scholarly journals On the pseudo drazin inverse of the sum of two elements in a Banach algebra

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 2011-2022 ◽  
Author(s):  
Honglin Zou ◽  
Jianlong Chen
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drazin Inverse ◽  
Necessary And Sufficient ◽  
Associative Rings

In this paper, some additive properties of the pseudo Drazin inverse are obtained in a Banach algebra. In addition, we find some new conditions under which the pseudo Drazin inverse of the sum a + b can be explicitly expressed in terms of a, az, b, bz. In particular, necessary and sufficient conditions for the existence as well as the expression for the pseudo Drazin inverse of the sum a+b are obtained under certain conditions. Also, a result of Wang and Chen [Pseudo Drazin inverses in associative rings and Banach algebras, LAA 437(2012) 1332-1345] is extended.

scholarly journals Derivations on Banach algebras

2003 ◽  
Vol 2003 (28) ◽  
pp. 1803-1806
Author(s):  
S. Hejazian ◽  
S. Talebi
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
Equivalent Conditions

LetDbe a derivation on a Banach algebra; by using the operatorD2, we give necessary and sufficient conditions for the separating ideal ofDto be nilpotent. We also introduce an idealM(D)and apply it to find out more equivalent conditions for the continuity ofDand for nilpotency of its separating ideal.

scholarly journals Derivations on the module extension Banach algebras

2021 ◽  
Vol 73 (4) ◽  
pp. 566-576
Author(s):  
A. Bodaghi ◽  
A. Teymouri ◽  
D. Ebrahimi Bagha
Keyword(s):  
Banach Algebra ◽  
Banach Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Closed Ideal ◽  
Closed Submodule ◽  
Module Extension ◽  
Necessary And Sufficient ◽  
Ideal Amenability

UDC 517.986 We correct some results presented in [M. Eshaghi Gordji, F. Habibian, A. Rejali, <em> Ideal amenability of module extension Banach algebras</em>, Int. J. Contemp. Math. Sci.,  <strong>2</strong>, No. 5, 213–219 (2007)] and, using the obtained consequences, we find necessary and sufficient conditions for the module extension to be -weakly amenable, where is a closed ideal of the Banach algebra and is a closed -submodule of the Banach -bimodule We apply this result to the module extension where are two Banach -bimodules.

scholarly journals On extension of bi-derivations to the bidual of Banach algebras

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2261-2267
Author(s):  
Attar Erfanian ◽  
S. Barootkoob ◽  
Vishki Ebrahimi
Keyword(s):  
Banach Algebras ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
C Algebras ◽  
Necessary And Sufficient

We present some necessary and sufficient conditions such that the (Arens) extensions of a bi-derivation on Banach algebras are again bi-derivations. We then examine our results for some Banach algebras. In particular, we show that the (Arens) extensions of a bi-derivation on C*-algebras are biderivations. Some results on extensions of an inner bi-derivation are also included.

scholarly journals The outer inverse f(2)T,S of a homomorphism of right R-modules

Filomat ◽  
2019 ◽  
Vol 33 (19) ◽  
pp. 6459-6468
Author(s):  
Zhou Wang
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Outer Inverse ◽  
Basic Properties ◽  
Definition Of ◽  
Necessary And Sufficient

In this paper, we introduce the definition of the generalized inverse f(2)T,S, which is an outer inverse of the homomorphism f of right R-modules with prescribed image T and kernel S. Some basic properties of the generalized inverse f(2)T,S are presented. It is shown that the Drazin inverse, the group inverse and the Moore-Penrose inverse, if they exist, are all the generalized inverse f 2) T,S. In addition, we give necessary and sufficient conditions for the existence of the generalized inverse f(1,2)T,S.

scholarly journals Representations of Generalized Inverses and Drazin Inverse of Partitioned Matrix with Banachiewicz-Schur Forms

2016 ◽  
Vol 2016 ◽  
pp. 1-14 ◽  
Author(s):  
Xiaoji Liu ◽  
Hongwei Jin ◽  
Jelena Višnjić
Keyword(s):  
Sufficient Conditions ◽  
Schur Complement ◽  
Block Matrix ◽  
Necessary And Sufficient Conditions ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Necessary And Sufficient ◽  
Generalized Schur Complement ◽  
Quotient Property

Representations of 1,2,3-inverses, 1,2,4-inverses, and Drazin inverse of a partitioned matrix M=ABCD related to the generalized Schur complement are studied. First, we give the necessary and sufficient conditions under which 1,2,3-inverses, 1,2,4-inverses, and group inverse of a 2×2 block matrix can be represented in the Banachiewicz-Schur forms. Some results from the paper of Cvetković-Ilić, 2009, are generalized. Also, we expressed the quotient property and the first Sylvester identity in terms of the generalized Schur complement.

scholarly journals The Generalized Inverse A(2)T, Sof a Matrix Over an Associative Ring

2007 ◽  
Vol 83 (3) ◽  
pp. 423-438 ◽  
Author(s):  
Yaoming Yu ◽  
Guorong Wang
Keyword(s):  
Associative Ring ◽  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drazin Inverse ◽  
Group Inverse ◽  
Arbitrary Matrix ◽  
Definition Of ◽  
Necessary And Sufficient ◽  
Explicit Expressions

AbstractIn this paper we establish the definition of the generalized inverse A(2)T, Swhich is a {2} inverse of a matrixAwith prescribed imageTand kernelsover an associative ring, and give necessary and sufficient conditions for the existence of the generalized inverseand some explicit expressions forof a matrix A over an associative ring, which reduce to the group inverse or {1} inverses. In addition, we show that for an arbitrary matrixAover an associative ring, the Drazin inverse Ad, the group inverse Agand the Moore-Penrose inverse. if they exist, are all the generalized inverse A(2)T, S.

scholarly journals Closed subalgebras of homogeneous Banach algebras

1975 ◽  
Vol 20 (3) ◽  
pp. 366-376 ◽  
Author(s):  
Ching-Nan Tseng ◽  
Hawi-Chiuan Wang
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Banach Algebras ◽  
Synthesis Method ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient ◽  
Closed Subalgebras ◽  
Segal Algebras

AbstractRudin's synthesis method for investigating closed subalgebras of L1(G), where G is an infinite compact abelian group, is extended to the study of closed subalgebras in homogeneous Banach algebras and Segal algebras. Necessary and sufficient conditions are given for the synthesis to hold in certain classes of homogeneous Banach algebras and it is proved that in the Ap(G) algebras the synthesis holds for 1 ≦ p 2 but fails for Ap(T), 2 < p < ∞.

scholarly journals *-DMP elements in *-semigroups and *-rings

Filomat ◽  
2018 ◽  
Vol 32 (9) ◽  
pp. 3073-3085 ◽  
Author(s):  
Yuefeng Gao ◽  
Jianlong Chen ◽  
Yuanyuan Ke
Keyword(s):  
Positive Integer ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Drazin Inverse ◽  
Complex Matrices ◽  
Arbitrary Ring ◽  
The Core ◽  
Core Inverse ◽  
Necessary And Sufficient

In this paper, we investigate *-DMP elements in *-semigroups and *-rings. The notion of *-DMP element was introduced by Patr?cio and Puystjens in 2004. An element a is *-DMP if there exists a positive integer m such that am is EP. We first characterize *-DMP elements in terms of the {1,3}-inverse, Drazin inverse and pseudo core inverse, respectively. Then, we characterize the core-EP decomposition utilizing the pseudo core inverse, which extends the core-EP decomposition introduced by Wang for complex matrices to an arbitrary *-ring; and this decomposition turns to be a useful tool to characterize *-DMP elements. Further, we extend Wang?s core-EP order from complex matrices to *-rings and use it to investigate *-DMP elements. Finally, we give necessary and sufficient conditions for two elements a,b in *-rings to have aaD = bbD, which contribute to study *-DMP elements.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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