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Perturbation analysis for the generalized inverses with prescribed idempotents in Banach algebras

155   0   0.0 ( 0 )
 Added by Yifeng Xue
 Publication date 2013
  fields
and research's language is English




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In this paper, we first study the perturbations and expressions for the generalized inverses $a^{(2)}_{p,q}$, $a^{(1, 2)}_{p,q}$, $a^{(2, l)}_{p,q}$ and $a^{(l)}_{p,q}$ with prescribed idempotents $p$ and $q$. Then, we investigate the general perturbation analysis and error estimate for some of these generalized inverses when $p,,q$ and $a$ also have some small perturbations.



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153 - Jianbing Cao , Yifeng Xue 2013
In this paper, the problems of perturbation and expression for the Moore--Penrose metric generalized inverses of bounded linear operators on Banach spaces are further studied. By means of certain geometric assumptions of Banach spaces, we first give some equivalent conditions for the Moore--Penrose metric generalized inverse of perturbed operator to have the simplest expression $T^M(I+ delta TT^M)^{-1}$. Then, as an application our results, we investigate the stability of some operator equations in Banach spaces under different type perturbations.
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We determine when contractive idempotents in the measure algebra of a locally compact group commute. We consider a dynamical version of the same result. We also look at some properties of groups of measures whose identity is a contactive idempotent.
127 - Fapeng Du , Yifeng Xue 2013
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154 - Rongwei Yang 2008
For a tuple $A=(A_0, A_1, ..., A_n)$ of elements in a unital Banach algebra ${mathcal B}$, its {em projective spectrum} $p(A)$ is defined to be the collection of $z=[z_0, z_1, ..., z_n]in pn$ such that $A(z)=z_0A_0+z_1A_1+... +z_nA_n$ is not invertible in ${mathcal B}$. The pre-image of $p(A)$ in ${cc}^{n+1}$ is denoted by $P(A)$. When ${mathcal B}$ is the $ktimes k$ matrix algebra $M_k(cc)$, the projective spectrum is a projective hypersurface. In infinite dimensional cases, projective spectrums can be very complicated, but also have some properties similar to that of hypersurfaces. When $A$ is commutative, $P(A)$ is a union of hyperplanes. When ${mathcal B}$ is reflexive or is a $C^*$-algebra, the {em projective resolvent set} $P^c(A):=cc^{n+1}setminus P(A)$ is shown to be a disjoint union of domains of holomorphy. Later part of this paper studies Maurer-Cartan type ${mathcal B}$-valued 1-form $A^{-1}(z)dA(z)$ on $P^c(A)$. As a consequence, we show that if ${mathcal B}$ is a $C^*$-algebra with a trace $phi$, then $phi(A^{-1}(z)dA(z))$ is a nontrivial element in the de Rham cohomology space $H^1_d(P^c(A), cc)$.
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