This paper presents two new theorems for multiplicative perturbations of C-regularized resolvent families, which generalize the previous related ones for the resolvent families.
Let T=(T(t))_(t≥0)be a bounded C-regularized semigroup generated by A on a Banach space X and R(C)be dense in X.We show that if there is a dense subspace Y of X such that for every x ∈ Y,σ_u(A,Cx),the set of all po...Let T=(T(t))_(t≥0)be a bounded C-regularized semigroup generated by A on a Banach space X and R(C)be dense in X.We show that if there is a dense subspace Y of X such that for every x ∈ Y,σ_u(A,Cx),the set of all points λ ∈ iR to which(λ-A)^(-1)Cx can not be extended holomorphically,is at most countable and σ_r(A)∩ iR=(?),then T is stable.A stability result for the case of R(C)being non-dense is also given.Our results generalize the work on the stability of strongly continuous semigroups.展开更多
We study a new class of group inverses determined by right c-regular elements.The new concept of right c-group inverses is introduced and studied.It is shown that every right c-group invertible element is group invert...We study a new class of group inverses determined by right c-regular elements.The new concept of right c-group inverses is introduced and studied.It is shown that every right c-group invertible element is group invertible,and an example is given to show that group invertible elements need not be right c-group invertible.The conditions that right c-group invertible elements are precisely group invertible elements are investigated.We also study the strongly clean decompositions of right c-group invertible elements.As applications,we give some new characterizations of abelian rings and directly finite rings from the point of view of right c-groupinverses.展开更多
We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R...We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R)[X]=W(R)[X]■Z(R[X]),where J(A),N^(*)(A),W(A),Z(A)are the Jacobson radical,upper nilradical,Wedderburn radical,and center of a given ring A,respectively,and A[X]denotes the polynomial ring with a set X of commuting indeterminates over A;we also prove that R is semiprime if and only if the right(left)singular ideal of R is zero.We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular,from any given ring.Moreover,for a C-regular ring R,the following are proved to be equivalent:(i)R is Abelian;(ii)every prime factor ring of R is a duo domain;(ii)R is quasi-duo;and(iv)R/W(R)is reduced.展开更多
文摘This paper presents two new theorems for multiplicative perturbations of C-regularized resolvent families, which generalize the previous related ones for the resolvent families.
基金supported by the NSF of Chinasupported by TRAPOYT and the NSF of China(No.10371046)
文摘Let T=(T(t))_(t≥0)be a bounded C-regularized semigroup generated by A on a Banach space X and R(C)be dense in X.We show that if there is a dense subspace Y of X such that for every x ∈ Y,σ_u(A,Cx),the set of all points λ ∈ iR to which(λ-A)^(-1)Cx can not be extended holomorphically,is at most countable and σ_r(A)∩ iR=(?),then T is stable.A stability result for the case of R(C)being non-dense is also given.Our results generalize the work on the stability of strongly continuous semigroups.
基金Supported by the National Natural Science Foundation of China(Grant No.12161049).
文摘We study a new class of group inverses determined by right c-regular elements.The new concept of right c-group inverses is introduced and studied.It is shown that every right c-group invertible element is group invertible,and an example is given to show that group invertible elements need not be right c-group invertible.The conditions that right c-group invertible elements are precisely group invertible elements are investigated.We also study the strongly clean decompositions of right c-group invertible elements.As applications,we give some new characterizations of abelian rings and directly finite rings from the point of view of right c-groupinverses.
文摘We study the structure of rings which satisfy the von Neumann regularity of commutators,and call a ring R C-regularif ab-ba ∈(ab-ba)R(ab-ba)for all a,b in R.For a C-regular ring R,we prove J(R[X])=N^(*)(R[X])=N^(*)(R)[X]=W(R)[X]■Z(R[X]),where J(A),N^(*)(A),W(A),Z(A)are the Jacobson radical,upper nilradical,Wedderburn radical,and center of a given ring A,respectively,and A[X]denotes the polynomial ring with a set X of commuting indeterminates over A;we also prove that R is semiprime if and only if the right(left)singular ideal of R is zero.We provide methods to construct C-regular rings which are neither commutative nor von Neumann regular,from any given ring.Moreover,for a C-regular ring R,the following are proved to be equivalent:(i)R is Abelian;(ii)every prime factor ring of R is a duo domain;(ii)R is quasi-duo;and(iv)R/W(R)is reduced.
