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

DOI:
https://doi.org/10.1090/S0002-9947-1972-0308829-2

MathSciNet review:
0308829

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Abstract: This article concerns two simple types of bounded operators with real spectrum on a Hilbert space . 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 where is selfadjoint, , and commutes with . A *sub-Jordan operator* is the restriction of a Jordan operator to an invariant subspace of . A *coadjoint operator* satisfies for some operators and or equivalently .

The main results are

**Theorem A**. *An operator is Jordan if and only if both and are coadjoint*.

**Theorem B**. *If is coadjoint, if has a cyclic vector, and if , then is unitarily equivalent to ' multiplication by ' on a weighted Sobolev space of order* 1 *which is supported on* .

**Theorem C**. *If is coadjoint and satisfies additional technical assumptions, then 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

DOI:
https://doi.org/10.1090/S0002-9947-1972-0308829-2

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