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In the setting of adjointable operators on Hilbert $ C^* $ C ∗ -modules, this paper deals with the polar decomposition of the product of three operators. The relationship between the polar decompositions associated with three operators is clarified. Based on this relationship, a formula for the polar decomposition of a multiplicative perturbation of an operator is provided. In addition, a full characterization is made on the polar decomposition of the product of three operators. Some applications of this characterization are also dealt with. [ABSTRACT FROM AUTHOR]

Let P and Q be projections on a Hilbert C * -module. This paper deals with the orthogonal complementarity of certain range closures associated with P and Q. Some necessary and sufficient conditions are derived under which (P, Q) is harmonious. [ABSTRACT FROM AUTHOR]

We introduce the notion of a tractable pair of operators as well as that of the generalized parallel sum in the setting of adjointable operators on Hilbert C ∗ -modules. Some significant results about the parallel sum known for matrices and Hilbert space operators are extended to the case of the generalized parallel sum. In particular, a factorization theorem on the parallel sum is proved, and a common upper bound of two positive operators is constructed in the Hilbert C ∗ -module case. The harmonic mean for positive operators on Hilbert C ∗ -modules is also dealt with. [ABSTRACT FROM AUTHOR]

A multiplicative perturbation M of a matrix T has the form M = E T F ∗ , where E and F are square matrices. It is proved that every acute perturbation is essentially a strong perturbation, which is a type of multiplicative perturbation. It is also proved that for every multiplicative perturbation M , M is a strong perturbation if and only if it is a weak perturbation and is rank-preserving. Some norm equations for the Moore-Penrose inverse are derived in the framework of the weak perturbation, through which some norm upper bounds for M † − T † are obtained. As an application, the perturbation estimation for the solution to the least squares problems is provided. The sharpness of the newly obtained upper bounds are illustrated by several numerical examples. [ABSTRACT FROM AUTHOR]

Let be the set of all complex matrices. For any Hermitian positive semi-definite matrices A and B in , their new common upper bound less than is constructed, where denotes the Moore–Penrose inverse of , and is the parallel sum of A and B. A factorization formula for is derived, where are any Hermitian positive semi-definite perturbations of A and B, respectively. Based on the derived factorization formula and the constructed common upper bound of X and Y, some new and sharp norm upper bounds of are provided. Numerical examples are also provided to illustrate the sharpness of the obtained norm upper bounds. [ABSTRACT FROM AUTHOR]

It is proved that for adjointable operators A and B between Hilbert -modules, certain majorization conditions are always equivalent without any assumptions on , where denotes the adjoint operator of A and is the norm closure of the range of . In the case that is not orthogonally complemented, it is proved that there always exists an adjointable operator B whose range is contained in that of A, whereas the associated equation for adjointable operators is unsolvable. [ABSTRACT FROM AUTHOR]

Let be the set of bounded linear operators on a Hilbert space H, and E be an anti-triangular operator matrix defined by where and I is the identity operator on H. Two associated operators are introduced, and it is proved that if and only if is invertible for any . A new kind of anti-triangular operator matrices is constructed such that the Drazin index of each member is exactly less than or equal to 2. Thus a generalization of He et al. [General exact solutions of certain second-order homogeneous algebraic differential equations. Linear Multilinear Algebra. 64;2016:1011-1031] is obtained. [ABSTRACT FROM AUTHOR]

A multiplicative perturbation of a matrixhas the formwithand, which can be expressed alternatively as, whereis the Moore–Penrose inverse ofT,andare introduced as In view of the aboveand, a new type of multiplicative perturbation called weak perturbation is introduced. A formula foris derived in the general case thatis a weak perturbation ofT. Based on this formula, an upper bound foris derived. The sharpness of the obtained upper bound is illustrated by some numerical examples. [ABSTRACT FROM AUTHOR]

The general exact solutions of the second-order homogeneous algebraic differential equation(Section.Display) is studied, whereandare three known bounded linear operators on a Banach space. Ifsuch thatis invertible, an anti-triangular operator matrixis defined. Whenis Drazin invertible, a necessary and sufficient condition is obtained in terms ofand its Drazin inverseunder whichbecomes a solution of (0.1). Specifically, a formula foris derived under the assumptions thatis a Hilbert space,andare positive semi-definite operators onwith the summationbeing positive definite. [ABSTRACT FROM PUBLISHER]

The second-order homogeneous algebraic differential equationis studied, whereare known such thatis non-singular for some. A formula for the general exact solutions of the above equation is provided under the condition that there exists, such that. For a special second-order homogeneous algebraic differential equation coming from a physics model investigated by Bhat and Bernstein, a detailed description is given about how to construct a particular solutionof the latter quadratic matrix equation. [ABSTRACT FROM AUTHOR]