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Open research questions in Holomorphic and Operator Theory

69 unresolved questions extracted from the limitations and future-work sections of 392 Holomorphic and Operator Theory papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • 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.

    Finite Dimensional Naimark Realizations and Choice Sequences · 2026 · DOI

  • The category error in the AMPS firewall argument, importing monogamy of entanglement from Type I von Neumann algebras into a Type III setting. The need for a new approach to resolving the paradox, one that takes into account the differences between Type I and Type III von Neumann algebras. The challenge of deriving a Lyapunov-type master equation governing the time evolution of the modular conjugation operator.

    Modular Flow, Hawking Evaporation, and the Firewall Paradox · 2026 · DOI

  • The lack of a characterization of the dual space of the matrix-weighted Hardy space H p W. The need for a necessary and sufficient condition for the boundedness of modified Calderón-Zygmund operators.

    Matrix-weighted Campanato spaces: duality and Calderón-Zygmund operators · 2026 · DOI

  • The lack of a comprehensive study on the Carleson condition in the tensor product of Hardy spaces. The need for a construction of a class of operators that satisfy the Carleson condition.

    Dynamical Representation of Frames in Tensor Product of Hardy Spaces · 2026 · DOI

  • Future research can build on the results of the paper to further study the properties of doubly commuting operators in the C1,r class and quantum annulus. Future research can apply the results of the paper to other areas of operator theory and its applications.

    On doubly commuting operators in C1,r class and quantum annulus · 2026 · DOI

  • The paper identifies a gap in the literature on dilation results for doubly commuting tuples of operators in quantum annulus as well as in C1,r class. The paper aims to fill this gap by providing a characterization of c.n.u. C1,r-contractions and extending dilation results.

    On doubly commuting operators in C1,r class and quantum annulus · 2026 · DOI

  • The lack of a characterization of Schrödinger type operators in terms of dynamical systems. The need for a way to realize these operators in a specific form.

    A model and characterization of a class of symmetric semibounded operators · 2026 · DOI

  • The validity of the adjunction inequality for spatially refined s-invariants is unclear. There is a lack of understanding of the properties of refined s-invariants.

    Adjunction inequality for spatially refineds-invariants · 2026 · DOI

  • Future research could investigate the boundedness of the operator Hg for other spaces of analytic functions. Future research could investigate the exact norm of H from H∞ to Qp spaces.

    Operators of Hilbert type acting on some spaces of analytic functions · 2026 · DOI

  • The prior work did not describe the Green's relations and ideals of the regular part of the transformation semigroup. The prior work did not investigate the kernel of the regular part.

    The regular part of transformation semigroups that preserve double direction equivalence relation · 2026 · DOI

  • The paper does not provide a complete solution to the Smith-Ward problem. The results are limited to certain types of operator systems and matrix convex sets. The paper does not explore the implications of the results for other areas of mathematics or physics.

    Geometry of essential matrix ranges and the Smith–Ward problem for operator systems · 2026 · DOI

  • There is a gap in the current understanding of the Smith-Ward property, particularly for d = 2. The paper identifies a need for new techniques and perspectives on the Smith-Ward problem. The authors highlight the importance of understanding the relation between the Smith-Ward property and geometric objects.

    Geometry of essential matrix ranges and the Smith–Ward problem for operator systems · 2026 · DOI

  • Generalizing the scheme to the case of unbounded Q. Applying the results to solve inverse problems in mathematical physics.

    A model and characterization of a class of symmetric semibounded operators · 2026 · DOI

  • Further study of the properties of refined s-invariants. Investigation of the adjunction inequality for other types of refined s-invariants.

    Adjunction inequality for spatially refineds-invariants · 2026 · DOI

  • The lack of a complete characterization of the boundedness of the operator Hg. The lack of a detailed proof of Theorem 2.

    Operators of Hilbert type acting on some spaces of analytic functions · 2026 · DOI

  • 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.

    Constraints on symplectic quasi-states · 2026 · DOI

  • The alignment procedure in Step 5 corrects misalignment between consecutive polydisk embeddings via translation by vector si, but the paper does not address what happens when M has non-trivial topology or when symplectic structure constraints prevent such translations from being feasible. The robustness of this alignment strategy under topological obstructions requires investigation.

    Constraints on symplectic quasi-states · 2026 · DOI

  • To study the properties of the proposed operators and their relation to the Landau singularities. To extend the results to more general classes of Feynman integrals.

    D-ideal of generic mass banana integrals in dimensional regularization · 2026 · DOI

  • The paper identifies a gap in the computation of Feynman integrals, which is addressed by proposing a set of ℓ+3 simple differential operators. The paper identifies a gap in the study of the differential properties of Feynman integrals.

    D-ideal of generic mass banana integrals in dimensional regularization · 2026 · DOI

  • To extend the results to asymptotically anti-de Sitter spacetimes, - To study the implications of the results for our understanding of the AdS/CFT correspondence and the nature of spacetime, - To apply the techniques developed in the paper to other areas of quantum gravity and cosmology

    Operator Algebraic AdS/CFT and Subregion Duality · 2026 · DOI

  • The paper identifies a gap in our understanding of the subregion-subregion duality, - The paper notes that prior work has not established an operator algebraic subregion-subregion duality of boundary causal diamonds and bulk causal wedges for Klein-Gordon fields

    Operator Algebraic AdS/CFT and Subregion Duality · 2026 · DOI

  • 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.

    On the geometry of polynomials acting on the unit circle · 2026 · DOI

  • 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 <i>m</i> -Isometric Weighted Composition Operators on Directed Graphs · 2026 · DOI

  • The lack of a generalized Daubechies' theorem for mixed-state localization operators. The need to establish a Daubechies-type result for localization operators with Reinhardt domains as masks.

    Extensions of Daubechies' theorem: Reinhardt domains, Hagedorn wavepackets and mixed-state localization operators · 2026 · DOI

  • The 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 transformations

    On Pauli matrices and the implied spacetime geometry · 2026 · DOI

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69 open questions have been extracted from the limitations and future-work passages of 392 Holomorphic and Operator Theory papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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