Several Complex Variables Problem List — Problem 27

Definitions and mathematical statement

Let $\Omega\subset{\mathbb{C}}$ be a bounded domain with smooth boundary, and use planar area measure ${\,\mathrm{d}} A$. Let $L^2(\Omega)$ denote the complex Hilbert space with inner product $\langle u,v\rangle=\int_\Omega u\overline v\,{\,\mathrm{d}} A$, and let \[ A^2(\Omega)=\{g\in L^2(\Omega):g\text{ is holomorphic on }\Omega\}. \] Interpret the Kohn solution operator $T:L^2(\Omega)\to L^2(\Omega)$ as the canonical solution operator characterized by \[ \frac{\partial(Tf)}{\partial\bar z}=f\quad\text{in distributions},\qquad Tf\perp A^2(\Omega). \] For $h\in L^\infty(\Omega)$, define \[ S_h f=\overline h\,Tf. \] The source describes $S_h$ as compact and non-self-adjoint; self-adjoint exceptional cases, such as $h\equiv0$, are possible.

The general question is to determine conditions on $\Omega$ and $h$ under which $S_h$ has eigenvalues. For the principal question, interpret this as asking for a nonzero eigenvalue, namely whether \[ \exists\lambda\in{\mathbb{C}}\setminus\{0\}\ \exists f\in L^2(\Omega)\setminus\{0\}:\quad \overline h\,Tf=\lambda f. \] This interpretation is necessary to qualify the source's unqualified use of "eigenvalues": when $h\equiv0$, zero is an eigenvalue even though $h$ has a single-valued holomorphic primitive.

In particular, suppose $h$ is bounded and holomorphic on $\Omega$. A single-valued holomorphic primitive is a holomorphic function $H:\Omega\to{\mathbb{C}}$ satisfying $H'=h$. The source states that the existence of such an $H$ excludes nonzero eigenvalues of $S_h$, and asks whether the converse holds. Equivalently, is the following implication valid for every such $\Omega$ and $h$? \[ \left[\exists\text{ a closed piecewise }C^1\text{ curve }\gamma\subset\Omega: \int_\gamma h(z)\,{\,\mathrm{d}} z\ne0\right] \quad\Longrightarrow\quad \left[\exists\lambda\in{\mathbb{C}}\setminus\{0\}\ \exists f\in L^2(\Omega)\setminus\{0\}:S_hf=\lambda f\right]. \] Here the source's phrase "periods of $H$" is interpreted as the additive monodromy of a locally defined primitive, measured by the periods $\int_\gamma h(z)\,{\,\mathrm{d}} z$. The source reports an affirmative answer for annuli. That special-case assertion and the primitive obstruction are supplied source claims, not independently verified results in this draft.

Short English statement

For the canonical $\bar\partial$-solution operator on a smooth bounded planar domain, determine when multiplication of its output by $\overline h$ produces a nonzero eigenvalue. When $h$ is bounded and holomorphic, must a nonzero period of $h(z)\,{\,\mathrm{d}} z$, equivalently the absence of a single-valued holomorphic primitive, force such an eigenvalue? The source reports that this holds for annuli. The qualification "nonzero" resolves an ambiguity in the supplied wording.

Sources

• Local UnsolvedMath registry: supplied complete record, permanent ID $800230$, exact source problem code catalogue entry, title "Several Complex Variables Problem List --- Problem 27." The registry supplies the status open; this label is not an independently verified current mathematical assessment.
• Buzzard, Fornaess, and Krantz, The Several Complex Variables Problem List (1995), Problem 27. Supplied source URL: https://arxiv.org/abs/math/9509202. The registry records extraction by arxiv_latex and access on 2026-10-16. This access date is supplied provenance and does not indicate independent source access or verification for this draft.
• The problem text attributes the question to D. Barrett, with the supplied contact barrett@math.lsa.umich.edu. The registry's proposed_by field instead names the list authors, Buzzard, Fornaess, and Krantz; both attribution levels are retained here.