On the Hu-Hurley-Tam conjecture concerning the generalized numerical range

✍ Scribed by Che-Man Cheng; Chi-Kwong Li

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 102 KB
• Volume: 305
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(99)00228-1

✦ Synopsis

Suppose m and n are integers such that 1 m n, and H is a subgroup of the symmetric group S m of degree m. Define the generalized matrix function associated with the principal character of the group H on an m × m matrix B = (b ij ) by

b jσ (j) , and define the generalized numerical range of an n × n matrix A associated with d H by

Hu, Hurley and Tam made the following conjecture:

Suppose H = S m , 2 m n with (m, n) / = (2, 2). Let A ∈ M n be a normal matrix. Then W H (A) is convex if and only if A is a multiple of a Hermitian matrix. In this note, counterexamples are given to show that the conjecture is not true when m < n. Some techniques are developed to show that the conjecture is valid under more restrictive assumptions.