On the algebra of smooth operators

Abstract

Let s be the space of rapidly decreasing sequences. We give the spectral representation of normal elements in the Fréchet algebra L(s ′ , s) of the so-called smooth operators. We also characterize closed commutative * -subalgebras of L(s ′ , s) and establish a Hölder continuous functional calculus in this algebra. The key tool is the property (DN ) of s.1 2010 Mathematics Subject Classification. Primary: 46H35, 46J25, 46H30. Secondary: 46H15, 46K10, 46A11, 46L05.Key words and phrases: Topological algebras of operators, topological algebras with involution, representations of commutative topological algebras, functional calculus in topological algebras, nuclear Fréchet spaces, C *algebras, smooth operators, space of rapidly decreasing smooth functions.

Commutative subalgebras of the algebra of smooth operators

“…However, in some situations the supremum norms | · | ∞,q (as they are relatively easy to compute) or the 1 -norms will be more convenient. 2 , and the same is true of −1 , i.e. maps the idempotents of λ ∞ (A) onto the idempotents of λ ∞ (B).…”

“…Now, we shall prove that (e N k ) k∈N is a Schauder basis of E. Choose ι ∈ I such that κ ∈ I ι and for k ∈ I ι let n k be an arbitrary element of N k . Then k∈I ι η n k e N k = ξ e M ι ∈ E. Consequently, by [3,Lemma 4.1], e N κ ∈ E. Since κ was arbitrarily choosen, each e N k is in E and it is a simple matter to show that (e N k ) k∈N is a Schauder basis of E.…”

“…In Sect. 5 we describe all closed * -subalgebras of L(s , s) as suitable Köthe sequence algebras (see Corollary 5.4 and compare with [3,Th.4.8]). …”

“…The present paper is a continuation of [3,8] and it focuses on descriptions of closed commutative * -subalgebras of L(s , s) (especially those with the property ( )). Most of the results have been already presented in the author's PhD dissertation [2].…”

“…• it is (properly) contained in the intersection of all Schatten classes S p ( 2 ) over p > 0; in particular L(s , s) is contained in the class HS( 2 ) of Hilbert-Schmidt operators, and thus it is a unitary space; • the operator C * -norm || · || 2 → 2 is the so-called dominating norm on that algebra (the dominating norm property is a key notion in the structure theory of nuclear Fréchet spaces -see [3,Prop. 3.2] and [13,Prop.…”