Abstract
Recently the authors characterized the Fredholmn properties of Toeplitz operators on weighted Fock spaces when the Laplacian of the weight function is bounded below and above. In the present work the authors extend their characterization to doubling Fock spaces with a subharmonic weight whose Laplacian is a doubling measure. The geometry induced by the Bergman metric for doubling Fock spaces is much more complicated than that of the Euclidean metric used in all the previous cases to study Fredholmness, which leads to considerably more involved calculations.
Highlights
Let be a domain of the complex plane C, and let μ be a positive Borel measure on
We investigate the properties of Toeplitz and Hankel operators acting on doubling Fock spaces with symbols of bounded and vanishing mean oscillation
To completely settle the theory of Fredholm Toeplitz operators T f with symbols in V M O1 on all Fock spaces Fφp with f ∈ V M O1, we prove that Theorem 1.2 remains true even for weights whose Laplacian is a doubling measure
Summary
Let be a domain of the complex plane C, and let μ be a positive Borel measure on. For 0 < p < ∞, the space L p( , dμ) consists of all Lebesgue measurable functions f on for which. To completely settle the theory of Fredholm Toeplitz operators T f with symbols in V M O1 on all Fock spaces Fφp with f ∈ V M O1, we prove that Theorem 1.2 remains true even for weights whose Laplacian is a doubling measure. This is our main result—see Theorem 4.9. We note that the nesting property Fφp ⊂ Fφq ( p ≤ q) known for weights whose Laplacian is bounded below and above is no longer true for doubling Fock spaces, which is yet another complication that we have to deal with
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