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.
We consider the Fréchet * -algebra L(s , s) ⊆ L( 2 ) of the so-called smooth operators, i.e. continuous linear operators from the dual s of the space s of rapidly decreasing sequences to s. This algebra is a non-commutative analogue of the algebra s. We characterize closed * -subalgebras of L(s , s) which are at the same time isomorphic to closed * -subalgebras of s and we provide an example of a closed commutative * -subalgebra of L(s , s) which cannot be embedded into s.
We describe the multiplier algebra of the noncommutative Schwartz space. This multiplier algebra can be seen as the largest * -algebra of unbounded operators on a separable Hilbert space with the classical Schwartz space of rapidly decreasing functions as the domain. We show in particular that it is neither a Q-algebra nor m-convex. On the other hand, we prove that classical tools of functional analysis, for example, the closed graph theorem, the open mapping theorem or the uniform boundedness principle, are still available.
For v ∈ R n let K be a compact set in R n containing a suitable smooth surface and such that the intersection {tv + x : t ∈ R} ∩ K is a closed interval or a single point for all x ∈ K. We prove that every linear first order differential operator with constant coefficients in direction v on space of Whitney functions E(K) admits a continuous linear right inverse.1 2010 Mathematics Subject Classification. Primary: 35E99, 35F05, 46E10. Secondary: 46A04.
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