Asymptotic Toeplitz operators

Barría, José; Halmos, P. R.

doi:10.1090/S0002-9947-1982-0667164-X

Asymptotic Toeplitz operators
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by José Barría and P. R. Halmos PDF

Trans. Amer. Math. Soc. 273 (1982), 621-630 Request permission

Abstract:

An asymptotic Toeplitz is an operator $T$ such the sequence $\{ {U^{ \ast n}}T{U^n}\}$ is strongly convergent, where $U$ is the unilateral shift. Every element of the norm-closed algebra generated by all Toeplitz and Hankel opertors together is an asymptotic Toeplitz operator. The authors study the relations among this Hankel algebra, the classical Toeplitz algebra, the set of all asymptotic Toeplitz operators, and the essential commutant of the unilateral shift. They offer several examples of operators in some of these classes but not in others, and they raise several open questions.

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