EP Operators and Generalized Inverses

1975 ◽  
Vol 18 (3) ◽  
pp. 327-333 ◽  
Author(s):  
Stephen L. Campbell ◽  
Carl D. Meyer
Keyword(s):  
Hilbert Space ◽  
Canonical Form ◽  
Generalized Inverse ◽  
Generalized Inverses ◽  
Space Setting ◽  
The Relationship

AbstractThe relationship between properties of the generalized inverse of A, A†, and of the adjoint of A, A*, are studied. The property that A†A and AA† commute, called (E4), is investigated. (E4) generalizes the property of A being EPr. A canonical form and a formula for A† are given if a matrix A is (E4). Results are in a Hilbert space setting whenever possible. Examples are given.

On the uniform boundedness and convergence of generalized, Moore-Penrose and group inverses

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 5993-6003 ◽  
Author(s):  
Lanping Zhu ◽  
Changpeng Zhu ◽  
Qianglian Huang
Keyword(s):  
Operator Theory ◽  
Generalized Inverse ◽  
Matrix Theory ◽  
Uniform Boundedness ◽  
Generalized Inverses ◽  
Convergence Theorems ◽  
Linear Subspaces ◽  
Group Inverses ◽  
Stable Perturbation ◽  
The Relationship

This paper concerns the relationship between uniform boundedness and convergence of various generalized inverses. Using the stable perturbation for generalized inverse and the gap between closed linear subspaces, we prove the equivalence of the uniform boundedness and convergence for generalized inverses. Based on this, we consider the cases for the Moore-Penrose inverses and group inverses. Some new and concise expressions and convergence theorems are provided. The obtained results extend and improve known ones in operator theory and matrix theory.

scholarly journals Local Spectral Properties Under Conjugations

2021 ◽  
Vol 18 (3) ◽  
Author(s):  
Pietro Aiena ◽  
Fabio Burderi ◽  
Salvatore Triolo
Keyword(s):  
Hilbert Space ◽  
Spectral Properties ◽  
Operator Matrices ◽  
Weyl Type Theorems ◽  
Triangular Operator ◽  
Analytic Core ◽  
Complex Symmetric ◽  
Symmetric Operators ◽  
The Relationship

AbstractIn this paper, we study some local spectral properties of operators having form JTJ, where J is a conjugation on a Hilbert space H and $$T\in L(H)$$ T ∈ L ( H ) . We also study the relationship between the quasi-nilpotent part of the adjoint $$T^*$$ T ∗ and the analytic core K(T) in the case of decomposable complex symmetric operators. In the last part we consider Weyl type theorems for triangular operator matrices for which one of the entries has form JTJ, or has form $$JT^*J$$ J T ∗ J . The theory is exemplified in some concrete cases.

scholarly journals TENSOR EXTENSION PROPERTIES OF C(K)-REPRESENTATIONS AND APPLICATIONS TO UNCONDITIONALITY

2010 ◽  
Vol 88 (2) ◽  
pp. 205-230 ◽  
Author(s):  
CHRISTOPH KRIEGLER ◽  
CHRISTIAN LE MERDY
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unconditional Basis ◽  
Compact Set ◽  
Bounded Operators ◽  
Space Setting ◽  
Uniformly Bounded

AbstractLet K be any compact set. The C*-algebra C(K) is nuclear and any bounded homomorphism from C(K) into B(H), the algebra of all bounded operators on some Hilbert space H, is automatically completely bounded. We prove extensions of these results to the Banach space setting, using the key concept ofR-boundedness. Then we apply these results to operators with a uniformly bounded H∞-calculus, as well as to unconditionality on Lp. We show that any unconditional basis on Lp ‘is’ an unconditional basis on L2 after an appropriate change of density.

Inverse complements and strongly unit regular elements

2021 ◽  
pp. 2250190
Author(s):  
Umashankara Kelathaya ◽  
Savitha Varkady ◽  
Manjunatha Prasad Karantha
Keyword(s):  
Partial Order ◽  
Regular Element ◽  
Generalized Inverse ◽  
Generalized Inverses ◽  
Order Unit ◽  
Regular Elements ◽  
Strongly Regular ◽  
Minus Partial Order

In this paper, the notion of “strongly unit regular element”, for which every reflexive generalized inverse is associated with an inverse complement, is introduced. Noting that every strongly unit regular element is unit regular, some characterizations of unit regular elements are obtained in terms of inverse complements and with the help of minus partial order. Unit generalized inverses of given unit regular element are characterized as sum of reflexive generalized inverses and the generators of its annihilators. Surprisingly, it has been observed that the class of strongly regular elements and unit regular elements are the same. Also, several classes of generalized inverses are characterized in terms of inverse complements.

scholarly journals Quasi-posinormal operators

2010 ◽  
Vol 7 (3) ◽  
pp. 1282-1287
Author(s):  
Baghdad Science Journal
Keyword(s):  
Hilbert Space ◽  
Normal Operator ◽  
Basic Properties ◽  
Hyponormal Operators ◽  
The Relationship

In this paper, we introduce a class of operators on a Hilbert space namely quasi-posinormal operators that contain properly the classes of normal operator, hyponormal operators, M–hyponormal operators, dominant operators and posinormal operators . We study some basic properties of these operators .Also we are looking at the relationship between invertibility operator and quasi-posinormal operator .

scholarly journals A Quantum Expectation Value Based Language Model with Application to Question Answering

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 533
Author(s):  
Qin Zhao ◽  
Chenguang Hou ◽  
Changjian Liu ◽  
Peng Zhang ◽  
Ruifeng Xu
Keyword(s):  
Hilbert Space ◽  
Density Matrix ◽  
Question Answering ◽  
Language Model ◽  
Language Models ◽  
Quantum Model ◽  
Expectation Value ◽  
Proposed Model ◽  
Matching Score ◽  
The Relationship

Quantum-inspired language models have been introduced to Information Retrieval due to their transparency and interpretability. While exciting progresses have been made, current studies mainly investigate the relationship between density matrices of difference sentence subspaces of a semantic Hilbert space. The Hilbert space as a whole which has a unique density matrix is lack of exploration. In this paper, we propose a novel Quantum Expectation Value based Language Model (QEV-LM). A unique shared density matrix is constructed for the Semantic Hilbert Space. Words and sentences are viewed as different observables in this quantum model. Under this background, a matching score describing the similarity between a question-answer pair is naturally explained as the quantum expectation value of a joint question-answer observable. In addition to the theoretical soundness, experiment results on the TREC-QA and WIKIQA datasets demonstrate the computational efficiency of our proposed model with excellent performance and low time consumption.

scholarly journals Primal Lower Nice Functions in Reflexive Smooth Banach Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2066
Author(s):  
Messaoud Bounkhel ◽  
Mostafa Bachar
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Space Setting ◽  
Nonlinear Anal ◽  
Finite Dimensional ◽  
In Trans ◽  
First Time ◽  
The Relationship ◽  
Primal Lower Nice Functions

In the present work, we extend, to the setting of reflexive smooth Banach spaces, the class of primal lower nice functions, which was proposed, for the first time, in finite dimensional spaces in [Nonlinear Anal. 1991, 17, 385–398] and enlarged to Hilbert spaces in [Trans. Am. Math. Soc. 1995, 347, 1269–1294]. Our principal target is to extend some existing characterisations of this class to our Banach space setting and to study the relationship between this concept and the generalised V-prox-regularity of the epigraphs in the sense proposed recently by the authors in [J. Math. Anal. Appl. 2019, 475, 699–29].

scholarly journals The Modal-Hamiltonian Interpretation of Quantum Mechanics as a Kind of “Atomic” Interpretation

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Juan Sebastián Ardenghi ◽  
Olimpia Lombardi
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum State ◽  
The Other ◽  
Interpretation Of Quantum Mechanics ◽  
Modal Interpretation ◽  
Atomic Systems ◽  
Modal Interpretations ◽  
The One ◽  
The Relationship

Modal interpretations are non-collapse interpretations, where the quantum state of a system describes its possible properties rather than the properties that it actually possesses. Among them, the atomic modal interpretation (AMI) assumes the existence of a special set of disjoint systems that fixes the preferred factorization of the Hilbert space. The aim of this paper is to analyze the relationship between the AMI and our recently presented modal-hamiltonian interpretation (MHI), by showing that the MHI can be viewed as a kind of “atomic” interpretation in two different senses. On the one hand, the MHI provides a precise criterion for the preferred factorization of the Hilbert space into factors representing elemental systems. On the other hand, the MHI identifies the atomic systems that represent elemental particles on the basis of the Galilei group. Finally, we will show that the MHI also introduces a decomposition of the Hilbert space of any elemental system, which determines with precision what observables acquire definite actual values.

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