Classification of nearly invariant subspaces of the backward shift

Author:
Eric Hayashi

Journal:
Proc. Amer. Math. Soc. **110** (1990), 441-448

MSC:
Primary 47A15; Secondary 30H05, 47B35, 47B38

DOI:
https://doi.org/10.1090/S0002-9939-1990-1019277-0

MathSciNet review:
1019277

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the backward shift operator on the Hardy space of the unit disk. A subspace of is called nearly invariant if is in whenever belongs to and . In particular, the kernel of every Toeplitz operator is nearly invariant. A function theoretic characterization is given of those nearly invariant subspaces which are the kernels of Toeplitz operators, and it is shown that they can be put into one-to-one correspondence with the Cartesian product of the set of exposed points of the unit ball of with the set of inner functions.

**[1]**V. M. Adamjan, D. Z. Arov, and M. G. Kreĭn,*Infinite Hankel matrices and generalized problems of Carathéodory-Fejér and F. Riesz*, Funkcional. Anal. i Priložen.**2**(1968), no. 1, 1–19 (Russian). MR**0234274****[2]**John B. Garnett,*Bounded analytic functions*, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR**628971****[3]**Eric Hayashi,*The solution sets of extremal problems in 𝐻¹*, Proc. Amer. Math. Soc.**93**(1985), no. 4, 690–696. MR**776204**, https://doi.org/10.1090/S0002-9939-1985-0776204-7**[4]**Eric Hayashi,*The kernel of a Toeplitz operator*, Integral Equations Operator Theory**9**(1986), no. 4, 588–591. MR**853630**, https://doi.org/10.1007/BF01204630**[5]**D. Hitt,*Invariant subspaces of**of an annulus*, preprint.**[6]**Donald Sarason,*Nearly invariant subspaces of the backward shift*, Contributions to operator theory and its applications (Mesa, AZ, 1987) Oper. Theory Adv. Appl., vol. 35, Birkhäuser, Basel, 1988, pp. 481–493. MR**1017680****[7]**-,*Exposed points in*I, Oper. Theory: Adv. Appl. (to appear).

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47A15,
30H05,
47B35,
47B38

Retrieve articles in all journals with MSC: 47A15, 30H05, 47B35, 47B38

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1990-1019277-0

Article copyright:
© Copyright 1990
American Mathematical Society