On properties of real selfadjoint operators

Abstract

In spite of the important applications of real selfadjoint operators and monotone operators, very few papers have dealt in depth with the properties of such operators. In the present paper, we follow A. Rhodius to define the spectrum $\sigma _{\mathbb {F}}(T)$ and the numerical range $W_{\mathbb {F}}(T)$ of a selfadjoint operator T acting on a Hilbert space H over the real/complex field $\mathbb {F}$, and study their topological and geometrical properties which are well known in the complex case ${\mathbb {F}}={\mathbb {C}}$. If $\mathbb F={\mathbb {R}}$, the results are new; if ${\mathbb {F}}={\mathbb {C}}$, the results constitute an expository body of results containing simple and short proofs of the known facts. The results are then applied to real selfadjoint operators and then to complex normal operators to sharpen their Borel functional calculi with new and shorter proofs avoiding the classical sophisticated Gelfand–Naimark theorem or the Berberian’s amalgamation theory. For such a real selfadjoint or complex normal operator N, a normed functional algebra $L^\infty _{\mathbb {F}}(N)$ consisting of certain Borel functions defined on $\sigma _{\mathbb {F}}(N)$ is constructed which inherits the isometric properties of the continuous functional calculus $f\mapsto f(N):C_{\mathbb {F}}(\sigma _{\mathbb {F}}(N))\rightarrow B(H)$.