Open research questions in Holomorphic and Operator Theory
111 gap statements mined from Holomorphic and Operator Theory papers in our 4.5M-paper local library, which holds 853 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.
What the literature leaves open
The paper suggests further research on symplectic quasi-states. It suggests exploring the implications of the main result for the study of symplectic geometry.
The lack of a general framework for studying composition operators on weighted modulation spaces. The need for a better understanding of the continuity of composition operators in different function spaces. The lack of explicit estimates for the constant C in the continuity result.
We plan to investigate this in future work. The properties of (the dual of) the bulk-to-boundary map ∂, together with the established inclusion of bulk wedge algebras into boundary diamond algebras (Proposition 4.4) suggests a potential connection with the notion of projective holography studied in (see for the notion of algebraic holography).
The problem of characterizing symmetric multi-affine polynomials is not fully understood. The paper identifies a gap in the literature on the characterization of symmetric multi-affine polynomials.
The k-quasi-m-isometric completion problem for weighted shift and composition operators on directed graphs is not well studied. The paper identifies a gap in the literature and provides a solution to this problem.
Completion Problem For Extension of m -Isometric Weighted Composition Operators on Directed Graphs · 2026 · DOIThe lack of a dynamical classification for cubic Blaschke products. The lack of understanding of the parameters that yield parabolic Blaschke products.
The lack of a complete characterization of the point spectrum for hyperbolic semigroups in the Dirichlet space. The need to understand the geometry of the Koenigs domain and its relation to the point spectrum.
The lack of a method to reconstruct a rational function from a positive skew symmetric choice sequence. The lack of a Routh-type test to determine if a rational matrix-valued function is bi-inner.
The paper suggests that future research directions include applying the theory of quantum doubly stochastic operators to non-commutative majorization and exploring connections to Connes' embedding and free entropy.
Quantum Doubly Stochastic Operators on Non-commutative $L_p$-Spaces · 2026The extension of doubly stochasticity to bounded linear operators on non-commutative Lp-spaces is a significant problem. The paper identifies a gap in the existing literature on quantum doubly stochastic operators. The paper aims to fill this gap by developing a foundational framework for quantum doubly stochastic operators on non-commutative Lp-spaces.
Quantum Doubly Stochastic Operators on Non-commutative $L_p$-Spaces · 2026The authors highlight the need for an explicit expression of the Fredholm transformation T involved.
Quantitative Fredholm backstepping and rapid stabilization · 2026Future research can focus on generalizing the results to more general classes of linear maps. Future research can focus on studying the properties of linear maps on second symmetric product spaces.
Positive Linear Maps on Second Symmetric Product Spaces · 2026The lack of understanding of harmonious quasi-projection pairs on Hilbert C∗-modules. The need for block matrix representations of these pairs and their associated projections.
The finite set Z of cube centers in Step 1 is obtained from Theorem 3.1, but the paper does not characterize how the cardinality |Z| = r depends on the manifold M's geometry, or whether there exist manifolds for which r must grow unbounded to approach measure 1-ε. This relationship between cube multiplicity and symplectic structure deserves explicit analysis.
It would be of interest to further investigate the dynamics of D-cyclic operators corresponding to various classes of subsets D ⊂ C, such as when D is compact, disconnected, totally disconnected, or possesses a fractal structure. In addition, it is natural to explore the possibility of formulating a D-cyclicity criterion analogous to Kitai’s Hypercyclicity Criterion.
In an n-dimensional polyhedral Banach space, does the preservation of Birkhoff-James orthogonality by a bounded linear operator at any n linearly independent extreme points necessarily imply that the operator is a scalar multiple of an isometry? In case of an arbitray Banach space the result is not necessarily correct, which follows from [12, Th.
The problem of characterizing linear maps that preserve positive decomposable elements is still open. Prior work has characterized linear maps that preserve decomposable elements, but the problem of preserving positive decomposable elements is still open.
Positive Linear Maps on Second Symmetric Product Spaces · 2026To study the continuity of composition operators on other function spaces. To establish explicit estimates for the constant C in the continuity result. To apply the paper's techniques to problems in signal processing and time-frequency analysis.
To fully characterize the parameters that yield parabolic Blaschke products. To study the dynamics of higher-degree Blaschke products. To apply the results of this paper to other areas of complex analysis.
The paper identifies a gap in the study of the condition (Ω) for weighted Fréchet spaces of holomorphic functions. The paper identifies a gap in the study of the surjectivity of the Cauchy-Riemann operator on certain weighted spaces of vector-valued smooth functions.
Further study of the point spectrum for parabolic semigroups in the Dirichlet space. Investigation of the geometry of the Koenigs domain for parabolic semigroups. Application of the results to other function spaces and contexts.
To further study the properties of finite rank de Branges-Rovnyak spaces. To apply the characterization of H(B)-spaces to other areas of function theory. To extend the results to more general classes of functions.
Unbounded Toeplitz operators and finite rank de Branges-Rovnyak spaces · 2026The gap is the lack of understanding of how Pauli matrices imply a manifold, - The gap is the lack of understanding of how Pauli matrices are inter-related by frame…
The paper identifies a gap in the study of approximating arbitrary increasing functions by the argument of a meromorphic inner function. The paper identifies a gap in the study of Beurling-Malliavin majorants in model spaces.
Inner approximations of doubling weights with applications to Beurling-Malliavin theory in Toeplitz kernels · 2026 · DOITo extend the results to irrational γ. To study the Bergman kernel on other classes of domains. To apply the results to other problems in complex analysis and geometry.
Most-cited papers in Holomorphic and Operator Theory
- On the asymmetry of the symmetric MAPE · International Journal of Forecasting · 1999 · 262 citations
- Deformation properties for continuous functionals and critical point theory · Topological Methods in Nonlinear Analysis · 1993 · 120 citations
- The Equivalence of Definitions of a Matric Function · American Mathematical Monthly · 1955 · 115 citations
- A Dual of Dilworth's Decomposition Theorem · American Mathematical Monthly · 1971 · 87 citations
- Closest Unitary, Orthogonal and Hermitian Operators to a Given Operator · Mathematics Magazine · 1975 · 85 citations
- A Simple Proof of the Fredholm Alternative and a Characterization of the Fredholm Operators · American Mathematical Monthly · 2001 · 83 citations
- A new concept of Lorenz domination · Economics Letters · 1987 · 82 citations
- A Remark on a Theorem of M. A. Krasnoselski · American Mathematical Monthly · 1966 · 63 citations
- Equational Characterization of the Subvarieties of BL Generated by t-norm Algebras · Studia Logica · 2004 · 42 citations
- Linear Transformations with Invariant Cones · American Mathematical Monthly · 1967 · 39 citations
Most recent work
- Non-Clifford Cost of Random Unitaries · PRX Quantum · 2026
- Parent Lindbladians for matrix product density operators · Physical Review Research · 2026
- Examples of \({p}\)-Harmonic Maps · SIAM Journal on Mathematical Analysis · 2026
- Cyclicity of Multipliers on the Unit Ball of $$\mathbb {C}^n$$: A Corona-Based Approach · Analysis and Mathematical Physics · 2026
- Block matrix representations for quasi-projection pairs on Hilbert $$C^*$$-modules · Banach Journal of Mathematical Analysis · 2026
- Centralizers of non-elliptic univalent self-maps and the embeddability problem in the unit disc · Transactions of the American Mathematical Society · 2026
- RSK Linear Operators and the Vershik-Kerov-Logan-Shepp Curve · The Electronic Journal of Combinatorics · 2026
- Differential operators on homogeneous plane curve singularities · Journal of Symbolic Computation · 2026
- Uniform ergodicity and the one-sided ergodic Hilbert transform · Journal of Mathematical Analysis and Applications · 2026
- Quasi-grammians and breathers of short pulse equation · Journal of Computational and Applied Mathematics · 2026
Find a gap in your own Holomorphic and Operator Theory sub-topic
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