2-nest algebras

Martim da Costa Graça Marques Ferreira

Key information

Authors:

Martim da Costa Graça Marques Ferreira (Martim Ferreira)

Supervisors:

Lina Maria Mateus de Oliveira (Lina Maria Mateus de Oliveira)

Published in

July 14, 2025

Abstract

We introduce a new class of operator algebras, named 2-nest algebras, determined by totally ordered families of partial isometries on a complex Hilbert space H. These algebras generalize classical nest algebras while also constituting left ideals of certain nest algebras. It is shown that any 2-nest algebra is decomposable and that the subset of finite-rank operators in its closed unit ball is strongly dense in the ball. Additionally, the commutant and diagonal of a 2-nest algebra are characterized, and a generalized version of the nest algebra Similarity Theorem is presented. Multiple interpolation problems are studied, notably the operator interpolation problem T_1X_1 + ... + T_nX_n = Y, with fixed X_1, ..., X_n, Y in B(H), and necessary and sufficient conditions to solve them in a 2-nest algebra are established. Furthermore, a generalized version of Ringrose's diagonal ideals is presented, and the Jacobson radical of a 2-nest algebra is characterized. Finally, the rank preserving maps on a mildly restricted subclass of 2-nest algebras are characterized and a relation is established between product preservation and rank-1 preservation, allowing the automorphisms of that same subclass of 2-nest algebras to be characterized.