On Certain Positivity Classes of Operators. (1st February 2016) - Kannan, M. Rajesh; Sivakumar, K. C.
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.