On the algebra of smooth operators

TL;DR

This paper analyzes the algebra of smooth operators, providing spectral representations, characterizations of subalgebras, and establishing a functional calculus within the context of the space of rapidly decreasing sequences.

Contribution

It introduces a spectral representation for normal elements and characterizes closed commutative *-subalgebras in the algebra of smooth operators, utilizing the property (DN) of s.

Findings

Spectral representation of normal elements in L(s',s)

Characterization of closed commutative *-subalgebras

Establishment of a Hölder continuous functional calculus

Abstract

Let s be the space of rapidly decreasing sequences. We give the spectral representation of normal elements in the Fr\'echet algebra L(s',s) of the so-called smooth operators. We also characterize closed commutative *-subalgebras of L(s',s) and establish a H\"older continuous functional calculus in this algebra. The key tool is the property (DN) of s.