Toeplitz operators on the space of real analytic functions: the Fredholm property. (English) Zbl 1489.47050

Domański, Pawel; Jasiczak, Michal

Banach J. Math. Anal. 12, No. 1, 31-67 (2018).

Summary: We completely characterize those continuous operators on the space of real analytic functions on the real line for which the associated matrix is Toeplitz (that is, we describe Toeplitz operators on this space). We also prove a necessary and sufficient condition for such operators to be Fredholm operators. While the space of real analytic functions is neither Banach space nor has a basis which makes available methods completely different from classical cases of Hardy spaces or Bergman spaces, nevertheless the results themselves show surprisingly strong similarity to the classical Hardy-space theory.

MSC:

47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
46E10	Topological linear spaces of continuous, differentiable or analytic functions
47A53	(Semi-) Fredholm operators; index theories
30E20	Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
30H10	Hardy spaces
30H50	Algebras of analytic functions of one complex variable
46A13	Spaces defined by inductive or projective limits (LB, LF, etc.)
47B38	Linear operators on function spaces (general)

Keywords:

Toeplitz operator; space of real analytic functions; Fredholm operator; Fredholm index; Cauchy transform