On the algebra of smooth operators (Q2856675)

The author deals with one particular Fréchet LMC algebra with involution -- the so-called algebra of smooth operators, denoted by \(L(s',s)\). This is the algebra of operators acting from the space \(s'\) of slowly increasing sequences into the spaces \(s\) of rapidly decreasing sequences. This algebra has several natural representations, e.g., as the algebra of operators from the space of tempered distributions to the space of smooth functions -- this representation suggests the name smooth or smoothing. There is also another name in use, i.e., the non-commutative Schwartz space. This algebra is also a linear subspace of the \(C^*\)-algebra of compact operators on the separable Hilbert space \(\ell_2\).NEWLINENEWLINEThe paper is divided into five parts. After the introduction and preliminaries, the author investigates the spectral properties of this algebra. He proves in Theorem 3.1 that every normal operator in \(L(s',s)\) has a unique spectral representation as it is the case for normal operators on \(\ell_2\). Although the result is analogous to the well-known one, the proof requires purely Fréchet space techniques -- crucial is the so-called \((DN)\) condition of Vogt-Zakharyuta. Corollary 3.6 gives several equivalent conditions for a compact Hilbert space operator to belong to \(L(s',s)\). Section~4 characterizes closed commutative \(^*\)-subalgebras of \(L(s',s)\). Such a subalgebra always has a Schauder basis and is generated by any single element belonging to this subalgebra. The last section shows that there is also a functional calculus available in the algebra of smooth operators. The description is a little bit more complicated than for \(C^*\)-algebras; however, it is still possible to have roots of positive elements.