## Abstract

Let A=(A
_{1},…,A
_{m}) be an m-tuple of bounded linear operators acting on a Hilbert space H. Their joint (p,q)-matricial range Λ
_{p,q}(A) is the collection of (B
_{1},…,B
_{m})∈M
_{q}
^{m}, where I
_{p}⊗B
_{j} is a compression of A
_{j} on a pq-dimensional subspace. This definition covers various kinds of generalized numerical ranges for different values of p,q,m. In this paper, it is shown that Λ
_{p,q}(A) is star-shaped if the dimension of H is sufficiently large. If dimH is infinite, we extend the definition of Λ
_{p,q}(A) to Λ
_{∞,q}(A) consisting of (B
_{1},…,B
_{m})∈M
_{q}
^{m} such that I
_{∞}⊗B
_{j} is a compression of A
_{j} on a closed subspace of H, and consider the joint essential (p,q)-matricial range Λ
_{p,q}
^{ess}(A)=⋂{cl(Λ
_{p,q}(A
_{1}+F
_{1},…,A
_{m}+F
_{m})):F
_{1},…,F
_{m} are compact operators}. Both sets are shown to be convex, and the latter one is always non-empty and compact.

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

Pages (from-to) | 2497-2515 |

Number of pages | 19 |

Journal | Journal of Functional Analysis |

Volume | 275 |

Issue number | 9 |

DOIs | |

Publication status | Published - 1 Nov 2018 |

## Keywords

- Bounded linear operators
- Higher rank numerical range
- Joint essential numerical range
- Joint matricial range

## ASJC Scopus subject areas

- Analysis