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On certain positivity classes of operatorsPublished in Taylor and Francis Inc.

2016

Volume: 37

Issue: 2

Pages: 206 - 224

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. © 2016 Taylor & Francis Group, LLC.

Topics: Operator theory (61)%61% related to the paper, Spectral theorem (59)%59% related to the paper, Invertible matrix (58)%58% related to the paper, Square matrix (58)%58% related to the paper and Operator norm (57)%57% related to the paper

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About the journal

Journal | Data powered by TypesetNumerical Functional Analysis and Optimization |
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Publisher | Data powered by TypesetTaylor and Francis Inc. |

ISSN | 01630563 |

Open Access | No |

Concepts (11)

- Banach spaces
- Vector spaces
- Infinite dimensional
- INVERTIBILITY
- MATRIX CASE
- PRINCIPAL MINOR
- PRINCIPAL PIVOT TRANSFORMS
- Schur complement
- SPACE OPERATORS
- Square matrices
- Matrix algebra