## Abstract

Let A= (A_{1}, … , A_{m}) , where A_{1}, … , A_{m} are n× n real matrices. The real joint (p, q)-matricial range of A, Λp,qR(A), is the set of m-tuple of q× q real matrices (B_{1}, … , B_{m}) such that (X^{∗}A_{1}X, … , X^{∗}A_{m}X) = (I_{p}⊗ B_{1}, … , I_{p}⊗ B_{m}) for some real n× pq matrix X satisfying X^{∗}X= I_{pq}. It is shown that if n is sufficiently large, then the set Λp,qR(A) is non-empty and star-shaped. The result is extended to bounded linear operators acting on a real Hilbert space H, and used to show that the joint essential (p, q)-matricial range of A is always compact, convex, and non-empty. Similar results for the joint congruence matricial ranges on complex operators are also obtained.

Original language | English |
---|---|

Pages (from-to) | 609-626 |

Number of pages | 18 |

Journal | Advances in Operator Theory |

Volume | 5 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1 Jul 2020 |

## Keywords

- Compact perturbation
- Congruence numerical range
- Convex
- Star-shaped

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory