ON THE REVERSE ORDER LAW (AB)D=BDAD

1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌ ｋｎｏｗｎｔｈａｔｔｈｅＤｒａｚｉｎｉｎｖｅｒｓｅｈａｓｂｅｅｎｗｉｄｅｌｙａｐｐｌｉｅｄｔｏｔｈｅｔｈｅｏｒｙｏｆｆｉｎｉｔｅＭａｒｋｏｖｃｈａｉｎｓａｎｄｓｉｎｇｕｌａｒｄｉｆｆｅｒｅｎｔｉａｌａｎｄｄｉｆｆｅｒｅｎｃｅｅｑｕａｔｉｏｎｓ[2 ] .Ｉｎａｃｌａｓｓｉｃｐａｐｅｒ [3],Ｇｒｅｖｉｌｌｅｇａｖｅｎｅｃｅｓｓａｒｙａｎｄｓｕｆｆｉｃｉｅｎｔｃｏｎｄｉｔｉｏｎｓｆｏｒｔｈｅｔｗｏｔｅｒｍｒｅ ｖｅｒｓｅｏｒｄｅｒｌａｗ (ＡＢ) + =Ｂ+ Ａ+ ｔｏｈｏ…     

ON THE STRUCTURES OF RANDOM MEASURE AND POINT PROCESS CONVOLUTION SEMIGROUPS

§0.ＩｎｔｒｏｄｕｃｔｉｏｎＫｅｎｄａｌｌ[1]ｆｏｕｎｄｅｄｔｈｅＤｅｌｐｈｉｃｓｅｍｉｇｒｏｕｐｔｈｅｏｒｙａｎｄｕｓｅｄｔｈｉｓｔｈｅｏｒｙｔｏｓｔｕｄｙｔｈｅｓｔｒｕｃｔｕｒｅｓｏｆｔｈｅｒｅｎｅｗａｌｓｅｑｕｅｎｃｅｓｅｍｉｇｒｏｕｐａｎｄｔｈｅｓｔａｎｄａｒｄｐ-ｆｕｎｃｔｉｏｎｓｅｍｉｇｒｏｕｐ.Ｄａｖｉｄｓｏｎ[2,3],ＲｕｚｓａａｎｄＳｚ啨ｋｅｌｙ[4]ａｎｄＨｅ[5,6]ｇｅｎｅｒａｌｉｚｅｄｔｈｅＤｅｌｐｈｉｃｓｅｍｉｇｒｏｕｐｔｈｅｏｒｙａｎｄｕｓｅｄｔｈｅｉｒｒｅｓｕｌｔｓｔｏｓｔｕｄ…     

Finite Rank Operators in Closed Maximal Triangular Algebras

Ｆｉｎｉｔｅｒａｎｋｏｐｅｒａｔｏｒｓａｎｄｒａｎｋｏｎｅｏｐｅｒａｔｏｒｓａｒｅｉｍｐｏｒｔａｎｔｔｏｔｈｅｔｈｅｏｒｙｏｆｎｅｓｔａｌｇｅｂｒａｓ .Ｉｎａｎｅｓｔａｌｇｅｂｒａ ,ｅａｃｈｆｉｎｉｔｅｒａｎｋｏｐｅｒａｔｏｒｃａｎｂｅｗｒｉｔｔｅｎａｓａｆｉｎｉｔｅｓｕｍｏｆｒａｎｋｏｎｅｏｐｅｒａｔｏｒｓｗｈｉｃｈｂｅｌｏｎｇｔｏｉｔｓｅｌｆ (Ｔｈｉｓｒｅｓｕｌｔｉｓｉｎ [1 ],ｂｕｔｂｅｌｏｎｇｓｔｏＪ .Ｒ .Ｒｉｎｇｒｏｓｅ) .Ｉｎａｎａｔｕｒａｌｗａｙ ,ｗｅｍａｙａｓｋ :　Ｑｕｅｓｔｉｏｎ…     

A New Class of Banach Spaces with Uniform Normal Structure

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＬｅｔＸｂｅａｎｏｒｍｅｄｌｉｎｅａｒｓｐａｃｅ ,ａｎｄｌｅｔＳ(Ｘ) ={ｘ∈Ｘ :ｘ =1 }ｂｅｔｈｅｕｎｉｔｓｐｈｅｒｅｏｆＸ .Ｉｎ 1 94 8,ＢｒｏｄｓｋｉｉａｎｄＭｉｌｍａｎ[1]ｉｎｔｒｏｄｕｃｅｄｔｈｅｆｏｌｌｏｗｉｎｇｇｅｏｍｅｔｒｉｃｃｏｎｃｅｐｔ:　Ｄｅｆｉｎｉｔｉｏｎ 1　ＡＢｏｕｎｄｅｄ ,ｃｏｎｖｅｘｓｕｂｓｅｔＫｏｆａＢａｎａｃｈｓｐａｃｅＸｉｓｓａｉｄｔｏｈａｖｅｎｏｒｍａｌｓｔｒｕｃｔｕｒｅｉｆｅｖｅｒｙｃｏｎｖｅｘｓｕｂｓｅｔＨｏｆＫｔｈａｔｃｏｎｔａ…     

Near Subnormality of Weighted Shifts and the Answer to the Hilbert Space Problem 160

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　Ｉｎ [1 ] ,ｔｗｏｏｐｅｒａｔｏｒｃｌａｓｓｅｓｏｆＤ ｎｅａｒｓｕｂｎｏｒｍａｌｏｐｅｒａｔｏｒｓａｎｄｎｅａｒｓｕｂｎｏｒｍａｌｏｐｅｒａｔｏｒｓｗｅｒｅｉｎｔｒｏｄｕｃｅｄ .ＴｈｅｅｌｅｍｅｎｔａｒｙｐｒｏｐｅｒｔｉｅｓｏｆｓｕｃｈｏｐｅｒａｔｏｒｓｗｅｒｅｏｂｔａｉｎｅｄｂｙｕｓｅｏｆｔｈｅＭ Ｐｇｅｎｅｒａｌｉｚｅｄｉｎｖｅｒｓｅ.Ｍｏｒｅｏｖｅｒ,ａｎｅｗｎｅｃｅｓｓａｒｙａｎｄｓｕｆｆｉｃｉｅｎｔｃｏｎｄｉｔｉｏｎｆｏｒａｎｏｐｅｒａｔｏｒｔｏｂｅｓｕｂｎ…     

A Note on “Multi-separation, Centrifugality and Centripetality Imply Chaos”

Ｉｎ [1 ] ,ｗｅｏｂｔａｉｎｅｄｔｈｅｆｏｌｌｏｗｉｎｇｔｗｏｔｈｅｏｒｅｍｓ:　Ｔｈｅｏｒｅｍ 4 .2　Ｌｅｔ(ｘ0 ,ｘ1,… ,ｘｎ)ｂｅａｒｅｔｕｒｎｔｒａｊｅｃｔｏｒｙｏｆｆ,ａｎｄｌｅｔＶ =[ｘｍ,ｘＭ]∩Ｆｉｘ(ｆ)ｂｅａｓｉｎＤｅｆｉｎｉｔｉｏｎ 2 .1 .Ｌｅｔｖ∈Ｖｂｅｇｉｖｅｎ .Ｉｆｔｈｅｔｒａｊｅｃｔｏｒｙ (ｘ0 ,ｘ1,… ,ｘｎ)ｃｏｎｔａｉｎｓｋｃｅｎｔｒｉｐｅｔａｌｐｏｉｎｔｓｒｅｌａｔｉｖｅｔｏｖ,(ｉ.ｅ.,ｔｈｅｃａｒｄｉｎａｌｎｕｍｂｅｒｏｆｔｈｅｓｅｔ{ｊ:ｘｊｉｓａｃｅｎｔｒｉｐｅｔａｌｐｏ…     

A Generalization of C—semigroup

§ 1.ＴｈｅＧｅｎｅｒａｌＣ ｓｅｍｉｇｒｏｕｐ　Ｉｎｔｈｉｓｐａｐｅｒ,ｗｅｌｅｔＸｂｅａＢａｎａｃｈｓｐａｃｅ,ａｎｄＢ(Ｘ)ｔｈｅｓｐａｃｅｏｆａｌｌｂｏｕｎｄｅｄｌｉｎｅａｒｏｐｅｒａｔｏｒｓｆｒｏｍＸｉｎｔｏＸ .　Ｄｅｆｉｎｉｔｉｏｎ 1 .1　ＳｕｐｐｏｓｅＣｉｓａｂｏｕｎｄｅｄｉｎｊｅｃｔｉｖｅｏｐｅｒａｔｏｒｆｒｏｍＸｉｎｔｏＸ .Ｔｈｅｆａｍｉｌｙｏｆｂｏｕｎｄｅｄｌｉｎｅａｒｏｐｅｒａｔｏｒｓ (Ｓ(ｔ) ) ｔ≥ 0 ｉｓａｇｅｎｅｒａｌＣ ｓｅｍｉｇｒｏｕｐｉｆｉｔｈａｓｔｈｅｆｏｌｌｏｗｉｎｇ…     

Inverse Limit of Spaces of Upper Semicontinuous Functions

ＳｕｐｐｏｓｅｔｈａｔＸｉｓａｃｏｍｐａｃｔＨａｕｓｄｏｒｆｆｓｐａｃｅ,ａｎｄＵ(Ｘ)ｄｅｎｏｔｅｓｔｈｅｓｅｔｏｆａｌｌｔｈｅｕｐｐｅｒｓｅｍｉｃｏｎｔｉｎｕｏｕｓｆｕｎｃｔｉｏｎｓｆｒｏｍＸｔｏ [0 ,1 ] .ＴｈｅｎＵ(Ｘ)ｉｓａｃｏｍｐｌｅｔｅｌａｔｔｉｃｅｕｎｄｅｒｔｈｅｐｏｉｎｔｗｉｓｅｏｒｄｅｒｉｎｇ ,ａｎｄｉｔｓｏｐｐｏｓｉｔｅｌａｔｔｉｃｅＵ(Ｘ) ｏｐｉｓａｃｏｎｔｉｎｕｏｕｓｌａｔｔｉｃｅ .ＴｈｕｓＵ(Ｘ)ｅｎｄｏｗｅｄｗｉｔｈｔｈｅｌｉｍ ｉｎｆｔｏｐｏｌｏｇｙｏｎＵ(Ｘ) ｏｐ,ｗｈｉｃ…     

有限单群中一类丢番图方程

§ 1.Ｉｎｔｒｏｄｕｃｔｉｏｎ　ＬｅｔＮ ,Ｎ0 ,ＱａｎｄＰｂｅｔｈｅｓｅｔｓｏｆｐｏｓｉｔｉｖｅｉｎｔｅｇｅｒｓ,ｎｏｎ ｎｅｇａｔｉｖｅｉｎｔｅｇｅｒｓ,ｒａｔｉｏｎａｌｎｕｍｂｅｒｓａｎｄｐｒｉｍｅｎｕｍｂｅｒｓｒｅｓｐｅｃｔｉｖｅｌｙ .Ｌｅｔ ｐ ,ｑ∈Ｐｗｉｔｈ ｐ≠ ｑ.ＴｈｅＤｉｏｐｈａｎｔｉｎｅｅｑｕａｔｉｏｎ1 +ｐａ =2 ｂｑｃ+2 ｄｐｅｑｆ,　　ａ ,ｂ,ｃ,ｄ ,ｅ,ｆ∈Ｎ0 (1 )ｉｓａｋｉｎｄｏｆｉｍｐｏｒｔａｎｔｅｘｐｏｎｅｎｔｉａｌＤｉｏｐｈａｎｔｉｎｅｅｑｕａｔｉｏｎｓｉｎｆｉｎｉｔｅｓｉｍ…     

A UNIFIED TREATMENT OF REGULARIZATION METHODS FOR LINEAR ILL-POSED PROBLEMS

1　ＩｎｔｒｏｄｕｃｔｉｏｎＬｅｔＡ :Ｘ→ＹｂｅａｂｏｕｎｄｅｄｌｉｎｅａｒｏｐｅｒａｔｏｒｂｅｔｗｅｅｎＨｉｌｂｅｒｔｓｐａｃｅｓＸａｎｄＹ ,ａｎｄｌｅｔＡ ｄｅ ｎｏｔｅｔｈｅＭｏｏｒｅ ＰｅｎｒｏｓｅｇｅｎｅｒａｌｉｚｅｄｉｎｖｅｒｓｅｏｆＡ (ｃｆ.[4 ]) .Ｔｈｉｓｐａｐｅｒｃｏｎｃｅｒｎｓｔｈｅｒｅｓｏｌｕｔｉｏｎｏｆｔｈｅｇｅｎｅｒａｌｉｚｅｄｓｏｌｕｔｉｏｎｘ :=Ａ ｙ0 ｏｆｔｈｅｅｑｕａｔｉｏｎＡｘ=ｙ0 (1.1)ｗｉｔｈｙ0 ∈Ｄ(Ａ ) :=Ｒ(Ａ) +Ｒ(Ａ) ⊥,ｗｈｅｒｅＤ(Ａ )ａｎｄＲ(Ａ)ｄｅｎｏｔ…     

Further Results on the Drazin Inverse of Tensors via Einstein Product

In this paper,we give further results on the Drazin inverse of tensors via the Einstein product.We give a limit formula for the Drazin inverse of tensors.By using this formula,the representations for the Drazin inverse of several block tensor are obtained.Further,we give the Drazin inverse of the sum of two tensors based on the representation for the Drazin inverse of a block tensor.     

Computing generalized inverses of a rational matrix and applications

In this paper we investigate symbolic implementation of two modifications of the Leverrier-Faddeev algorithm, which are applicable in computation of the Moore-Penrose and the Drazin inverse of rational matrices. We introduce an algorithm for computation of the Drazin inverse of rational matrices. This algorithm represents an extension of the papers [11] and [14]. and a continuation of the papers [15, 16]. The symbolic implementation of these algorithms in the package mathEmatica is developed. A few matrix equations are solved by means of the Drazin inverse and the Moore-Penrose inverse of rational matrices.     

A finite algorithm for the Drazin inverse of a polynomial matrix

Based on Greville's finite algorithm for Drazin inverse of a constant matrix we propose a finite numerical algorithm for the Drazin inverse of polynomial matrices. We also present a new proof for Decell's finite algorithm through Greville's finite algorithm.     

Formulae for the generalized Drazin inverse of a block matrix in terms of Banachiewicz–Schur forms

We introduce new expressions for the generalized Drazin inverse of a block matrix with the generalized Schur complement being generalized Drazin invertible in a Banach algebra under some conditions. We generalized some recent results for Drazin inverse and group inverse of complex matrices.     

A weighted Drazin inverse and applications

In this paper the notation of the Cline–Greville W-weighted Drazin inverse of a rectangular matrix is extended to bounded linear operators between Banach spaces. We give new characterizations of the W-weighted Drazin inverse, and we study the perturbations and the the continuity of the W-weighted Drazin inverse.     

Jacobson’s lemma for the generalized Drazin inverse

0truemm0truemm We study properties of elements in a ring which admit the generalized Drazin inverse. It is shown that the element 1-ab is generalized Drazin invertible if and only if so is 1-ba and a formula for the generalized Drazin inverse of 1-ba in terms of the generalized Drazin inverse and the spectral idempotent of 1-ab is provided. Further, recent results relating to the Drazin index can be recovered from our theorems.     

On the Generalized Drazin Inverse and Generalized Resolvent

We investigate the generalized Drazin inverse and the generalized resolvent in Banach algebras. The Laurent expansion of the generalized resolvent in Banach algebras is introduced. The Drazin index of a Banach algebra element is characterized in terms of the existence of a particularly chosen limit process. As an application, the computing of the Moore-Penrose inverse in >C ^*-algebras is considered. We investigate the generalized Drazin inverse as an outer inverse with prescribed range and kernel. Also, 2 × 2 operator matrices are considered. As corollaries, we get some well-known results.     

Left and right generalized Drazin invertible operators

In this paper, we define and study the left and the right generalized Drazin inverse of bounded operators in a Banach space. We show that the left (resp. the right) generalized Drazin inverse is a sum of a left invertible (resp. a right invertible) operator and a quasi-nilpotent one. In particular, we define the left and the right generalized Drazin spectra of a bounded operator and also show that these sets are compact in the complex plane and invariant under additive commuting quasi-nilpotent perturbations. Furthermore, we prove that a bounded operator is left generalized Drazin invertible if and only if its adjoint is right generalized Drazin invertible. An equivalent definition of the pseudo-Fredholm operators in terms of the left generalized Drazin invertible operators is also given. Our obtained results are used to investigate some relationships between the left and right generalized Drazin spectra and other spectra founded in Fredholm theory.     

On Drazin inverse of operator matrices

In this short paper, we offer (another) formula for the Drazin inverse of an operator matrix for which certain products of the entries vanish. We also give formula for the Drazin inverse of the sum of two operators under special conditions.     

The (2,2,0) Drazin inverse problem

We consider the additive Drazin problem and we study the existence of the Drazin inverse of a two by two matrix with zero (2,2) entry.