Asymptotic Toeplitz operators

Authors:
José Barría and P. R. Halmos

Journal:
Trans. Amer. Math. Soc. **273** (1982), 621-630

MSC:
Primary 47B35

DOI:
https://doi.org/10.1090/S0002-9947-1982-0667164-X

MathSciNet review:
667164

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An asymptotic Toeplitz is an operator such the sequence is strongly convergent, where 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.

**[1]**Arlen Brown, P. R. Halmos, and A. L. Shields,*Cesàro operators*, Acta Sci. Math. (Szeged)**26**(1965), 125–137. MR**0187085****[2]**Kenneth R. Davidson,*On operators commuting with Toeplitz operators modulo the compact operators*, J. Functional Analysis**24**(1977), no. 3, 291–302. MR**0454715****[3]**Ronald G. Douglas,*Banach algebra techniques in operator theory*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 49. MR**0361893****[4]**R. G. Douglas,*Banach algebra techniques in the theory of Toeplitz operators*, American Mathematical Society, Providence, R.I., 1973. Expository Lectures from the CBMS Regional Conference held at the University of Georgia, Athens, Ga., June 12–16, 1972; Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 15. MR**0361894****[5]**Paul Richard Halmos,*A Hilbert space problem book*, 2nd ed., Graduate Texts in Mathematics, vol. 19, Springer-Verlag, New York-Berlin, 1982. Encyclopedia of Mathematics and its Applications, 17. MR**675952****[6]**T. L. Kriete III and David Trutt,*The Cesàro operator in 𝑙² is subnormal*, Amer. J. Math.**93**(1971), 215–225. MR**0281025**, https://doi.org/10.2307/2373458

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
47B35

Retrieve articles in all journals with MSC: 47B35

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1982-0667164-X

Article copyright:
© Copyright 1982
American Mathematical Society