On (m, P)-expansive operators: products, perturbation by nilpotents, Drazin invertibility

IF 0.3 Q4 MATHEMATICS
B. Duggal
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引用次数: 1

Abstract

Abstract A generalisation of m-expansive Hilbert space operators T ∈ B(ℋ) [18, 20] to Banach space operators T ∈ B(𝒳) is obtained by defining that a pair of operators A, B ∈ B(𝒳) is (m, P)-expansive for some operator P ∈ B(𝒳) if Δ A,Bm(P)= (I-LARB)m(P)=∑j=0m(-1)j(jm){\left( {I - {L_A}{R_B}} \right)^m}\left( P \right) = \sum\nolimits_{j = 0}^m {{{\left( { - 1} \right)}^j}\left( {_j^m} \right)}AjPBj≤0; LA(X) = AX and RB(X)=XB. Unlike m-isometric and m-left invertible operators, commuting products and perturbations by commuting nilpotents of (m, I)-expansive operators do not result in expansive operators: using elementary algebraic properties of the left and right multiplication operators, a sufficient condition is proved. For Drazin invertible A and B ∈ B(ℋ), with Drazin inverses Ad and Bd, a sufficient condition proving (Ad, Bd) ^ (A, B) is (m − 1, P)-isometric (resp., (m − 1, P)-contractive) for m even (resp., m odd) is given, and a Banach space analogue of this result is proved.
关于(m,P)-扩张算子:乘积,幂零算子的扰动,Drazin可逆性
摘要通过定义一对算子A,B∈B(f)是(m, P),对于某些算子P∈B(f)是可扩张的,如果Δ A,Bm(P)= (I- larb)m(P)=∑j=0m(-1)j(jm) {\left ({I-{ L_AR_B }{}}\right)^m }\left (P \right)= \sum\nolimits _j{ =0 ^m }{{{\left (-1 {}\right)}^j }\left (_j{^m}\right)Aj}PBj≤0,得到了可扩张的Hilbert空间算子T∈B(f)的推广;LA(X) = AX, RB(X)=XB。与m-等距算子和m-左可逆算子不同,(m, I)-膨胀算子的交换积和交换幂零扰动不产生膨胀算子:利用左右乘法算子的初等代数性质,证明了一个充分条件。对于Drazin可逆的A和B∈B(h),且Drazin逆Ad和Bd,证明(Ad, Bd) ^ (A, B)是(m−1,P)-等距(P = 1)的充分条件。, (m−1,P)-压缩), m奇数),并证明了这一结果的一个巴拿赫空间模拟。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Concrete Operators
Concrete Operators MATHEMATICS-
CiteScore
1.00
自引率
16.70%
发文量
10
审稿时长
22 weeks
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