Nijenhuis operators on Banach homogeneous spaces
Tomasz Goli\'nski, Gabriel Larotonda, Alice Barbora Tumpach

TL;DR
This paper characterizes when bounded operators on Lie algebras induce homogeneous vector bundle maps on Banach homogeneous spaces, extending previous results and linking Nijenhuis torsion to integrability of almost complex structures.
Contribution
It establishes general conditions for bounded operators to define homogeneous vector bundle maps on Banach homogeneous spaces, including cases with non-complemented subalgebras, and relates Nijenhuis torsion to integrability.
Findings
Conditions for bounded operators to define bundle maps
Equivalence of Nijenhuis torsion vanishing and values in Lie(K)
Characterization of integrability of almost complex structures
Abstract
For a Banach--Lie group and an embedded Lie subgroup we consider the homogeneous Banach manifold . In this context we establish the most general conditions for a bounded operator acting on to define a homogeneous vector bundle map . In particular our considerations extend all previous settings on the matter and are well-suited for the case where is not complemented in . We show that the vanishing of the Nijenhuis torsion for a homogeneous vector bundle map (defined by an admissible bounded operator on ) is equivalent to the Nijenhuis torsion of having values in . As an application, we consider the question of integrability of an almost complex structure on induced by an admissible bounded operator , and…
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Taxonomy
TopicsAdvanced Banach Space Theory · advanced mathematical theories · Holomorphic and Operator Theory
