Infinite dimensional Jordan operators and Sturm-Liouville conjugate point theory

Abstract:

This article concerns two simple types of bounded operators with real spectrum on a Hilbert space $H$. The purpose of this note is to suggest an abstract algebraic characterization for these operators and to point out a rather unexpected connection between such algebraic considerations and the classical theory of ordinary differential equations. Now some definitions. A Jordan operator has the form $S + N$ where $S$ is selfadjoint, ${N^2} = 0$, and $S$ commutes with $N$. A sub-Jordan operator is the restriction of a Jordan operator $J$ to an invariant subspace of $J$. A coadjoint operator $T$ satisfies ${e^{ - is{T^ \ast }}}{e^{isT}} = I + {A_1}s + {A_2}{s^2}$ for some operators ${A_1}$ and ${A_2}$ or equivalently ${T^{ \ast 3}} - 3{T^{ \ast 2}}T + 3{T^ \ast }{T^2} - {T^3} = 0$. The main results are Theorem A. An operator $T$ is Jordan if and only if both $T$ and ${T^ \ast }$ are coadjoint. Theorem B. If $T$ is coadjoint, if $T$ has a cyclic vector, and if $\sigma (T) = [a,b]$, then $T$ is unitarily equivalent to ’ multiplication by $x$’ on a weighted Sobolev space of order 1 which is supported on $[a,b]$. Theorem C. If $T$ is coadjoint and satisfies additional technical assumptions, then $T$ is a sub-Jordan operator. Let us discuss Theorem C. Its converse, every sub-Jordan operator is coadjoint, is easy to prove. The proof of Theorem C consists of using Theorem B to reduce Theorem C to a question about ordinary differential equations which can be solved by an exacting application of the Jacobi conjugate point theorem for Sturm-Liouville operators. The author suspects that Theorem C is itself related to the conjugate point theorem.