Journal ArticleDOI
On Certain Positivity Classes of Operators
M. Rajesh Kannan,K. C. Sivakumar +1 more
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In this paper, the notion of a P-operator is extended to infinite dimensional spaces and relations between invertibility of some subsets of intervals of operators and certain P-operators are established.Abstract:
A real square matrix A is called a P-matrix if all its principal minors are positive. Such a matrix can be characterized by the sign non-reversal property. Taking a cue from this, the notion of a P-operator is extended to infinite dimensional spaces as the first objective. Relationships between invertibility of some subsets of intervals of operators and certain P-operators are then established. These generalize the corresponding results in the matrix case. The inheritance of the property of a P-operator by the Schur complement and the principal pivot transform is also proved. If A is an invertible M-matrix, then there is a positive vector whose image under A is also positive. As the second goal, this and another result on intervals of M-matrices are generalized to operators over Banach spaces. Towards the third objective, the concept of a Q-operator is proposed, generalizing the well known Q-matrix property. An important result, which establishes connections between Q-operators and invertible M-op...read more
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Journal ArticleDOI
Theoretical and computational properties of semi-infinite quasi-Toeplitz M-matrices
TL;DR: In this paper , it was shown that under a mild and easy-to-check condition, an invertible quasi-Toeplitz matrix has a unique square root that is an M-matrix possessing quasi-toplitz structure.
Book ChapterDOI
M-Matrices over Infinite Dimensional Spaces
TL;DR: In this article, the authors present an overview of some very recent results on three classes of operators over Hilbert spaces, extending the corresponding matrix results, including a relationship between Q-operators and M-operator.
Journal ArticleDOI
Algorithms for square root of semi-infinite quasi-Toeplitz M-matrices
TL;DR: In this paper , the square root of invertible quasi-Toeplitz $M$-matrices is computed at the $m$ roots of unity, where the correction part can be approximated by solving a nonlinear matrix equation with coefficients of finite size followed by extending the solution to infinity.
P-Operators on Hilbert Spaces
A. A. Rashid,P. Sam Johnson +1 more
TL;DR: The notion of P-matrix has been recently extended by Kannan and Sivakumar to infinite-dimensional Banach spaces relative to a given Schauder basis as mentioned in this paper .
References
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Book
Nonnegative Matrices in the Mathematical Sciences
TL;DR: 1. Matrices which leave a cone invariant 2. Nonnegative matrices 3. Semigroups of non negative matrices 4. Symmetric nonnegativeMatrices 5. Generalized inverse- Positivity 6. M-matrices 7. Iterative methods for linear systems 8. Finite Markov Chains
Book
Topological Vector Spaces
TL;DR: In this paper, the authors present the most basic results on topological vector spaces, including the uniformity of vector spaces over non-discrete valuated fields, and the notion of boundedness of these fields.
Book
The Linear Complementarity Problem
TL;DR: In this article, the authors present an overview of existing and multiplicity of degree theory and propose pivoting methods and iterative methods for degree analysis, including sensitivity and stability analysis.
Journal ArticleDOI
On matrices with non-positive off-diagonal elements and positive principal minors
Miroslav Fiedler,Vlastimil Pták +1 more
BookDOI
A basis theory primer
TL;DR: In this paper, the Fourier Transform on the Real Line is used to transform the real line into a Fourier series, and Wavelet Bases and Frames are used in applied harmonic analysis.