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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Infinite dimensional Jordan operators and Sturm-Liouville conjugate point theory

Author: J. William Helton
Journal: Trans. Amer. Math. Soc. 170 (1972), 305-331
MSC: Primary 47A65; Secondary 34B25, 47B40, 47E05
MathSciNet review: 0308829
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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.

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