ON THE INVERTIBILITY OF HILBERT SPACE IDEMPOTENTS

This note is to present some results on the group invertibility of linear combina- tions of idempotents when the difference of two idempotents is group invertible.     

Invertibility of linear combinations of two idempotents

Let and be two idempotents on a Hilbert space. In this note, we prove that the invertibility of the linear combination is independent of the choice of , if and

     

Strongly smooth paths of idempotents

It is shown that a curve q(t), t∈I (0∈I) of idempotent operators on a Banach space X, which verifies that for each ξ∈X, the map t?q(t)ξ∈X is continuously differentiable, can be lifted by means of a regular curve Gt, of invertible operators in X:

     

Nonsingularity and group invertibility of linear combinations of two k-potent matrices

An n × n complex matrix A is said to be k-potent if A ^k = A. Let T ₁ and T ₂ be k-potent and c ₁ and c ₂ be two nonzero complex numbers. We study the range space, null space, nonsingularity and group invertibility of linear combinations T = c ₁ T ₁ + c ₂ T ₂ of two k-potent matrices T ₁ and T ₂.     

Path connectivity of idempotents on a Hilbert space

Let and be two idempotents on a Hilbert space. In 2005, J. Giol in [Segments of bounded linear idempotents on a Hilbert space, J. Funct. Anal. 229(2005) 405-423] had established that, if is invertible, then and are homotopic with In this paper, we have given a necessary and sufficient condition that where denotes the minimal number of segments required to connect not only from to , but also from to in the set of idempotents.

     

Segments of bounded linear idempotents on a Hilbert space

Let H be a separable Hilbert space. We prove that any two homotopic idempotents in the algebra L(H) may be connected by a piecewise affine idempotent-valued path consisting of 4 segments at most. Moreover, we show that this constant is optimal provided H has infinite dimension. We also explain how this result is linked to the problem of finding common complements for two closed subspaces of H.     

On the invertibility of the operator A–XB

For a given pair of (A, B) and an arbitrary operator X, expressions for the inverse, the Moore–Penrose inverse and the generalized Drazin inverse of the operator A–XB are derived under some conditions. Copyright © 2009 John Wiley & Sons, Ltd.     

On the Moore-Penrose inverse, EP Banach space operators, and EP Banach algebra elements

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.     

Weighted weak group inverse for Hilbert space operators

We present the weighted weak group inverse, which is a new generalized inverse of operators between two Hilbert spaces, and we extend the notation of the weighted weak group inverse for rectangular matrices. Some characterizations and representations of the weighted weak group inverse are investigated. We also apply these results to define and study the weak group inverse for a Hilbert space operator. Using the weak group inverse, we define and characterize various binary relations.     

两个幂等算子线性组合的Drazin逆

利用空间分解的技巧,在条件PQP=QPQ下,得到两个幂等算子P和Q的多线性组合aP+bQ+cPQ+dQP+ePQP的Drazin逆的表达式.     

On the infimum problem of Hilbert space effects

The quantum effects for a physical system can be described by the set ε(H) of positive operators on a complex Hilbert space H that are bounded above by the identity operator I. The infimum problem of Hilbert space effects is to find under what condition the infimum A∧B exists for two quantum effects A and B∈ε(H). The problem has been studied in different contexts by R. Kadison, S. Gudder, M. Moreland, and T. Ando. In this note, using the method of the spectral theory of operators, we give a complete answer of the infimum problem. The characterizations of the existence of infimum A∧B for two effects A. B∈ε(H) are established.     

Integral representation of the W-weighted Drazin inverse for Hilbert space operators

This paper studies the integral representation of the W-weighted Drazin inverse for bounded linear operators between Hilbert spaces. By using operator matrix blocks, some integral representations of the W-weighted Drazin inverse for Hilbert space operators are established.     

左中心rpp半群的半直积

讨论了幂等元都是左中心元的rpp半群的半直积,给出了这种半群半直积的充要条件,推广了一些已知的结果.     

On the definition of Hilbert space

The projection theorem expresses a central feature of classical Hilbert space. Do other infinite dimensional sesquilinear spaces share this property? We show here that this is not the case for several prominent candidates; in particular Kalish's p-adic Hilbert spaces, Springers non archimedean normed spaces, the positive definite spaces over ordered fields. This yields interesting characterizations of classical Hilbert space.     

Matrices over a commutative ring as sums of three idempotents or three involutions

Motivated by Hirano-Tominaga’s work on rings for which every element is a sum of two idempotents and by de Seguins Pazzis’s results on decomposing every matrix over a field of positive characteristic as a sum of idempotent matrices, we address decomposing every matrix over a commutative ring as a sum of three idempotent matrices and, respectively, as a sum of three involutive matrices.