Representation for the W-weighted Drazin inverse of linear operators

In this paper we study the W-weighted Drazin inverse of the bounded linear operators between Banach spaces and its representation theorem. Based on this representation, utilizing the spectral theory of Banach space operators, we derive an approximating expression of the W-weighted Drazin inverse and an error bound. Also, a perturbation theorem for the W-weighted Drazin inverse is uniformly obtained from the representation theorem.     

A note on the perturbation bounds of W-weighted Drazin inverse

We investigate the perturbation bound for the W-weighted Drazin inverse of a rectangular matrix and present two explicit expressions for the W-weighted Drazin inverse under the one-sided condition, which extends the results in Appl. Math. Comput. 2004;149:423–430.     

The representation and perturbation of theW-weighted Drazin inverse

LetA andE bem x n matrices andW an n xm matrix, and letA _d,W denote the W-weighted Drazin inverse ofA. In this paper, a new representation of the W-weighted Drazin inverse ofA is given. Some new properties for the W-weighted Drazin inverseA _d,W and B_d,W are investigated, whereB =A+E. In addition, the Banach-type perturbation theorem for the W-weighted Drazin inverse ofA andB are established, and the perturbation bounds for ∥B_d,W∥ and ∥B_{d, W}, -A_d,W∥/∥A_d,W∥ are also presented. WhenA andB are square matrices andW is identity matrix, some known results in the literature related to the Drazin inverse and the group inverse are directly reduced by the results in this paper as special cases.     

The reduced minimum modulus of Drazin inverses of linear operators on Hilbert spaces

In this article, we study the reduced minimum modulus of the Drazin inverse of an operator on a Hilbert space and give lower and upper bounds of the reduced minimum modulus of an operator and its Drazin inverse, respectively. Using these results, we obtain a characterization of the continuity of Drazin inverses of operators on a Hilbert space.

     

Hilbert空间中算子广义逆的积分表示

利用算子矩阵分块的技巧,得到了Hilbert空间中算子的Moore-Penrose逆和Drazin逆的积分表示.给出了较为简洁的证明,同时将有限维的结论推广到无限维的情形.     

Comment on “Condition number of W-weighted Drazin inverse and their condition numbers of singular linear systems”

In this paper, we establish the explicit condition number formulas for the W-weighted Drazin inverse of a singular matrix A, where A∈? ^m×n , W∈? ^n×m , ?((AW) ^k )=?((AW) ^{k *}), ?((WA) ^k )=?((WA) ^{k *}), and k=max{index(AW), index(WA)}, by the Schur decomposition of A and W. The sensitivity for the W-weighted Drazin-inverse solution of singular systems is also discussed. Based on this form of Schur decomposition, the explicit condition number formulas for the W-weighted Drazin inverse are given by the spectral norm and Frobenius norm instead of the ‖?‖ _P,W -norm, where P is a transformation matrix of the Jordan canonical form of AW, thereby improving the earlier work of Lei et al. (Appl. Math. Comput. 165:185–194, [2005]) and Wang et al. (Appl. Math. Comput. 162:434–446, [2005]).     

Hilbert空间上算子W-加权Drazin逆的刻画及表示

该文研究了Hilbert空间上线性算子的W-加权Drazin逆,利用算子的分块矩阵表示,给出了W-加权Drazin逆的刻画及表示,所获结果推广了魏益民等的相关结果.     

Representations for the weighted Drazin inverse of a sum of matrices

Three representations for the W-weighted Drazin inverse of a matrix A?CWB have been developed under some conditions where A,B,C∈? ^m×n , and W∈? ^n×m . The results of this paper not only extend the earlier works about the Drazin inverse and group inverse, but also weaken the assumed condition of a result of the Drazin inverse to the case where Γ _d ZZ _g =ZZ _g Γ _d is substituted with C(Γ _d ZZ _g ?ZZ _g Γ _d )B=0. Numerical examples are given to illustrate some new results.     

On the Drazin inverses involving power commutativity

We explore the Drazin inverses of bounded linear operators with power commutativity (PQ=QmP) in a Hilbert space. Conditions on Drazin invertibility are formulated and shown to depend on spectral properties of the operators involved. Moreover, we prove that P±Q is Drazin invertible if P and Q are dual power commutative (PQ=QmP and QP=PnQ) and show that the explicit representations of the Drazin inverse D(P±Q) depend on the positive integers m,n?2.     

Additive results for the Wg-Drazin inverse

In this paper we prove the formula for the expression (A+B)^d,W in terms of A,B,W,A^d,W,B^d,W, assuming some conditions for A,B and W. Here S^d,W denotes the generalized W-weighted Drazin inverse of a linear bounded operator S on a Banach space.     

A characteristic operator function for the class of n-hypercontractions

We consider a class of bounded linear operators on Hilbert space called n-hypercontractions which relates naturally to adjoint shift operators on certain vector-valued standard weighted Bergman spaces on the unit disc. In the context of n-hypercontractions in the class C0⋅ we introduce a counterpart to the so-called characteristic operator function for a contraction operator. This generalized characteristic operator function Wn,T is an operator-valued analytic function in the unit disc whose values are operators between two Hilbert spaces of defect type. Using an operator-valued function of the form Wn,T, we parametrize the wandering subspace for a general shift invariant subspace of the corresponding vector-valued standard weighted Bergman space. The operator-valued analytic function Wn,T is shown to act as a contractive multiplier from the Hardy space into the associated standard weighted Bergman space.     

New additive results for the generalized Drazin inverse

In this paper, we investigate additive properties of generalized Drazin inverse of two Drazin invertible linear operators in Banach spaces. Under the commutative condition of PQ=QP, we give explicit representations of the generalized Drazin inverse d(P+Q) in term of P, Pd, Q and Qd. We consider some applications of our results to the perturbation of the Drazin inverse and analyze a number of special cases.     

Area Integral Functions and H Functional Calculus for Sectorial Operators on Hilbert Spaces

Area integral functions are introduced for sectorial operators on Hilbert spaces. We establish the equivalence relationship between the square and area integral functions. This immediately extends McIntosh/Yagi's results on H^∞ functional calculus of sectorial operators on Hilbert spaces to the case when the square functions are replaced by the area integral functions.     

Representations and properties of the W-Weighted Drazin inverse

Several new representations of the W-weighted Drazin inverse are introduced. These representations are expressed in terms of various matrix powers as well as in terms of matrix products involving the Moore–Penrose inverse and the usual matrix inverse. Also, the properties of various generalized inverses which arise from derived representations are investigated. The computational complexity and efficiency of the proposed representations are considered. Representations are tested and compared among themselves in a substantial number of randomly generated test examples.     

Wandering r-tuples for unitary systems

The properties of the set Wr(U) of all complete wandering r-tuples for a system U of unitary operators acting on a Hilbert space are investigated by parameterizing Wr(U) in terms of a fixed wandering r-tuple Ψ and the set of all unitary operators which locally commute with U at Ψ. The special case of greatest interest is the system 〈D,T〉 of dilation (by 2) and translation (by 1) unitary operators acting on L²(R), for which the complete wandering r-tuples are precisely the orthogonal multiwavelets with multiplicity r. We also give some examples for its application.     

Weighted binary relations for operators on Banach spaces

Using the Wg-Drazin inverses, we introduce and characterize new weighted pre-orders on the set of all bounded linear operators between two Banach spaces. As an application, we present two generalized Drazin pre-orders and an extension of the generalized Drazin order to a partial order.     

On k-hyperexpansive operators

This paper considers the k-hyperexpansive Hilbert space operators T (those satisfying , 1?n?k) and the k-expansive operators (those satisfying the above inequality merely for n=k). It is known that if T is k-hyperexpansive then so is any power of T; we prove the analogous result for T assumed merely k-expansive. Turning to weighted shift operators, we give a characterization of k-expansive weighted shifts, and produce examples showing the k-expansive classes are distinct. For a weighted shift W that is k-expansive for all k (that is, completely hyperexpansive) we obtain results for k-hyperexpansivity of back step extensions of W. In addition, we discuss the completely hyperexpansive completion problem which is parallel to Stampfli's subnormal completion problem.     

Core–EP pre-order of Hilbert space operators

A new binary relation associated with the core–EP inverse is presented and studied on the corresponding subset of all generalized Drazin invertible bounded linear Hilbert space operators. Using the (dual) core partial order between core parts of operators and the minus partial order between quasinilpotent parts of operators, new pre-orders and partial orders are also introduced and characterized.     

On the Drazin inverse of the sum of two operators and its application to operator matrices

Given two bounded linear operators F,G on a Banach space X such that G²F=GF²=0, we derive an explicit expression for the Drazin inverse of F+G. For this purpose, firstly, we obtain a formula for the resolvent of an auxiliary operator matrix in the form . From the provided representation of D(F+G) several special cases are considered. In particular, we recover the case GF=0 studied by Hartwig et al. [R.E. Hartwig, G. Wang, Y. Wei, Some additive results on Drazin inverse, Linear Algebra Appl. 322 (2001) 207-217] for matrices and by Djordjevi? and Wei [D.S. Djordjevi?, Y. Wei, Additive results for the generalized Drazin inverse, J. Aust. Math. Soc. 73 (1) (2002) 115-126] for operators. Finally, we apply our results to obtain representations for the Drazin inverse of operator matrices in the form which are extensions of some cases given in the literature.     

Boundedness of fractional integral operators on α-modulation spaces

In this paper, we obtain the boundedness of the fractional integral operators, the bilinear fractional integral operators and the bilinear Hilbert transform on α-modulation spaces.