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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Infinite dimensional Jordan operators and Sturm-Liouville conjugate point theory
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by J. William Helton PDF
Trans. Amer. Math. Soc. 170 (1972), 305-331 Request permission

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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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 170 (1972), 305-331
  • MSC: Primary 47A65; Secondary 34B25, 47B40, 47E05
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0308829-2
  • MathSciNet review: 0308829