Infinite dimensional Jordan operators and SturmLiouville conjugate point theory
Author:
J. William Helton
Journal:
Trans. Amer. Math. Soc. 170 (1972), 305331
MSC:
Primary 47A65; Secondary 34B25, 47B40, 47E05
DOI:
https://doi.org/10.1090/S00029947197203088292
MathSciNet review:
0308829
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Abstract  References  Similar Articles  Additional Information
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 subJordan 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 subJordan operator. Let us discuss Theorem C. Its converse, every subJordan 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 SturmLiouville operators. The author suspects that Theorem C is itself related to the conjugate point theorem.

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Additional Information
Keywords:
Multipliers on a Sobolev space,
nonselfadjoint spectral representations,
generalized spectral operators,
Jordan operators on a Hilbert space,
SturmLiouville conjugate point theory
Article copyright:
© Copyright 1972
American Mathematical Society