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.     

Perturbations of Invertible Operators and Stability of g-Frames in Hilbert Spaces

In this paper, we study the perturbations of invertible operators and stability of g-frames in Hilbert spaces. In particular, we obtain some conditions under which the perturbations of an invertible operator are still an invertible operator, the perturbations of a right invertible operator or a surjective operator are still a right invertible operator or surjective operator. Then we apply the perturbations of invertible operators to study the stability of g-frames which is close related with the invertibility (or right invertibility) property of operators.     

Lipschitz-α算子的M-谱理论

本文运用一个选定的可逆Lip-α算子M作为尺度算子(称为谱尺度),引入两个Banach空间之间的非线性Lip-α算子的M-豫解集、M-谱集、M-谱半径、豫解集、谱集及谱半径,证明了它们的一列系重要性质,给出了M-谱的一个摄动定理,初步建立了Lip-α算子的M-谱理论,使得现有的谱理论成为其特例.     

Invertibility Conditions for Second-Order Differential Operators in the Space of Continuous Bounded Functions

We prove the invertibility of second-order differential operators with constant operator coefficients acting on the Banach space of bounded continuous functions on the real line under the condition that they are uniformly injective (in particular, left invertible) or surjective (in particular, right invertible). We show that if these operators are considered on the space of periodic functions, then the unilateral invertibility does not imply the invertibility of such operators. We obtain criteria for the injectivity, surjectivity, and invertibility of differential operators on the space of periodic functions.     

Invertible sequences of bounded linear operators

In this paper, we study the invertibility of sequences consisting of finitely many bounded linear operators from a Hilbert space to others. We show that a sequence of operators is left invertible if and only if it is a g-frame. Therefore, our result connects the invertibility of operator sequences with frame theory.     

Generalized projection operators in Geometric Algebra

Given an automorphism and an anti-automorphism of a semigroup of a Geometric Algebra, then for each element of the semigroup a (generalized) projection operator exists that is defined on the entire Geometric Algebra. A single fundamental theorem holds for all (generalized) projection operators. This theorem makes previous projection operator formulas [2] equivalent to each other. The class of generalized projection operators includes the familiar subspace projection operation by implementing the automorphism ‘grade involution’ and the anti-automorphism ‘inverse’ on the semigroup of invertible versors. This class of projection operators is studied in some detail as the natural generalization of the subspace projection operators. Other generalized projection operators include projections ontoany invertible element, or a weighted projection ontoany element. This last projection operator even implies a possible projection operator for the zero element.     

无穷维Hilbert空间上框架的算子与范数

在框架理论研究中,哪类可逆算子能使得某些框架性质保持不变这个问题是基本和重要的,本文在无穷维Hilbert空间上对下述两个问题进行研究.问题1:哪类可逆算子能使得框架算子保持不变;问题2:哪类可逆算子能使得框架范数只相差一列常数.本文从抽象的算子理论和具体的构造方法两方面对问题1给出解答.利用框架的相容算子的概念,当把问题2中的可逆算子集换成一类较小的算子集时,得到了问题2的回答.     

Spectral Shorted Operators

If $$\mathcal{H}$$ is a Hilbert space, $$\mathcal{S}$$ is a closed subspace of $$\mathcal{H},$$ and A is a positive bounded linear operator on $$\mathcal{H},$$ the spectral shorted operator $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ is defined as the infimum of the sequence $$\sum (\mathcal{S},A^n )^{1/n} ,$$ where denotes $$\sum \left( {\mathcal{S},B} \right)$$ the shorted operator of B to $$\mathcal{S}.$$ We characterize the left spectral resolution of $$\rho \left( {\mathcal{S},\mathcal{A}} \right)$$ and show several properties of this operator, particularly in the case that dim $${\mathcal{S} = 1.}$$ We use these results to generalize the concept of Kolmogorov complexity for the infinite dimensional case and for non invertible operators.     

一致可逆性质及广义(ω)性质的摄动

Hilbert空间算子T∈B(H)称为是一致可逆的,若对任意的S∈B(H),TS与ST的可逆性相同.本文中根据一致可逆性质定义了一个新的谱集,用该谱集来研究广义(ω)性质的稳定性,即考虑了Hilbert空间上有界线性算子的有限秩摄动、幂零摄动以及Riesz摄动的广义(ω)性质.之后研究了能分解成有限个正规算子乘积的一类算子的广义(ω)性质的稳定性.     

On factoring an operator using elements of its kernel

A well-known theorem factors a scalar coefficient differential operator given a linearly independent set of functions in its kernel. The goal of this paper is to generalize this useful result to other types of operators. In place of the derivation ? acting on some ring of functions, this paper considers the more general situation of an endomorphism 𝔇 acting on a unital associative algebra. The operators considered, analogous to differential operators, are those which can be written as a finite sum of powers of 𝔇 followed by left multiplication by elements of the algebra. Assume that the set of such operators is closed under multiplication and that a Wronski-like matrix produced from some finite list of elements of the algebra is invertible (analogous to the linear independence condition). Then, it is shown that the set of operators whose kernels contain all of those elements is the left ideal generated by an explicitly given operator. In other words, an operator has those elements in its kernel if and only if it has that generator as a right factor. Three examples demonstrate the application of this result in different contexts, including one in which 𝔇 is an automorphism of finite order.     

On invertible contractions of quotients generated by a differential expression and by a nonnegative operator function

In the present paper, we describe invertible contractions of the maximal quotient generated by a differential expression with bounded operator coefficients and by a nonnegative operator function. We show that the operators inverse to such contractions are integral operators and prove a criterion for such operators to be holomorphic. Using the results obtained, we describe the generalized resolvents of symmetric quotients.     

ON THE PUTNAM-FUGLEDE THEOREM OF NON-NORMAL OPERATORS

In this paper we have extended the Putnam-Fuglede Theorem of nomal operators anddiscussed the condition for the Putnam-Fuglede Theorem holding.We have proved that ifA and B~* are hyponomal operators and AX=XB,then A~*X=XB~*;that if A and B~* aresemi-hyponomal operators and X is     

Operators on Differential Form Spaces for Riemann Surfaces

In the present paper, a problem of Ioana Mihaila is negatively answered on the invertibility of composition operators on Riemann surfaces, and it is proved that the composition operator Cp is Predholm if and only if it is invertible if and only if p is invertible for some special cases. In addition, the Toeplitz operators on ∧1 2, a（M） for Riemann surface M are defined and some properties of these operators are discussed.     

可亚正规化的算子

最近,Ky Fan在[1]中就有限维希尔伯特空间引进了所谓可正规化算子的概念,考察了这类算子的一系列特征,其实在[1]中给出的可正规化算子的概念推广到无限维的Hilbert空间时,所列出的绝大部分特征也是成立的,希氏空间H上的线性有界算子T称为可正规化的,是指存在H上的可逆算子S,使S~*TS是正规算子。     

Perturbing Fredholm operators to obtain invertible operators

A condition is given on a set $\text{Ol}$ of operators on Hilbert space that guarantees it has the following property: For any Fredholm operator T of index zero there exists anA?$\text{A}$ such that T + ?A is invertible for all sufficiently small nonzero ?. As a corollary one obtains in a quite general setting the density of the invertible Toeplitz operators in the set of Fredholm Toeplitz operators of index zero.     

Generalized resolvents of linear relations generated by a nonnegative operator function and a differential elliptic-type expression

We study invertible extensions of the minimal relation generated by a nonnegative operator function and a differential elliptic-type expression. We prove that the operators inverse to such extensions are integral operators and describe such integral operators. We obtain a formula for generalized resolvents of the minimal relation.     

Wiener-Hopf operators on a subsemigroup of a discrete torsion free abelian group

In the paper Wiener-Hopf operators on a semigroup of nonnegative elements of a linearly quasi-ordered torsion free Abelian group are considered. Wiener-Hopf factorization of an invertible element of the group algebra is constructed, notions of a topological index and a factor index are introduced. It turns out that the set of factor indices for invertible elements of the group algebra is a linearly ordered group. It is shown that Wiener-Hopf operator with an invertible symbol is an one-side invertible operator and its invertibility properties are defined by the sign of the factor index of its symbol. Groups on which there exist nontrivial Fredholm Wiener-Hopf operators are described. As an example, all linear quasi-orders on the group ⁿ are found and corresponding Wiener-Hopf operators are considered.     

On the Spectrum of Invertible Semi-hyponormal Operators

It is known that for a semi-hyponormal operator, the spectrum of the operator is equal to the union of the spectra of the general polar symbols of the operator. The original proof of this theorem involves the so-called singular integral model. The purpose of this paper is to give a different proof of the same theorem for the case of invertible semi-hyponormal operators without using the singular integral model.      

Invertibility of 2 × 2 operator matrices

Properties of right invertible row operators, i.e., of 1 × 2 surjective operator matrices are studied. This investigation is based on a specific space decomposition. Using this decomposition, we characterize the invertibility of a 2 × 2 operator matrix. As an application, the invertibility of Hamiltonian operator matrices is investigated.