0.b?M Lat(DN+RN),?N?N?KM?k?N M N.?JM = NUM = N,?M N.?N M N,u?Xj?x M ?NMy N ? M,?x y + y x DN.?y = (x y + y x)x M,?q?M = N N.?WM?Lat(DN+ RN) N.
9、?a?N Lat(DN+ RN),?U?(2)?N N?M?M?K?N?JM ?= N,?j?x N ?M.?M0?M?Kx?k?n?u2.2,?y N?xy RN=RM,?y M0.?N M0,?M0 N.?Wx MMx M0,G?M?K?M0?MM?M?K?N?G?N = M M,k?N M.tB?M N.?M = N.SY3.6?D?X?eT?Von Neumann?D DN,?wDN= T (N) T (N),?(RN+ D) (RN+ D)= D.hR?A = R + D RN+ D?R= D + R D T (N),k?R DN.?WRN?f?e?G?R = 0,?A = D D.
10、o3.7?M3.5M3.6?RNMJN?x?o3.8?I?T (N)?KRN?(?)ug?Y?q?JN?INTRh?x?o3.9?A?q?E?H?Q?o?JN.?A= X B(H) : AmXAm?z?0.?Y6?AnoussisMKatsoulis?A?T (N)?KRN?ug?AMJN?fB?ZM?q?E?A?A?JN?118?ww?41o?K?1 Ringgrose J R. On some algebras of operators. Proc London Math Soc, 1965, 15(3): 6183 2 Larson D R. Nest algebras and simi
11、larity transforms. Ann of Math, 1985, 121: 409427 3 Erdos J A. On some ideals of nest algebra. Proc London Math Soc, 1982, 44(3): 143160 4 Hopenwasser A. Hypercausal ideals in algebras. J London Math Soc, 1986, 34(2): 129138 5 Erdos J A, Power C. Weakly closed ideals of nest algebras. J Operator The
12、ory, 1982, 7: 219235 6 Anoussis M, Katsoulis E G. Descriptions of nest algebras. Proc Amer Math Soc, 1990, 109(3): 7397457 Davidson K R. Nest Algebras, Pitman Research Notes in Mathmatics Series 191, Longman Scientific and Technical, Burnt Mill Harlow, Essex, UK, 1988 8 Ringgrose J R. On some algebras of operators II. Proc London Math Soc, 1966, 16(3): 3854029 Erdos J A, Operators of finite rank in nest algebras. J London Math Soc, 1968, 43: 391397 10 Dai X D, Lu S J. Isomorphisms of radicals of nest algebras. JCU (to appear)
