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Journal ArticleDOI

On Certain Positivity Classes of Operators

TL;DR: 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...
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Journal ArticleDOI
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.

3 citations

Book ChapterDOI
01 Jan 2016
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.
Abstract: The intention here is to present an overview of some very recent results on three classes of operators, extending the corresponding matrix results. The relevant notions that are generalized here are that of a P-matrix, a Q-matrix, and an M-matrix. It is widely known (in the matrix case) that these notions coincide for Z-matrices. While we are not able to prove such a relationship between these classes of operators over Hilbert spaces, nevertheless, we are able to establish a relationship between Q-operators and M-operators, extending an analogous matrix result. It should be pointed out that, in any case, for P-operators, some interesting generalizations of results for P-matrices vis-a-vis invertibility of certain intervals of matrices have been obtained. These were proved by Rajesh Kannan and Sivakumar [92]. Since these are new, we include proofs for some of the important results. The last section considers a class of operators that are more general than M-operators. In particular, we review results relating to the nonnegativity of the Moore–Penrose inverse of Gram operators over Hilbert spaces, reporting the work of Kurmayya and Sivakumar [61] and Sivakumar [125]. These results find a place here is due to the reason that they extend the applicability of results for certain subclasses of M-matrices to infinite dimensional spaces.
Journal ArticleDOI
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.
Abstract: A quasi-Toeplitz $M$-matrix $A$ is an infinite $M$-matrix that can be written as the sum of a semi-infinite Toeplitz matrix and a correction matrix. This paper is concerned with computing the square root of invertible quasi-Toeplitz $M$-matrices which preserves the quasi-Toeplitz structure. We show that the Toeplitz part of the square root can be easily computed through evaluation/interpolation at the $m$ roots of unity. This advantage allows to propose algorithms solely for the computation of correction part, whence we propose a fixed-point iteration and a structure-preserving doubling algorithm. Additionally, we show that the correction part can be approximated by solving a nonlinear matrix equation with coefficients of finite size followed by extending the solution to infinity. Numerical experiments showing the efficiency of the proposed algorithms are performed.
05 May 2022
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 .
Abstract: A real square matrix A is called a P-matrix if all its principal minors are positive. Using the sign non-reversal property of matrices, the notion of P-matrix has been recently extended by Kannan and Sivakumar to infinite-dimensional Banach spaces relative to a given Schauder basis. Motivated by their work, we discuss P-operators on separable real Hilbert spaces. We also investigate P-operators relative to various orthonormal bases.
References
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Book
01 Aug 1979
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
Abstract: 1. Matrices which leave a cone invariant 2. Nonnegative matrices 3. Semigroups of nonnegative matrices 4. Symmetric nonnegative matrices 5. Generalized inverse- Positivity 6. M-matrices 7. Iterative methods for linear systems 8. Finite Markov Chains 9. Input-output analysis in economics 10. The Linear complementarity problem 11. Supplement 1979-1993 References Index.

6,572 citations


"On Certain Positivity Classes of Op..." refers background in this paper

  • ...For matrices, it is known that any matrix in the subset r(A,B) is invertible if and only if BA−1 is a P-matrix [8]....

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  • ...If s > ρ(B), thenA is invertible, and in this caseA−1 ≥ 0 [2]....

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  • ...HenceA is a P-matrix....

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  • ...This result is quite well known in the theory of linear complementarity problems [2]....

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  • ...It is well known that the linear complementarity problem de ned by a matrix A has a unique solution if and only if A is a P-matrix [4]....

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Book
08 Feb 2014
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.
Abstract: This chapter presents the most basic results on topological vector spaces. With the exception of the last section, the scalar field over which vector spaces are defined can be an arbitrary, non-discrete valuated field K; K is endowed with the uniformity derived from its absolute value. The purpose of this generality is to clearly identify those properties of the commonly used real and complex number field that are essential for these basic results. Section 1 discusses the description of vector space topologies in terms of neighborhood bases of 0, and the uniformity associated with such a topology. Section 2 gives some means for constructing new topological vector spaces from given ones. The standard tools used in working with spaces of finite dimension are collected in Section 3, which is followed by a brief discussion of affine subspaces and hyperplanes (Section 4). Section 5 studies the extremely important notion of boundedness. Metrizability is treated in Section 6. This notion, although not overly important for the general theory, deserves special attention for several reasons; among them are its connection with category, its role in applications in analysis, and its role in the history of the subject (cf. Banach [1]). Restricting K to subfields of the complex numbers, Section 7 discusses the transition from real to complex fields and vice versa.

4,183 citations

Book
18 Feb 1992
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.
Abstract: Introduction. Background. Existence and Multiplicity. Pivoting Methods. Iterative Methods. Geometry and Degree Theory. Sensitivity and Stability Analysis. Chapter Notes and References. Bibliography. Index.

2,897 citations

Journal ArticleDOI

823 citations


"On Certain Positivity Classes of Op..." refers background in this paper

  • ...Fiedler and Ptak [5] have shown that A is a P-matrix if and only if A does not reverse the sign of any non-zero vector....

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  • ...AmatrixA ∈ Rn×n is said to be a P-matrix [5] if all its principal minors are positive....

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  • ...A real square matrix A is called a P-matrix if all its principal minors are positive....

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  • ...P-operators As mentioned earlier, a square matrix A is called a P-matrix if all its principal minors are positive....

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  • ...More importantly, such a matrix is, in fact, a P-matrix....

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BookDOI
01 Jan 2011
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.
Abstract: ANHA Series Preface.- Preface.- General Notation.- Part I. A Primer on Functional Analysis .- Banach Spaces and Operator Theory.- Functional Analysis.- Part II. Bases and Frames.- Unconditional Convergence of Series in Banach and Hilbert Spaces.- Bases in Banach Spaces.- Biorthogonality, Minimality, and More About Bases.- Unconditional Bases in Banach Spaces.- Bessel Sequences and Bases in Hilbert Spaces.- Frames in Hilbert Spaces.- Part III. Bases and Frames in Applied Harmonic Analysis.- The Fourier Transform on the Real Line.- Sampling, Weighted Exponentials, and Translations.- Gabor Bases and Frames.- Wavelet Bases and Frames.- Part IV. Fourier Series.- Fourier Series.- Basic Properties of Fourier Series.- Part V. Appendices.- Lebesgue Measure and Integration.- Compact and Hilbert-Schmidt Operators.- Hints for Exercises.- Index of Symbols.- References.- Index.

345 citations