Extended Aluthge Transforms and Applications (Research on structure of operators by order and related topics)

This is a joint work with Raul E. CurtoGiven a bounded linear operator T with canonical polar decomposition T = V|T|, the Aluthge transform of T is the operator Δ(T) := √|T|V √|T|. For P an arbitrary positive operator such that VP = T, we define the extended Aluthge transform of T associated with P by Δp(T) := √PV √P. First, we establish some basic properties of Δp; second, we study the fixed points of the extended Aluthge transform; third, we consider the case when T is an idempotent; next, we discuss whether Δp leaves invariant the class of complex symmetric operators. We also study how Δp transforms the numerical radius and numerical range. As a key application, we prove that the spherical Aluthge transform of a commuting pair of operators corresponds to the extended Aluthge transform of a 2 x 2 operator matrix built from the pair; thus, the theory of extended Aluthge transforms yields results for spherical Aluthge transforms

Tracheobronchial calcifications in a child

Three months old child was admitted in the department of pediatrics I for respiratory distress, fever and laryngeal stridor. He was born to consanguineous parents. The prenatal history was unremarkable. However, he was hospitalized at birth for respiratory distress which improved quickly with oxygen nasal cannula. Since one month old, he was presenting a laryngeal stridor, persistent cough and chest congestion. Clinical examination found a dyspnea, stridor occuring in both phases of respiration, signs of retractions, a dysmorphic face with mid-facial hypoplasia and brachytelephalengia. The chest x-ray showed calcifications involving the entire tracheobronchial tree. A chest computed tomography revealed bilateral and symmetrical calcifications involving the anterior cartilaginous part of the trachea and the stem bronchi. The transthoracic ultrasound was normal. Routine physicochemical examinations found a low prothrombin time with decrease in the levels of vitamin-k dependent coagulation factors. The clinical course was favorable with oxygen nasal cannula and respiratory physiotherapy. The diagnosis of Keutel syndrome was made on calcifications of the tracheal cartilage associated to brachytelephalengia, and also facial dysmorphism

Joint spectra of spherical Aluthge transforms of commuting n-tuples of Hilbert space operators

Let be a commuting n-tuple of operators on a Hilbert space , and let be its canonical joint polar decomposition (i.e. ⁎ ⁎ , a joint partial isometry, and ). The spherical Aluthge transform of T is the (necessarily commuting) n-tuple . We prove that , where denotes the Taylor spectrum. We do this in two stages: away from the origin, we use tools and techniques from criss-cross commutativity; at the origin, we show that the left invertibility of T or implies the invertibility of P. As a consequence, we can readily extend our main result to other spectral systems that rely on the Koszul complex for their definitions