Fapeng Du, Yifeng Xue
Let $K,\,H$ be Hilbert spaces and let $L(K,H)$ denote the set of all bounded linear operators from $K$ to $H$. Let $A \in L(H)\triangleq L(H,H)$ with $R(A)$ closed and $X,Y \in L(K,H)$ with $R(X)\subseteq R(A),R(Y)\subseteq R(A^*)$. In this short note, we give some new expressions of the Moore--Penrose inverse $(A-XY^*)^+$ of $A-XY^*$ under certain suitable conditions.
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The Moore-Penrose inverses of unbounded closable operators and the Cartesian product of closed operators in Hilbert spaces
该研究针对希尔伯特空间中的算子理论问题，给出了刻画无界可闭算子的Moore-Penrose逆与闭算子笛卡尔积的相关结果。
Berezin number and Berezin norm inequalities via Moore-Penrose inverse
该研究利用Moore-Penrose逆推导再生核Hilbert空间上有界线性算子的Berezin数与Berezin范数不等式，改进并推广了已有结果。
In this paper we characterize Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space using indefinite matrix multiplication. This characterization includes the acuteness (or obtuseness) of certain closed convex cones.
In this paper we derive necessary and sufficient conditions for the nonnegativity of Moore-Penrose inverses of unbounded Gram operators between real Hilbert spaces. These conditions include statements on acuteness of certain closed convex cones. The main result generalizes the existing result for bounded operators [11, Theorem 3.6].
The weighted Moore-Penrose inverse will be introduced and studied in the context of Banach algebras. In addition, weighted EP Banach algebra elements will be characterized. The Banach space operator case will be also considered.
Expressions and characterizations for the Moore-Penrose inverse
本文证明了算子和的 Moore-Penrose 逆在特定条件下等于各自逆之和，并推导了算子、有限矩阵、循环矩阵及特定图距离矩阵的 Moore-Penrose 逆表达式与刻画，为其计算提供了实用方法。
In this article we explore several aspects concerning to the Moore-Penrose inverse of a bounded linear operator. On the one hand, we study monotonicity properties of the Moore-Penrose inverse with respect to the Löwner, star, minus, sharp and diamond orders. On the other hand, we analyze the validity of the reverse order law, $B^\dagger A^\dagger=(AB)^\dagger$, under hypothesis of operator ranges and also under hypothesis of order operators. Finally, we study the operator $B^\dagger A^\dagger$ as different weighted inverses of $AB$.
The main concern of this note is the Moore-Penrose inverse in the context of Banach spaces and algebras. Especially attention will be given to a particular class of elements with the aforementioned inverse, namely EP Banach space operators and Banach algebra elements, which will be studied and characterized extending well-known results obtained in the frame of Hilbert space operators and $C^*$-algebra elements.
In this article properties of the $(b, c)$-inverse, the inverse along an element, the outer inverse with prescribed range and null space $A^{(2)}_{T, S}$ and the Moore-Penrose inverse will be studied in the contexts of Banach spaces operators, Banach algebras and $C^*$-algebras. The main properties to be considered are the continuity, the differentiability and the openness of the sets of all invertible elements defined by all the aforementioned outer inverses but the Moore-Penrose inverse. The relationship between the $(b, c)$-inverse and the outer inverse $A^{(2)}_{T, S}$ will be also charact…
Inner product bounds via the Moore-Penrose inverse with applications
该研究针对希尔伯特空间算子，利用Moore-Penrose逆推导柯西-施瓦茨型内积界，并将其推广至涉及算子不同组合的数值半径-范数界，提供了新的精细结果。
Estimation of $\mathrm{A}$-Berezin number and $\mathrm{A}$-Berezin norm inequalities via Moore-Penrose inverse
利用Moore-Penrose逆建立再生核希尔伯特空间上有界线性算子的A-Berezin数和范数不等式，并扩展到两个算子之和的情况，还讨论了块矩阵相关上界，完善并推广了先前结果。
Let $H_1,H_2$ be complex Hilbert spaces and $T:H_1\rightarrow H_2$ be a densely defined closed operator with domain $D(T)\subseteq H_1$ and $T^{\dagger}$ be the Moore-Penrose inverse of $T$. Let $S:H_1\rightarrow H_2$ be a bounded operator. In this article we focus our attention on the following questions: $1.$ Under what conditions closedness of range of $T$ will imply the closedness of range of $T+S$? $2.$ What is the relation between $T^{\dagger}$ and $(T+S)^{\dagger}$? $3.$ What is the relation between $T^{\dagger}$ and $S^{\dagger}$?.
Antitonicity property of the Moore-Penrose inverse for selfadjoint operators
本文研究自伴算子在Löwner序下Moore-Penrose逆的反单调性，利用正分解并关联弱版本与Thompson分量。
A Krein space H and bounded linear operators B, C on H are given. Then, some min and max problems about the operators (BX - C)^{#}(BX -C), where X runs over the space of all bounded linear operators on H, are discussed. In each case, a complete answer to the problem, including solvability conditions and characterization of the solutions, is presented. Also, an adequate decomposition of B is considered and the min-max problem is addressed. As a by-product the Moore-Penrose inverse of B is characterized as the only solution of a variational problem. Other generalized inverses are described in a …
Spectral Properties of Off Diagonal Block Linear Relations via Moore Penrose Inverses in Hilbert Spaces
本文研究了非对角块线性关系的谱性质，通过Moore-Penrose逆和AB、BA的谱特性建立精确的谱关系。
Every bounded linear operator on a Hilbert space which is invertible modulo compact operators has a closed range and is, thus, generalized invertible. We consider the analogue question in general $C^*$-algebras and describe the closed ideals (called Moore-Penrose ideals in what follows) with the property that whenever an element is invertible modulo that ideal, then it is generalized invertible. In particular, we will see that the class of Moore-Penrose ideals coincides with the class of the dual ideals. Finally, we study some questions related with the projection lifting property of Moore-Pen…
In the present paper we investigate Moore-Penrose inverse and characteristic matrix of unbounded WCT operators on the Hilbert space $L^2(μ)$. Also, we obtain some applications of the Moore-Penrose inverse of unbounded operators on the Hilbert space $\mathcal{H}$ to variational regularization problem. Moreover,some examples are provided to illustrate the applications of our results in linear equations and specially Fredholm integral equations.
$k$-quasi $n$-power posinormal Weighted Composition and Cauchy Dual of Moore-Penrose inverse of Lambert Operators
本文刻画了希尔伯特空间上的$k$-拟$n$幂正正规复合与加权复合算子，给出Lambert算子Cauchy对偶成为此类算子的充要条件，并构造了树上的显式例子。
Let $t$ be a regular operator between Hilbert $C^*$-modules and $t^†$ be its Moore-Penrose inverse. We investigate the Moore-Penrose invertibility of the Gram operator $t^*t$. More precisely, we study some conditions ensuring that $t^† = (t^* t)^† t^*= t^* (t t^*)^†$ and $(t^*t)^†=t^†t^{* †}$ hold. As an application, we get some results for densely defined closed operators on Hilbert $C^*$-modules over $C^*$-algebras of compact operators.
In this paper, we establish a link between the Hyers-Ulam stability and the Moore--Penrose inverse, that is, a closed operator has the Hyers-Ulam stability if and only if it has a bounded Moore-Penrose inverse. Meanwhile, the stability constant can be determined in terms of the Moore-Penrose inverse. Based on this result, some conditions for the perturbed operators having the Hyers--Ulam stability are obtained and the Hyers-Ulam stability constant is expressed explicitly in the case of closed operators. In the case of the bounded linear operators we obtain some characterizations for the Hyers-…
We propose a method and algorithm for computing the weighted Moore-Penrose inverse of one-variable rational matrices. Continuing this idea, we develop an algorithm for computing the weighted Moore-Penrose inverse of one-variable polynomial matrix. These methods and algorithms are generalizations of the method for computing the weighted Moore-Penrose inverse for constant matrices, originated in Wang and Chen [G.R. Wang, Y.L. Chen, A recursive algorithm for computing the weighted Moore-Penrose inverse AMN, J. Comput. Math. 4 (1986) 74-85], and the partitioning method for computing the Moore-Penr…
We introduce a method and an algorithm for computing the weighted Moore-Penrose inverse of multiple-variable polynomial matrix and the related algorithm which is appropriated for sparse polynomial matrices. These methods and algorithms are generalizations of algorithms developed in [M.B. Tasic, P.S. Stanimirovic, M.D. Petkovic, Symbolic computation of weighted Moore-Penrose inverse using partitioning method, Appl. Math. Comput. 189 (2007) 615-640] to multiple-variable rational and polynomial matrices and improvements of these algorithms on sparse matrices. Also, these methods are generalizatio…
Perturbation analysis for Moore-Penrose inverse of closed operators on Hilbert spaces
本文通过定义新的内积，给出了Hilbert空间上闭算子的Moore-Penrose逆的表达式及其范数和差的上界，推广和改进了该领域的许多相关结果。
We establish some relationships between an m-accretive operator and its Moore-Penorse inverse. We derive some perturbation result of the Moore-Penorse inverse of a maximal accretive operator. As an application we give a factorization theorem for a quadratic pencil of accretive operators. Also, we study a result of existence, uniqueness, and maximal regularity of the strict solution for complete abstract second order differential equation. Illustrative examples are also given.
Perturbations and expressions of the Moore--Penrose metric generalized inverses and applications to the stability of some operator equations
研究Banach空间上有界线性算子的Moore-Penrose度量广义逆在扰动下的表达式，并应用于算子方程的稳定性分析。
Numerical range for weighted Moore-Penrose inverse of tensor
本文引入张量加权奇异值分解，用于计算任意阶张量的加权Moore-Penrose逆，并研究偶数阶方阵张量该逆的数值域性质，推广了矩阵情形下的现有结果。
We present more than 50 results including some range inclusion results to characterize reverse order law for Moore-Penrose inverse of closed range Hilbert space operators. We use basic properties of Moore-Penrose inverse to prove the results. Some examples are also provided to illustrate failure cases to hold the reverse order law in infinite dimensional settings.
Geometric approach to the Moore-Penrose inverse and the polar decomposition of perturbations by operator ideals
本文用几何方法研究Hilbert空间闭值域算子经对称范数理想扰动后的Moore-Penrose逆连续性，证明其定义域构成实解析Banach流形且映射为实双解析，并将该框架推广到极分解，证明相关映射为实解析纤维丛。
A new formula for the weighted Moore-Penrose inverse and its applications
本文在 Hilbert $C^*$-模可伴随算子框架下给出加权 M-P 逆的新公式，并据此证明其可归约为正定权重情形、推广极限公式、刻画其代数归属与连续性。
About the generalized LM-inverse and the weighted Moore-Penrose inverse
本文提出了一种递归计算广义LM逆的算法，并证明该算法与Wang和Chen提出的加权Moore-Penrose逆算法等价，通过MATHEMATICA实现并比较了CPU时间。
Using a factorization theorem of Douglas, we prove functional characterizations of trace spaces $H^s(\partial Ω)$ involving a family of positive self-adjoint operators. Our method is based on the use of a suitable operator by taking the trace on the boundary $\partial Ω$ of a bounded Lipschitz domain $Ω\subset \mathbb R^d$ and applying Moore--Penrose pseudo-inverse properties together with a special inner product on $H^1(Ω)$. Moreover, generalized results of the Moore--Penrose pseudo-inverse are also established.
A generalization with singular weights of Moore-Penrose generalized inverses of closed range operators in Hilbert spaces is studied using the notion of compatibility of subspaces and positive operators.
An algorithm for computing {2, 3}, {2, 4}, {1, 2, 3}, {1, 2, 4} -inverses and the Moore-Penrose inverse of a given rational matrix A is established. Classes A(2, 3)s and A(2, 4)s are characterized in terms of matrix products (R*A)+R* and T*(AT*)+, where R and T are rational matrices with appropriate dimensions and corresponding rank. The proposed algorithm is based on these general representations and the Cholesky factorization of symmetric positive matrices. The algorithm is implemented in programming languages MATHEMATICA and DELPHI, and illustrated via examples. Numerical results of the alg…
Spectral Characterizations of Schatten-class perturbations of Partial isometries
研究刻画部分等距算子标量倍数的紧或Schatten类扰动的有界算子，通过\(T^*T\)本质谱等进行，给出相关充要条件及等价准则，还应用于等距算子扰动刻画、Moore - Penrose逆行为描述及闭值域算子分解。
We address the question as to when it is true that $(A+B)^{\dagger}=A^{\dagger}+B^{\dagger},$ where $\dagger$ denotes the Moore-Penrose inverse. A similar question is addressed for the group inverse.
Cauchy dual and Wold-type decomposition for bi-regular covariant representations
本文利用 Moore–Penrose 逆将 Cauchy 对偶推广至闭值域协变表示，建立正则完全有界情形的 Wold 型分解，并证明凹协变表示模特定空间的 Cauchy 对偶具有次正规性。
Proper splittings of Hermitian operators
本文深化Hilbert空间算子proper分裂研究，改进Hermitian算子极与Moore-Penrose proper分裂的收敛性刻画，并引入新分裂比较其收敛性。
Generalized weighted EP elements in Banach algebras
本文提出一类新的广义加权逆，作为Banach *-代数中EP和*-DMP元素的推广，研究了其特征、core-EP分解及类极化性质，并揭示了*-DMP元素的新性质。
In this paper, necessary and sufficient conditions are given for the existence of Moore-Penrose inverse of a product of two matrices in an indefinite inner product space (IIPS) in which reverse order law holds good. Rank equivalence formulas with respect to IIPS are provided and an open problem is given at the end.
Several characterizations of EP and normal Moore-Penrose invertible Banach algebra elements will be considered. The Banach space operator case will be also studied. The results of the present article will extend well known facts obtained in the frames of matrices and Hilbert space operators.
Let $n$ be any natural number. The $n$-centered operator is introduced for adjointable operators on Hilbert $C^*$-modules. Based on the characterizations of the polar decomposition for the product of two adjointable operators, $n$-centered operators, centered operators as well as binormal operators are clarified, and some results known for the Hilbert space operators are improved. It is proved that for an adjointable operator $T$, if $T$ is Moore-Penrose invertible and is $n$-centered, then its Moore-Penrose inverse is also $n$-centered. A Hilbert space operator $T$ is constructed such that $T…
Fuglede-Putnam theorem is not true in general for $ EP $ operators on Hilbert spaces. We prove that under some conditions the theorem holds good. If the adjoint operation is replaced by Moore-Penrose inverse in the theorem, we get Fuglede-Putnam type theorem for $ EP $ operators -- however proofs are totally different. Finally, interesting results on $ EP $ operators have been proved using several versions of Fuglede-Putnam type theorems for $ EP $ operators on Hilbert spaces.
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