arXiv subjects

Shih-Yu Chang

Publications and source records attributed to Shih-Yu Chang.

At least 19 recordsLinked to original sources

arXiv2026

Intrinsic Geometry of Categorified Spectral Objects

This paper develops the intrinsic geometry of the categorified spectral object $\mathfrak{Spec}(A)$ associated with an admissible operator-semantic system $A$ in the Categorified Spectral Duality (CSD) framework. We prove that the tangent complex, singular locus, inertia stack, and contextual curvature class are canonically determined by the duality adjunction between $\mathfrak{Spec}$ and the global sections functor, together with the reconstruction theorem identifying $A$ with the global sections of $\mathfrak{Spec}(A)$. The Canonical Geometry Theorem establishes that any CSD-compatible geometric structure is induced by the canonical datum consisting of $\mathfrak{Spec}(A)$, its tangent complex, its singular locus, its inertia stack, and its contextual curvature class; hence the geometry of $\mathfrak{Spec}(A)$ is intrinsic to the semantic structure of $A$. We prove that the tangent complex controls the deformation theory of $\mathfrak{Spec}(A)$ and satisfies a Hochschild realization, establishing a direct bridge between geometry and algebra. The assignment sending $A$ to its canonical geometric datum is functorial and Morita invariant, with explicit computations for the complex numbers and matrix algebras demonstrating that noncommutativity, detected by the inertia stack, is distinct from contextuality, which requires additional structures. Thus semantics determines geometry, and the intrinsic geometry of $\mathfrak{Spec}(A)$ provides a canonical geometric encoding of $A$ up to Morita equivalence.

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arXiv2026

Operator Inequalities in $\Phi$-Product Tensor Algebras: Invariance and Transform Sensitivity

We study classical operator inequalities in $\Phi$-product tensor algebras (a transformation $\Phi$-based generalization of the $t$-product framework) for third-order tensors. Although these algebras are algebraically isomorphic under different unitary transforms $\Phi$, we show that their quantitative behavior is not invariant. We prove that fundamental inequalities, including Golden--Thompson, Jensen, Klein, and Lieb, extend to the $\Phi$-product setting with the same constants as in the matrix case. However, the associated defect -- the slack between the two sides of the inequality -- depends explicitly on transform-domain noncommutativity. In particular, we establish a sharp characterization of the defect in terms of slice-wise commutators, revealing that inequality tightness is governed by transform-induced noncommutativity. We further demonstrate strong transform sensitivity by constructing explicit tensor pairs for which the defect vanishes under one transform (e.g., discrete Fourier transform) but grows linearly with the tensor depth under another transform (e.g., discrete Cosine transform), yielding an $\Omega(p)$ separation where $p$ is the matrix dimension of $\Phi$. Moreover, we prove that no transform is universally optimal: for any pair of transforms, there exist tensors for which each is strictly better than the other. These results show that the choice of transform defines a coordinate system in which commutativity is measured, inducing a nontrivial geometry of inequality tightness. Consequently, optimal transform selection is inherently data-dependent and can be formulated as an optimization problem over the unitary group.

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arXiv2026

Categorified Spectral Duality: From Operator Systems to Spectral Stacks and Back

Classical Gelfand duality provides an equivalence between commutative C-star algebras and topological spaces, but fails to furnish a geometric object for noncommutative operator systems. To address this, we introduce a categorified notion of spectrum that captures the full operator-semantic structure. For an operator system A, we construct a spectral stack, denoted Spec(A), over the site of its commutative contexts. The construction proceeds in three stages. First, we encode the syntax of A via a colored operad called the synergy operad. Second, we aggregate local semantic data using a left Kan extension, providing an explicit coend formula. Third, we enforce descent via sheafification, yielding a stack that satisfies higher gluing conditions. We prove that Spec(A) satisfies a Yoneda-style universal property, making it the initial descent-complete semantic realization of A. This yields a contravariant functor from operator systems to spectral objects that admits a right adjoint, the global sections functor. Under semantic generation and descent completeness, the counit of this adjunction is an equivalence, establishing a reconstruction theorem: A is isomorphic to the endomorphisms of the structure sheaf on its spectrum. We further provide a recognition theorem characterizing the essential image of this functor and prove Morita invariance of the associated quasi-coherent sheaf categories. The construction naturally recovers classical Gelfand spectra and Bohrification as truncations of a Postnikov tower. Explicit computations for matrix algebras, Pauli systems, and the Mermin-Peres square yield a quantitative invariant of contextuality extracted from the inertia stack of Spec(A). Comparison theorems show that our framework subsumes Gelfand duality, Bohrification, and Tannaka reconstruction as special cases.

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arXiv2026

A Universal Theory of Spectral Propagation for Compositional Operator Networks

Classical spectral theory lacks a framework for understanding how spectra propagate through compositional systems like deep neural networks, feedback control loops, and quantum circuits. This paper develops a universal theory governed by three invariants: the operadic spectrum (local spectral data), spectral derivatives (perturbation sensitivity), and interaction residue (emergent interface-generated content). We prove three main theorems: the Spectral Propagation Theorem decomposes global output into propagated local spectra, residues, and derivative corrections; the Stability Theorem introduces the SOC stability radius and condition number; and the Universality Theorem shows any reasonable propagation rule is uniquely determined by the three invariants. These results provide a coordinate-free, representation-invariant language for spectral analysis of compositional operator systems.

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arXiv2026

Interaction Residues and Localized Spectral Defects in Stratified Operadic Systems

We develop a framework for studying how global spectral structure emerges from interacting local sectors in stratified operadic systems. The central object is the interaction residue, which measures the failure of exact spectral decomposition across interfaces. Under suitable localization assumptions, the global spectrum decomposes into local spectral sectors together with interface-generated residue contributions. The theory introduces a classification of spectral defects based on interface geometry and algebraic structure. Point interfaces produce isolated spectral contributions; line and surface interfaces produce extended spectral regimes. Non-semisimple operator structure generates nilpotent defects associated with Jordan blocks and generalized eigenspaces, yielding a two-dimensional defect taxonomy combining geometric localization with Jordan complexity. Several structural results are established, including interface localization, rigidity and vanishing criteria, refinement functoriality, and deformation stability. Under a local triviality condition, the residue is homotopy invariant, preserving its homology and Betti numbers throughout admissible deformations. The framework is illustrated through explicit operator and block matrix examples demonstrating how localized interactions generate localized spectral defects. The nilpotent sector connects the theory with classical operator theory via generalized eigenvectors, functional calculus, and perturbative Jordan splitting. Overall, the framework provides a unified viewpoint for understanding how interface interactions and non-semisimple structure influence global spectral behavior.

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arXiv2026

Tensor-Lifted Multivariate Functional Calculus Beyond Commutativity and Boundedness

Classical functional calculus is primarily spectral, capturing eigenvalue information through resolvent methods while largely ignoring nilpotent structure. Building on the projector-nilpotent characterization developed in our companion work, we introduce a multivariate functional calculus for arbitrary operators that incorporates both spectral and algebraic information. The framework has three main components. First, nilpotent derivative terms are explicitly included in the functional expansion, allowing the calculus to capture generalized eigenspaces and Jordan structures beyond classical resolvent methods. Second, tensor lifting treats non-commuting operators by embedding them into a commuting system on a tensor-product space. Third, a two-level convergence theory is established: Level 1 proves existence through strong resolvent convergence implying strong operator topology convergence, while Level 2 provides stability through norm resolvent convergence implying operator norm convergence with explicit error bounds. The framework simultaneously handles discrete, continuous, and hybrid spectra. Results are developed for bounded operators, unbounded self-adjoint operators recovering the classical spectral theorem, and unbounded non-self-adjoint operators with compact resolvent. For operators without compact resolvent, we introduce a compactifying regularization method based on perturbation by a positive self-adjoint operator with compact resolvent. The proposed framework is compatible with existing functional calculi and recovers their behavior under corresponding assumptions. To our knowledge, this is the first unified framework simultaneously addressing non-commutativity, non-self-adjointness, and unboundedness while explicitly preserving nilpotent structure and providing convergence guarantees.

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arXiv2026

Spectral Operadic Calculus: Norm-Analytic Functor Calculus

Classical spectral theory provides powerful tools for analyzing linear operators, but does not extend naturally to nonlinear or compositional settings. In particular, there is no general way to transport spectral invariants in a functorial manner across structured categories. In earlier work, we showed that this failure is fundamental and introduced an operadic notion of spectrum that provides a canonical replacement. In this paper, we develop the analytic consequences of this construction and show that the operadic spectrum acts as a control parameter for a calculus of functors. We establish a criterion for polynomial behavior based on higher cross-effects, and prove convergence results for the associated Taylor tower, including explicit exponential error bounds. We further show that the derivatives of a functor form a structured algebraic object with symmetric and operadic features, and satisfy a chain rule governed by a natural composition operation (operadic plethysm). This leads to a reconstruction theorem, showing that analytic functors are completely determined by their derivative data, and hence to a classification in terms of algebraic structures. Compared with classical Goodwillie calculus, which is governed by homotopy-theoretic conditions, the present framework is analytic and quantitative in nature, providing explicit control over convergence and approximation. These results place functor calculus in a setting that combines spectral ideas, analytic methods, and operadic algebra, and suggest further connections with deformation theory and geometry.

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arXiv2026

The Operadic Spectrum and Obstructions to Spectral Base Change

We introduce an operadic notion of spectrum for algebras over colored operads in a symmetric monoidal category. The construction is defined via a canonical Hochschild-type object together with an operadic residue, which together encode spectral information in a manner compatible with operadic composition. A central result of this work is that classical spectral invariants do not, in general, admit a natural base change in the operadic setting. More precisely, we show that there is no functorial procedure that transports spectra along strong monoidal functors while preserving their expected structural properties. This establishes a fundamental obstruction to spectral base change. To address this issue, we construct a universal operadic residue object and show that it induces a well-defined and functorial notion of operadic spectrum. We further prove that this construction is canonical and reduces to the classical spectrum in the case of the trivial operad. These results provide a conceptual foundation for spectral theory in operadic and higher algebraic contexts, and clarify the limitations of extending classical spectral invariants beyond the linear setting.

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arXiv2026

The Unitary Conjugation Groupoid as a Universal Mediator of the Baum--Connes Assembly Map

We show that the Baum--Connes assembly map factors canonically through the unitary conjugation groupoid, which serves as a universal mediator among groupoid models that are Morita equivalent to a given transformation groupoid. This establishes a structural link between groupoid-based index theory and the Baum--Connes program at the level of K-theory. Building on our previous development of unitary conjugation groupoids and their associated index theory, we extend the $K_1$ index framework beyond the Type I setting to non-Type I examples, including the irrational rotation algebra and amenable crossed products. Using Morita equivalence, we relate unitary conjugation groupoids to transformation and action groupoids, enabling the transfer of descent-type index constructions to these settings. Our main result shows that, among all groupoid realizations that are Morita equivalent to a transformation groupoid, the factorization through the unitary conjugation groupoid is canonical at the level of K-theory. This identifies the unitary conjugation groupoid as a universal intermediary for the Baum--Connes assembly map. As applications, we recover the classical index pairing with the tracial state for the irrational rotation algebra in the sense of Connes, and we prove that for amenable crossed products the descent construction agrees with the analytic Baum--Connes assembly map under Morita equivalence. These results provide a conceptual interpretation of the assembly map in terms of internal symmetries of crossed product algebras and suggest a unified framework connecting Fredholm-type index data with equivariant K-theory via groupoid methods.

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arXiv2026

An Index Theorem for Fredholm Operators via the Unitary Conjugation Groupoid

In a previous paper we introduced the unitary conjugation groupoid associated to any unital separable Type I C*-algebra. This groupoid encodes the representation-theoretic structure of the algebra through the action of its unitary group on the characters of commutative subalgebras and admits a canonical embedding of the algebra into its groupoid C*-algebra. In this paper we apply this framework to two fundamental operator algebras: the algebra of all bounded operators on a separable Hilbert space and the unitization of the compact operators. For any Fredholm operator in these algebras we construct a natural equivariant KK-theory class using the phase of the operator. Applying Kasparov descent for groupoids produces a class in the K-theory of the associated groupoid C*-algebra. Using the Morita equivalences established in the previous work, this descended class can be identified with the symbol class of the operator. We prove that composing this class with the boundary map of the Calkin extension recovers the classical Fredholm index. In particular, the construction yields index minus one for the unilateral shift and index zero for compact perturbations of the identity. This provides a groupoid equivariant formulation of the Fredholm index and connects the unitary conjugation groupoid with classical operator index theory. This perspective suggests a general approach to index theory through equivariant K-theory of groupoids associated with operator algebras.

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arXiv2026

The Unitary Conjugation Groupoid of a Type I C*-Algebra: Topology, Fell Continuity, and the Canonical Diagonal Embedding

This paper introduces a canonical Polish groupoid associated to any separable unital C*-algebra, termed the unitary conjugation groupoid. It is defined as the semidirect product of the algebra's dual space by its unitary group, acting by conjugation. Classical groupoid models for C*-algebras typically require additional structure such as a Cartan subalgebra and rely on the locally compact Hausdorff framework. In contrast, our construction is entirely canonical but forces a paradigm shift: the natural topologies on the dual space and the unitary group are not locally compact. To address this, we equip the dual space with a Polish topology derived from the weak-star topology on pure states and the unitary group with the strong operator topology. This yields a Polish groupoid admitting a continuous Haar system. We prove that the associated reduced groupoid C*-algebra is Morita equivalent to the original algebra tensored with the compact operators, establishing that the groupoid encodes the K-theory of the algebra. A key feature is a canonical diagonal embedding of the original algebra into the groupoid C*-algebra, which is a unital injective star-homomorphism for separable Type I C*-algebras. We characterize commutativity via this embedding and establish functoriality of the construction under appropriate star-homomorphisms. The theory is illustrated with detailed computations for three fundamental classes: finite-dimensional matrix algebras, commutative algebras over compact metrizable spaces, and the unitized compact operators. These examples demonstrate that our general constructions reduce to familiar objects. We also discuss limitations by analyzing the irrational rotation algebra, a non-Type I algebra not covered by our construction, highlighting directions for future research.

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arXiv2026

Categorified Spectral Sheaves and Homotopical Invariants for Noncommuting Operators

Classical spectral theory gives a complete description of a single normal operator, but it fails for noncommuting operators, where no canonical joint spectrum or simultaneous diagonalization exists. Existing approaches provide only partial solutions: joint spectra apply mainly to commuting families, noncommutative geometry emphasizes global invariants that obscure local structure, and topos-theoretic methods capture contextuality while losing higher coherence information. This paper proposes a geometric and higher-categorical reformulation of the spectral problem for noncommuting operators. Local classical spectra associated with commutative subalgebras are organized into a stack-valued object, called a spectral stack, which retains automorphism and unitary equivalence data between contexts. Noncommutativity is thereby interpreted as nontrivial descent data rather than a breakdown of spectral theory. Using homotopical and derived constructions, we define functorial invariants that measure obstructions to global spectral assembly and extend classical index-theoretic ideas. The resulting framework views noncommutative operator algebras as geometric objects equipped with a spectral atlas, providing a concise bridge between operator theory, homotopy theory, and higher geometry.

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arXiv2025

Multicategorical Adjoints, Monadicity, and Quantum Resources

This paper is the third part of a program aimed at building a unified operadic and multicategorical foundation for operator theory and quantum processes. Building on the multicategory HilbMult and the previously introduced Synergy Operad, we investigate the intrinsic algebraic structure underlying compositional quantum dynamics. Three main themes are developed. First, we introduce n-adjoints, a hierarchy of generalized adjoint operations that extend classical adjoints to multicategorical settings and capture higher-order operator interactions. We prove a multicategorical version of Stinespring's theorem formulated in terms of n-adjoints. Second, we study the algebraic semantics of quantum processes and show that completely positive trace-preserving maps admit a canonical monadic description. In particular, we prove a monadicity theorem identifying quantum processes, up to operadic operational equivalence, with algebras of a synergy monad induced by a symmetric quantum interaction operad, yielding a faithful representation that uniformly accounts for sequential and parallel composition. Third, we present an algebraic formulation of quantum no-go principles by encoding constraints such as the no-cloning theorem as operadic ideals. Together, these results provide a coherent algebraic framework that unifies operator theory, category theory, and quantum information.

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arXiv2025

HilbMult: A Banach-Enriched Multicategory for Operator Algebras

Category and multicategory theory provide abstract frameworks for describing structures and their compositions, with multicategories extending traditional categories to handle multi-input operations. These theories enable modular reasoning and coherent composition of complex systems, and have found applications in computer science, physics, and mathematics, including programming language semantics, quantum processes, tensor networks, operads, and higher algebra. Operator theory, in contrast, studies linear and multilinear transformations in functional spaces, forming the analytic backbone of modern analysis and quantum mechanics, with applications ranging from signal processing and control theory to data science. This paper explores the synergy between these two areas by showing how operator theory provides concrete analytic structures that naturally enrich multicategories, while multicategory theory supplies a unifying framework for organizing multi-input operators and ensuring coherence in complex networks. We develop a comprehensive categorical framework integrating operator theory with multicategories, introducing a symmetric monoidal multicategory of Hilbert spaces with bounded multilinear maps and establishing its enrichment and coherence properties. The framework includes a functorial spectral theorem, covariance under unitary transformations, and universality as a semantic target, providing a unified language for linking analytic structure, categorical semantics, and operator representations. This work lays the foundation for a broader research program uniting operator theory, category theory, and noncommutative geometry.

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arXiv2025

Composition and Coherence: The Syntax of Operator Networks

Coherence is a central issue in category theory and multicategory theory, ensuring that formally distinct compositions of morphisms, such as tensor reorderings or diagrammatic rewiring, represent the same underlying transformation. In operator theory and in Hilbert space based systems, coherence guarantees that equivalent operator networks produce identical effects on states and signals, maintaining both mathematical consistency and computational interpretability. This paper, the second installment of the Categorical Spectral Architecture (CSA) program, develops a unified framework that integrates operator theory, spectral analysis, and categorical algebra. Building on the functorial and spectral foundations established in the first paper, we introduce the Synergy Operad, a syntactic framework that extends the multicategory of Hilbert spaces and bounded multilinear maps. The Canonical Representation Theorem provides a structure-preserving functor that aligns syntactic compositions with operator semantics. Another central component, the Coherence Theorem, ensures that grammatical equivalences correspond to identical semantic results. Together these contributions identify a syntactic-semantic duality at the core of CSA and provide a rigorous basis for analyzing and constructing complex systems of operators with coherent and predictable meaning.

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arXiv2025

Existence and Enumeration of Polynomially Transformed Matrices under Spectral and Nilpotent Constraints

Matrix functions extend scalar function concepts to linear operators, offering a unified framework with broad applications in mathematics, science, and engineering. Classical definitions--via power series, spectral calculus, or Jordan form--capture both diagonalizable and defective matrices, revealing insights into dynamics, stability, and modal interactions. Building on this foundation, this study examines how matrix functions behave under nilpotency and diagonalizability constraints, inspired by the Jordan decomposition into spectral and nilpotent parts. The spectral part encodes eigenstructure, while the nilpotent part reflects deviations from diagonalizability. Two main problems are addressed: (1) identifying when a polynomial-transformed matrix becomes nilpotent, and (2) determining when it becomes diagonalizable. The analysis considers two scenarios--matrices with one distinct eigenvalue and those with multiple distinct eigenvalues--each posing unique algebraic and combinatorial challenges. Depending on whether the polynomial transform is sparse or dense, distinct tools are applied: combinatorial root-counting techniques such as Descartes' rule and Sturm's theorem for sparse cases, and analytic or algebraic methods like the argument principle, Rouche's theorem, and square-free factorization for dense ones. Together, these methods establish a systematic framework for studying the existence and classification of matrices with specific structural properties under polynomial transformations. The findings deepen understanding of spectral transformations and provide new perspectives for operator theory, dynamical systems, and computational linear algebra.

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arXiv2025

Spectral and Nilpotent Matrix Orderings: Comparison and Applications in Dynamic Systems

In our earlier work, we proposed the \emph{Spectral and Nilpotent Ordering} (SNO) as a new framework that extends matrix comparison beyond the Hermitian setting by incorporating both spectral and nilpotent structures. Building on that foundation, the present paper develops concrete certificates and applications of SNO. First, we employ generalized Gershgorin theorems to design certificates for spectral ordering that avoid direct eigenvalue computation and analyze their robustness under perturbations. Second, we introduce rank-based criteria that provide certificates for ordering the nilpotent parts of matrices without requiring a full Jordan decomposition. Finally, we apply the SNO framework to linear dynamical systems, where we construct a hierarchy of stability orderings that capture both asymptotic and transient behaviors. These contributions advance the theoretical underpinnings of SNO and demonstrate its potential as a versatile tool for operator analysis, computational methods, and stability studies in complex systems.

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arXiv2025

Generalized Multiple Operator Integrals and Perturbation Theory for Operators with Continuous Spectra

Operators with continuous spectra naturally arise in spectral theory, quantum mechanics, automorphic forms, and noncommutative geometry. However, analyzing such operators, particularly in the non-selfadjoint setting, remains challenging due to spectral instability and the lack of an orthonormal basis. This work advances the theory of Multiple Operator Integrals (MOIs) by developing a unified framework for generalized MOIs (GMOIs) associated with general (non-normal, non-selfadjoint) operators possessing continuous spectra. Building on prior work in Generalized Double Operator Integrals (GDOIs) and finite dimensional GMOIs, we extend the theory to include: the formulation of GMOIs in the continuous spectrum setting, their algebraic structure, continuity properties, norm and Lipschitz estimates, and a perturbation formula that generalizes classical results. As a key application, we derive a Krein-type spectral shift formula for GDOIs in the continuous spectrum setting and further extend it to arbitrary-order approximations. These contributions provide a foundation for broader developments in spectral theory, operator algebras, noncommutative geometry, and noncommutative analysis.

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