Open composition-operator questions of Dellepiane and Seco for de Branges–Rovnyak spaces

Given Schur-class functions $b_1,b_2$ on the unit disk $\mathbb{D}$ and an analytic self-map $\varphi:\mathbb{D}\to\mathbb{D}$ , characterize when the composition operator $C_\varphi f=f\circ\varphi$ defines a bounded operator $C_\varphi:\mathcal{H}(b_1)\to\mathcal{H}(b_2)$ between the corresponding de Branges–Rovnyak spaces.

References

Primary source

arXiv

Additional references

Littlewood subordination for de Branges--Rovnyak spaces — arXiv

functional-analysis operator-theory

Posted 4 weeks ago · edited 4 weeks ago

Progress summary

Refreshed 4 weeks ago

Claimed progress

A September 2026 paper claims advances on these questions, but it does not specify which questions it settles.

The problem tracks open classification questions posed by Dellepiane and Seco about bounded composition operators on de Branges–Rovnyak spaces. No date for the original questions is given.

September 2026 claimed advance

The arXiv work Littlewood subordination for de Branges–Rovnyak spaces is reported to answer open questions through cross-space analysis using reproducing kernels and Aleksandrov–Clark measures. Its abstract does not identify which individual questions are answered, so the advance remains unverified here.

Current status (as of September 2026): A recent work claims progress on the questions, but the exact questions resolved and the validity of the claimed results remain unconfirmed.

Sources

No solutions have been posted yet.