Inner functions are fundamental in complex analysis, but their impact on Toeplitz

operators acting in the Hardy space $H^1(\mathbb{C}^+)$ presents intriguing questions:

Is the Toeplitz operator $T_{\overline\Theta}$, with ${\overline\Theta}$ being the conjugate of an inner function,

bounded in $H^1(\mathbb{C}^+)$?

If not, what is the most natural space for its definition?

In this talk, we will explore these questions, revealing their connections to the duals

of 1-de Branges spaces.

This work stems from an ongoing collaboration with Marco Peloso.

References

[1] Bellavita C., Peloso M. On Toeplitz operators on H1(C+), Proc. Amer. Math. Soc..

[2] Bellavita C., Peloso M. Duality, BM O and Hankel operators on Bernstein spaces, J. Funct. Anal. 288-2 (2025).