On Certain Positivity Classes of Operators |
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Authors: | M. Rajesh Kannan K. C. Sivakumar |
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Affiliation: | 1. Department of Mathematics, Technion, Israel Institute of Technology, Haifa, Israelrajeshkannan1.m@gmail.com;3. Department of Mathematics, Indian Institute of Technology Madras, Chennai, India |
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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-operators, is proved for Hilbert space operators. |
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Keywords: | Interval of operators M-operators P-operators Q-operators |
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