关于p-弱亚正规算子的一个注记

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Furuta showed that if A≥B≥0,then for each r≥0,f（p）=（A^r/2 B^p A^r/2）^t＋r/p＋r is decreasing for p≥t≥0.Using this result,the following inequality（C^r/2（AB^2A）^δC^ r/2）^ p-1＋r/4δ＋r ≤C^p-1＋r is obtained for 0〈p ≤1,r≥1,1/4≤δ≤1 and three positive operators A, B, C satisfy（A^1/2BA^1/2）^p/2≤A^p,（B^1/2AB^1/2）^p/2≥B^p,（C^1/2AC^1/2）^p/2≤C^p,（A^1/2CA^1/2）^p/2≥A^p.     

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IllltllodllCtIOllLetHdenoteacomplexHllbertspace．LetB（H）beacomplexBanachspaceofallboundedlinearoperatorsonH，andB（H）”，thecolljugatespaceofB（H）．TherealandimaginarypartsofanoperatorAonXaredenotedbyReAandImA，respectively，l．e。D。A一T，、、A＿一Ti，＊＊，。、．．＿TT。－＿，。，、，、＿＿。、、。＿、。＾。＿，、、＿。，，、。＿＾。nireH”一十，lmH“＝－－－－－－．rortWOInirm1LlanOperaLOrSH．nOllrr。weWriteH5otoIndicatethatB—AIsapositiveoperator．i．e，（（B—A）。，…     

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Let X be a Banach space and {e_j}_(j=1)~∞ be a sequence in X. The author showsthat {e_j}_(j=1)~∞ is a basic sequence if and only if ∑_(n=1)~∞, r_nα_(nj) converges for every j≥1 and∑_(n=1)~∞ r_n ∑_(j=1)~∞, α_(nj)e_j=∑_(j=1)~∞,(∑_(n=1)~∞ r_nα_(nj))e_j holds for every choice of scalar variables{α_(nj)} such that ∑_(j=1)~∞ α_(nj)e_j converges for each n≥1 and any choice of scalar variables{r_n} such that ∑_(n=1)~∞ ∑_(j=1)~∞, r_nα_(nj)e_j converges. Moreover, some applications about theresult are given.