# Kernel maps and operator decomposition

## TL;DR

This paper introduces kernel maps and sets for operators on Hilbert spaces, showing that certain algebraic structures are decomposable, with limitations when the nest is not continuous.

## Contribution

It defines kernel maps and sets for operators relative to subspace lattices and proves decomposability of norm closed Lie modules in continuous nest algebras.

## Key findings

- Kernel maps and sets characterize finite rank operators.
- Every norm closed Lie module of a continuous nest algebra is decomposable.
- Continuity of the nest is essential for the results.

## Abstract

We introduce the notions of kernel map and kernel set of a bounded linear operator on a Hilbert space relative to a subspace lattice. The characterization of the kernel maps and kernel sets of finite rank operators leads to showing that every norm closed Lie module of a continuous nest algebra is decomposable. The continuity of the nest cannot be lifted, in general.