Drazin Inverse Research Papers - Academia.edu (original) (raw)

This paper studies additive properties of the generalized Drazin inverse (g-Drazin inverse) in a Banach algebra and finds an explicit expression for the g-Drazin inverse of the sum a + b in terms of a and b and their g-Drazin inverses... more

This paper studies additive properties of the generalized Drazin inverse (g-Drazin inverse) in a Banach algebra and finds an explicit expression for the g-Drazin inverse of the sum a + b in terms of a and b and their g-Drazin inverses under fairly mild conditions on a and b.

The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated spectral point ofthe operator. This inverse is useful for instance in the solution of... more

The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated spectral point ofthe operator. This inverse is useful for instance in the solution of differential equations formulated in a Banach space X. Since the elements of X rarely enter into our considerations, the exposition seems to gain in clarity when the operators are regarded as elements of the Banach algebra L(X).

In this talk results on isolated spectral points, group, Drazin and Koliha-Drazin invertible elements in the context of quotient Banach algebras or in ranges of (not necessarily continuous) homomorphism between complex unital Banach... more

In this talk results on isolated spectral points, group, Drazin and Koliha-Drazin invertible elements in the context of quotient Banach algebras or in ranges of (not necessarily continuous) homomorphism between complex unital Banach algebras will be presented. The case of Calkin algebras on Banach and Hilbert spaces will be also studied. In addition, (generalized) B-Fredholm elements in Banach algebras will be considered.

Given Banach spaces XXX and YYY and Banach space operators AinL(X)A\in L(X)AinL(X) and BinL(Y)B\in L(Y)BinL(Y), let rhocolonL(Y,X)toL(Y,X)\rho\colon L(Y,X)\to L(Y,X)rhocolonL(Y,X)toL(Y,X) denote the generalized derivation defined by AAA and BBB, i.e., rho(U)=AU−UB\rho (U)=AU-UBrho(U)=AU−UB ($U\in L(Y,X)$). The main... more

Given Banach spaces XXX and YYY and Banach space operators AinL(X)A\in L(X)AinL(X) and BinL(Y)B\in L(Y)BinL(Y), let rhocolonL(Y,X)toL(Y,X)\rho\colon L(Y,X)\to L(Y,X)rhocolonL(Y,X)toL(Y,X) denote the generalized derivation defined by AAA and BBB, i.e., rho(U)=AU−UB\rho (U)=AU-UBrho(U)=AU−UB ($U\in L(Y,X)$). The main objective of this article is to study Weyl and Browder type theorems for rhoinL(L(Y,X))\rho\in L(L(Y,X))rhoinL(L(Y,X)). To this end, however, first the isolated points of the spectrum and the Drazin spectrum of rhoinL(L(Y,X))\rho\in L(L(Y,X))rhoinL(L(Y,X)) need to be characterized. In addition, it will be also proved that if AAA and BBB are polaroid (respectively isoloid), then rho\rhorho is polaroid (respectively isoloid).

Given a (not necessarily continuous) homomorphism between Banach algebras TcolonAtoBT\colonA\toBTcolonAtoB, an element ainAa\inAainA will be said to be B-Fredholm (respectively generalized B-Fredholm) relative to TTT, if T(a)inBT(a)\in \BT(a)inB is Drazin invertible... more

Given a (not necessarily continuous) homomorphism between Banach algebras TcolonAtoBT\colonA\toBTcolonAtoB, an element ainAa\inAainA will be said to be B-Fredholm (respectively generalized B-Fredholm) relative to TTT, if T(a)inBT(a)\in \BT(a)inB is Drazin invertible (respectively Koliha-Drazin invertible). In this article, the aforementioned elements will be characterized and their main properties will be studied. In addition, perturbation properties will be also considered.

Given a Banach Algebra AAA and ainAa\in AainA, several relationships among the Drazin spectrum of aaa and the ascent, the descent and the Drazin spectra of the multiplication operators LaL_aLa and RaR_aRa will be presented; the Banach space... more

Given a Banach Algebra AAA and ainAa\in AainA, several relationships among the Drazin spectrum of aaa and the ascent, the descent and the Drazin spectra of the multiplication operators LaL_aLa and RaR_aRa will be presented; the Banach space operator case will be also examined. In addition, a characterization of the spectrum of aaa in terms of the Drazin spectrum and the poles of the resolvent of aaa will be considered. Furthermore, several basic properties of the Drazin spectrum in Banach algebras will be studied.

Given a Banach Algebra A and ainAa\in AainA, several relations among the Drazin spectrum of aaa and the Drazin spectra of the multiplication operators LaL_aLa and RaR_aRa will be stated. The Banach space operator case will be also examined.... more

Given a Banach Algebra A and ainAa\in AainA, several relations among the Drazin spectrum of aaa and the Drazin spectra of the multiplication operators LaL_aLa and RaR_aRa will be stated. The Banach space operator case will be also examined. Furthermore, a characterization of the Drazin spectrum will be considered.

In this article poles, isolated spectral points, group, Drazin and Koliha-Drazin invertible elements in the context of quotient Banach algebras or in ranges of (not necessarily continuous) homomorphism between complex unital Banach... more

In this article poles, isolated spectral points, group, Drazin and Koliha-Drazin invertible elements in the context of quotient Banach algebras or in ranges of (not necessarily continuous) homomorphism between complex unital Banach algebras will be characterized using Fredholm and Riesz Banach algebra elements. Calkin algebras on Banach and Hilbert spaces will be also considered.

In this paper we introduce an interpolation method for computing the Drazin inverse of a given polynomial matrix. This method is an extension of the known method from [A. Schuster, P. Hippe, Inversion of polynomial matrices by... more

In this paper we introduce an interpolation method for computing the Drazin inverse of a given polynomial matrix. This method is an extension of the known method from [A. Schuster, P. Hippe, Inversion of polynomial matrices by interpolation, IEEE Trans. Automat. Control 37 (3) (1992) 363–365], applicable to usual matrix inverse. Also, we improve our interpolation method, using a more

Linear time-invariant networks are modelled by linear differential-algebraic equations with constant coefficients. These equations can be represented by a matrix pencil. Many publications on this subject are restricted to regular matrix... more

Linear time-invariant networks are modelled by linear differential-algebraic equations with constant coefficients. These equations can be represented by a matrix pencil. Many publications on this subject are restricted to regular matrix pencils. In particular, the influence of the Weierstrass structure of a regular pencil on the poles of its inverse is well known. In this paper we investigate singular matrix pencils. The relations between the Kronecker structure of a singular matrix pencil and the multiplicity of poles at zero of the Moore–Penrose inverse and the Drazin inverse of the rational matrix are investigated. We present example networks whose circuit equations yield singular matrix pencils.

We present a unified representation theorem for the class of all outer generalized inverses of a bounded linear operator. Using this representation we develop a few specific expressions and computational procedures for the set of outer... more

We present a unified representation theorem for the class of all outer generalized inverses of a bounded linear operator. Using this representation we develop a few specific expressions and computational procedures for the set of outer generalized inverses. The obtained result is a generalization of the well-known representation theorem of the Moore--Penrose inverse as well as a generalization of the

The transfer property for the generalized Browder's theorem both of the tensor product and of the left-right multiplication operator will be characterized in terms of the B-Weyl spectrum inclusion. In addition, the isolated points of... more

The transfer property for the generalized Browder's theorem both of the tensor product and of the left-right multiplication operator will be characterized in terms of the B-Weyl spectrum inclusion. In addition, the isolated points of these two classes of operators will be fully characterized.

The paper introduces and studies the weighted g-Drazin inverse for bounded linear operators between Banach spaces, extending the concept of the weighted Drazin inverse of Rakočević and Wei (Linear Algebra Appl. 350 (2002), 25–39) and of... more

The paper introduces and studies the weighted g-Drazin inverse for bounded linear operators between Banach spaces, extending the concept of the weighted Drazin inverse of Rakočević and Wei (Linear Algebra Appl. 350 (2002), 25–39) and of Cline and Greville (Linear Algebra Appl. 29 (1980), 53–62). We use the Mbekhta decomposition to study the structure of an operator possessing the weighted g-Drazin inverse, give an operator matrix representation for the inverse, and study its continuity. An open problem of Rakočević and Wei is solved.

In this paper we introduce an interpolation method for computing the Drazin inverse of a given polynomial matrix. This method is an extension of the known method from [A. Schuster, P. Hippe, Inversion of polynomial matrices by... more

In this paper we introduce an interpolation method for computing the Drazin inverse of a given polynomial matrix. This method is an extension of the known method from [A. Schuster, P. Hippe, Inversion of polynomial matrices by interpolation, IEEE Trans. Automat. Control 37 (3) (1992) 363–365], applicable to usual matrix inverse. Also, we improve our interpolation method, using a more effective estimation of degrees of polynomial matrices generated in Leverrier–Faddeev method. Algorithms are implemented and tested in the symbolic package MATHEMATICA.