Abstract
We prove that every continuous linear operator acting on a stable, nuclear power series space is a commutator, so in particular our theorem holds for the space of all entire functions, holomorphic functions on the unit disc or smooth functions. We also show that on the product of Frechet spaces \(\prod _{i=1}^\infty X\) all operators are commutators.
Highlights
A commutator of a pair of elements A and B in the algebra L(X ) of linear, continuous operators on a locally convex space X is given by [A, B] := AB − B A.The problem of representing operators as commutators comes from quantum mechanics, where the so-called commutator relation plays an important role
Commutators are connected with derivations in algebra as they are the main examples of derivations
We will show that in many classical Fréchet spaces every operator is a commutator. More precisely this is true for every nuclear stable power series space, both of finite and infinite type
Summary
A commutator of a pair of elements A and B in the algebra L(X ) of linear, continuous operators on a locally convex space X is given by [A, B] := AB − B A. The problem of representing operators as commutators comes from quantum mechanics, where the so-called commutator relation plays an important role. Commutators are connected with derivations in algebra as they are the main examples of derivations (the so-called inner derivations). As shown by Wintner [11], on a Banach space not every operator is a commutator. All elements of the form λ1l + M, where λ = 0, M lies in a proper, closed ideal M of a Banach algebra A cannot be commutators
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