Aluthge operator field and its numerical range and spectral properties

Gilles Cassier (1, 2) , Thomas Perrin (3)

1 ICJ - Institut Camille Jordan
2 EDPA - Équations aux dérivées partielles, analyse
3 ENS Paris Saclay - Ecole Normale Supérieure Paris-Saclay

For an arbitrary operator T acting on a Hilbert space we consider a field of operators (∆ z (T)) called the Aluthge operator field associated withT . After giving preliminary results, we establish that two fields (left and right), canonically linked to the Altuthge field (∆ z (T)) and a support subspace, are constant on each horizontal segment where they are defined. This result leads to a positive solution of a conjecture stated by Jung-Ko-Pearcy in 2000. Then we do a detailed spectral study of (∆ z (T)) and we give a Yamazaki type formula in this context.

Mots clés: Aluthge transform; numerical range; spectral properties.

Domaines: Analyse fonctionnelle [math.FA]

Gilles Cassier, Thomas Perrin. Aluthge operator field and its numerical range and spectral properties. Integral Equations and Operator Theory, 2021, 93 (4), Article 41 (20 p.). ⟨10.1007/s00020-021-02656-2⟩. ⟨hal-02995331⟩