HARNACK PARTS FOR 4-BY-4 TRUNCATED SHIFT WITH NUMERICAL RADIUS ONE

Gilles Cassier (1) , Mehdi Naimi (2) , Mohammed Benharrat (3)

1 EDPA - Équations aux dérivées partielles, analyse
2 Université Abdelhamid Ibn Badis de Mostaganem
3 ENP - École nationale polytechnique [Alger, Algérie]

Let S be a n-by-n truncated shift whose numerical radius equal one. First, Cassier, Benharrat and Belmouhoub in [12] proved that the Harnack part of S is trivial if n = 2, while, if n = 3, is an orbit associated with the action of a group of unitary diagonal matrices , see [12, Theorem 3.1 and Theorem 3.3]. Second, Cassier and Benharrat in [7] described elements of the Harnack part of the truncated n-by-n shift S under an extra assumption. In Section 2, we present useful results in the general finite dimensional situation. In Section 3, we give a complete description of the Harnack part of S for n = 4, the answer is surprising and instructive. It shows that, even when the dimension is an odd number, the Harnack part is bigger than conjectured in [7, Question 2.]. We also give a negative answer to [7, Question 1.] when ρ = 2.

Mots clés: Truncated shift, ρ-contractions, Harnack parts, Operator kernel, Numerical range, Numerical radius

Domaines: Mathématiques [math]

Gilles Cassier, Mehdi Naimi, Mohammed Benharrat. HARNACK PARTS FOR 4-BY-4 TRUNCATED SHIFT WITH NUMERICAL RADIUS ONE. 2023. ⟨hal-04168163v1⟩