Abstract Cone Operators

• Abstract cone operators are constructions that encode cone geometry and singularity properties using differential, matrix, and order-theoretic frameworks.
• They integrate methodologies from microlocal analysis, operator systems, and scaling-invariant extension theory to address problems in geometric PDEs and boundary asymptotics.
• Applications include maximal Lq-regularity in PDEs, refined classifications in quantum information, and the development of index-theoretic rigidity theorems on manifolds with conical singularities.

Searching arXiv for the cited works and closely related papers. arXiv search: "Abstract cone operators" Abstract cone operators are a family of constructions in which cone geometry is encoded by linear, unbounded, matrix-ordered, or order-theoretic operators. In current literature, the expression is used for several related but nonidentical objects: cone differential operators on manifolds with conical singularities, scaling-invariant indicial operators on generalized tangent cones, abstract operator-system realizations of convex cones and linear matrix inequalities, linear Scott-continuous maps between complete positive cones, set-valued residuation operators built from upper and lower cones in posets, and geometric cone operators on currents and forms. A common theme is that a classical cone or conic singularity is replaced by a structured operator framework whose domain, positivity, asymptotics, or cohomological action can be analyzed abstractly (Schrohe et al., 2017, Krainer, 2021, Berger et al., 2023, Ehrhard, 2020, Cecchini et al., 20 May 2025).

1. Terminological scope and recurring structures

The term has no single universally standardized meaning. In microlocal analysis and geometric PDE, a cone operator is typically a differential or first-order operator with radial-singular form. In noncommutative convexity, the relevant object is an operator system over a classical cone, with matrix levels controlled by complete positivity. In semantics and order theory, operators are maps or set-valued constructions defined directly on positive cones or posets.

Context	Underlying object	Representative form
Cone analysis	differential operator near a cone tip	$A=t^{-\mu}\sum_{k=0}^{\mu} a_k(t)(-t\partial_t)^k$
Scaling-invariant extension theory	symmetric operator on $L^2(\mathbb{R}_+;E_0)$	$A^\circ=x^{-m}\sum_{j=0}^p a_j(xD_x)^j$
Singular Dirac theory	abstract Hilbert-space cone operator	$\partial_r+\frac1r(S_0+S_1(r))$
Operator systems and LMIs	matrix-ordered cone at all levels	free spectrahedra, CP maps, complete order isomorphisms
Cone semantics	linear Scott-continuous map	$f:P\to Q$ in $\mathbf{CLin}$
Residuated posets	set-valued cone operators	$M(x,y),R(x,y)$ built from $L$ and $U$

This multiplicity is not merely terminological. It reflects several mathematically distinct ways of replacing a cone by an operator-theoretic object: through radial-singular differential structure, through matrix convexity, through order-enriched categorical structure, or through cone-valued boundary data. A plausible implication is that “abstract cone operator” functions less as a single definition than as a pattern for transferring cone geometry into operator form.

2. Matrix-ordered and operator-system formulations

In the operator-system approach, an abstract operator system on a finite-dimensional complex $*$-vector space $L^2(\mathbb{R}_+;E_0)$0 is a sequence of proper cones $L^2(\mathbb{R}_+;E_0)$1 closed under conjugation by matrices, often with a distinguished order unit. By the Choi–Effros theorem, such a system admits a concrete realization by a $L^2(\mathbb{R}_+;E_0)$2-linear unital map $L^2(\mathbb{R}_+;E_0)$3, and in the finite-dimensional case one obtains free spectrahedra defined by Hermitian pencils. Containment and equivalence of such models are governed by completely positive maps and unital complete order isomorphisms (Berger et al., 2023).

For a proper convex cone $L^2(\mathbb{R}_+;E_0)$4, the smallest and largest operator systems over $L^2(\mathbb{R}_+;E_0)$5 are

$L^2(\mathbb{R}_+;E_0)$6

and

$L^2(\mathbb{R}_+;E_0)$7

Free duality interchanges these extremal systems. If $L^2(\mathbb{R}_+;E_0)$8 and $L^2(\mathbb{R}_+;E_0)$9 are free spectrahedral pencils, then

$A^\circ=x^{-m}\sum_{j=0}^p a_j(xD_x)^j$0

for all $A^\circ=x^{-m}\sum_{j=0}^p a_j(xD_x)^j$1, so inclusion is equivalent to a CP Kraus factorization by sums of compressions. Equality is characterized by mutual dominance in this sense (Berger et al., 2023).

This framework yields precise classification results. For polyhedral cones, $A^\circ=x^{-m}\sum_{j=0}^p a_j(xD_x)^j$2 is a free spectrahedron if and only if $A^\circ=x^{-m}\sum_{j=0}^p a_j(xD_x)^j$3 is polyhedral, while $A^\circ=x^{-m}\sum_{j=0}^p a_j(xD_x)^j$4 is a free spectrahedron if and only if $A^\circ=x^{-m}\sum_{j=0}^p a_j(xD_x)^j$5 is a simplex cone. Every polyhedral cone therefore has a weakest LMI model, namely the diagonal pencil determined by its facet-defining linear forms. For the $A^\circ=x^{-m}\sum_{j=0}^p a_j(xD_x)^j$6-dimensional Lorentz cone

$A^\circ=x^{-m}\sum_{j=0}^p a_j(xD_x)^j$7

the canonical $A^\circ=x^{-m}\sum_{j=0}^p a_j(xD_x)^j$8 pencil

$A^\circ=x^{-m}\sum_{j=0}^p a_j(xD_x)^j$9

defines the cone, and every defining pencil is obtained from it by sums of compressions. Equality $\partial_r+\frac1r(S_0+S_1(r))$0 holds exactly when

$\partial_r+\frac1r(S_0+S_1(r))$1

for some $\partial_r+\frac1r(S_0+S_1(r))$2 (Berger et al., 2023).

Over the PSD cone, the operator-system viewpoint recovers familiar quantum-information classes. The systems $\partial_r+\frac1r(S_0+S_1(r))$3, $\partial_r+\frac1r(S_0+S_1(r))$4, $\partial_r+\frac1r(S_0+S_1(r))$5, $\partial_r+\frac1r(S_0+S_1(r))$6, $\partial_r+\frac1r(S_0+S_1(r))$7, and $\partial_r+\frac1r(S_0+S_1(r))$8 correspond respectively to entanglement-breaking, completely positive, completely copositive, doubly completely positive, decomposable, and positive maps. Their finitary behavior is sharply separated: $\partial_r+\frac1r(S_0+S_1(r))$9 and $f:P\to Q$0 are finitely generated and finite-dimensional realizable; $f:P\to Q$1 is finite-dimensional realizable but not finitely generated; $f:P\to Q$2 is finitely generated but not finite-dimensional realizable; and $f:P\to Q$3 is neither (Berger et al., 2021). Closely related finite-dimensional results identify mapping cones on $f:P\to Q$4 with super-homogeneous operator systems on $f:P\to Q$5, uniquely determined by their Choi cones and characterized by invariance under pre- and post-composition by CP maps (Johnston et al., 2011).

3. Cone differential operators and parameter-elliptic analysis

In the analytic theory of manifolds with conical singularities, a cone differential operator of order $f:P\to Q$6 has the collar form

$f:P\to Q$7

with $f:P\to Q$8. Freezing coefficients at $f:P\to Q$9 gives the model operator

$\mathbf{CLin}$0

on the open infinite cone $\mathbf{CLin}$1 (Schrohe et al., 2017).

Three symbolic objects control the operator: the interior principal symbol $\mathbf{CLin}$2, the rescaled principal symbol $\mathbf{CLin}$3, and the conormal Mellin symbol

$\mathbf{CLin}$4

Parameter-ellipticity with respect to a sector $\mathbf{CLin}$5 consists of three conditions: invertibility of $\mathbf{CLin}$6 and $\mathbf{CLin}$7, invertibility of $\mathbf{CLin}$8 on the two critical lines

$\mathbf{CLin}$9

and minimal growth of the model operator $M(x,y),R(x,y)$0 on the weighted $M(x,y),R(x,y)$1 cone space (Schrohe et al., 2017).

The natural functional setting is the Mellin-Sobolev scale $M(x,y),R(x,y)$2 and the corresponding weighted spaces $M(x,y),R(x,y)$3. Minimal and maximal domains differ by a finite-dimensional asymptotics space spanned by terms of the form

$M(x,y),R(x,y)$4

where $M(x,y),R(x,y)$5 runs over poles of $M(x,y),R(x,y)$6 in the relevant strip. Closed extensions are therefore

$M(x,y),R(x,y)$7

and the same pattern holds for the model operator. The residue calculus for the Mellin symbol produces the asymptotics spaces and a bijection between the closed extensions of $M(x,y),R(x,y)$8 and those of $M(x,y),R(x,y)$9 (Schrohe et al., 2017).

Under parameter-ellipticity, the resolvent belongs to the parameter-dependent cone calculus:

$L$0

where $L$1 is a cone pseudodifferential family and $L$2 is a Green family. This yields sectorial estimates

$L$3

for $L$4 large, and hence a bounded $L$5-calculus for $L$6 on $L$7. The Laplace–Beltrami operator on a warped conical metric and the porous medium equation on warped cones are the main applications. In particular, for the Laplacian with a canonical asymptotics choice, $L$8 has a bounded $L$9-calculus, and this supplies maximal $U$0-regularity for the quasilinear porous medium problem (Schrohe et al., 2017).

4. Scaling-invariant extension theory and indicial operators

A complementary abstraction starts from a separable Hilbert space $U$1 equipped with a unitary, strongly continuous dilation action $U$2, and a densely defined symmetric operator satisfying the scaling law

$U$3

For the quotient $U$4, the induced action has infinitesimal generator $U$5, and the adjoint pairing

$U$6

descends to a nondegenerate Hermitian form on $U$7 (Krainer, 2021).

The spectrum of $U$8 is symmetric with respect to reflection across the line $U$9, and $*$0 decomposes into generalized eigenspaces. The resulting Canonical Form Theorem gives a Green formula, Jordan-type normal forms for the boundary pairing, and the signature formula

$*$1

A symmetric operator has selfadjoint extensions if and only if this total sum is zero, and invariant selfadjoint extensions exist under the corresponding pointwise condition on the critical line (Krainer, 2021).

In the semibounded case, the Friedrichs and Krein extensions are invariant. Under the sign condition, the critical generalized eigenspaces admit canonical invariant Lagrangian subspaces $*$2, and these determine the boundary quotients of the Friedrichs and Krein domains. The order relation between invariant selfadjoint extensions can then be expressed entirely in terms of inclusions between the corresponding boundary subspaces (Krainer, 2021).

For Fuchs-type differential operators

$*$3

on $*$4, with holomorphic indicial family

$*$5

the abstract extension theory becomes a Mellin-symbol calculus. The maximal domain splits as

$*$6

and every element has a finite polyhomogeneous expansion

$*$7

The boundary pairing admits the residue formula

$*$8

and the signature equals the spectral flow of the indicial family along $*$9 (Krainer, 2021).

5. Positive cones, linear logic, and residuated order structures

In the semantics of probabilistic higher-order computation, a complete positive cone is a cancellative $L^2(\mathbb{R}_+;E_0)$00-semimodule equipped with a norm whose unit ball has least upper bounds for bounded nondecreasing $L^2(\mathbb{R}_+;E_0)$01-chains. The primitive abstract cone operators are then the linear Scott-continuous maps $L^2(\mathbb{R}_+;E_0)$02. They form the internal hom

$L^2(\mathbb{R}_+;E_0)$03

with operator norm

$L^2(\mathbb{R}_+;E_0)$04

and these operators organize into the category $L^2(\mathbb{R}_+;E_0)$05 (Ehrhard, 2020).

The category has products and equalizers, and the Special Adjoint Functor Theorem yields a tensor product $L^2(\mathbb{R}_+;E_0)$06 left adjoint to $L^2(\mathbb{R}_+;E_0)$07. Bilinear Scott-continuous maps are classified by the adjunction

$L^2(\mathbb{R}_+;E_0)$08

With this tensor, internal hom, and evaluation, $L^2(\mathbb{R}_+;E_0)$09 is symmetric monoidal closed. A linear exponential comonad $L^2(\mathbb{R}_+;E_0)$10 is characterized by

$L^2(\mathbb{R}_+;E_0)$11

where $L^2(\mathbb{R}_+;E_0)$12 consists of bounded, totally monotone, Scott-continuous stable maps. Probabilistic coherence spaces embed fully faithfully and densely into this setting, so the cone calculus extends the earlier PCS tensor and exponential to general cones (Ehrhard, 2020).

A different order-theoretic use of cone operators appears in complemented posets. For a poset $L^2(\mathbb{R}_+;E_0)$13, the upper and lower cone operators are

$L^2(\mathbb{R}_+;E_0)$14

An operator left residuated poset is a bounded poset with unary operation and set-valued operators $L^2(\mathbb{R}_+;E_0)$15 satisfying

$L^2(\mathbb{R}_+;E_0)$16

$L^2(\mathbb{R}_+;E_0)$17

In a bounded relatively pseudocomplemented poset one may take

$L^2(\mathbb{R}_+;E_0)$18

and obtain an operator residuated poset satisfying divisibility. In a Boolean poset,

$L^2(\mathbb{R}_+;E_0)$19

again yielding divisibility. For pseudo-orthomodular posets, the operators

$L^2(\mathbb{R}_+;E_0)$20

define an operator left residuated structure (Chajda et al., 2018).

6. Geometric cone operators and singular Dirac applications

A recent functional-analytic formulation defines an abstract cone operator as a closable, densely defined unbounded operator

$L^2(\mathbb{R}_+;E_0)$21

between separable Hilbert spaces with orthogonal decompositions

$L^2(\mathbb{R}_+;E_0)$22

where the cone summands are identified with $L^2(\mathbb{R}_+;E_0)$23, and on compactly supported smooth cone sections

$L^2(\mathbb{R}_+;E_0)$24

Here $L^2(\mathbb{R}_+;E_0)$25 is an essentially self-adjoint link operator, $L^2(\mathbb{R}_+;E_0)$26 is a measurable bounded perturbation, and the crucial spectral-gap hypothesis is

$L^2(\mathbb{R}_+;E_0)$27

Under axioms (AC0)–(AC6), the cone domain is

$L^2(\mathbb{R}_+;E_0)$28

with graph norm equivalent to the weighted first-order norm involving $L^2(\mathbb{R}_+;E_0)$29 and $L^2(\mathbb{R}_+;E_0)$30. Explicit parametrices built from Hilbert–Schmidt kernels $L^2(\mathbb{R}_+;E_0)$31 and $L^2(\mathbb{R}_+;E_0)$32 then yield Fredholmness (Cecchini et al., 20 May 2025).

This abstraction is applied to twisted Dirac operators on spherical suspensions and on manifolds with cone-like singularities. Near a cone point, the twisted Dirac operator takes the form

$L^2(\mathbb{R}_+;E_0)$33

with $L^2(\mathbb{R}_+;E_0)$34. The integrated Schrödinger–Lichnerowicz formula extends to these singular settings, and the index is

$L^2(\mathbb{R}_+;E_0)$35

This analytic package yields Llarull-type rigidity theorems: for odd-dimensional closed spin manifolds with $L^2(\mathbb{R}_+;E_0)$36 metrics and scalar curvature lower bound, a nonzero-degree area non-increasing Lipschitz map to the sphere is a metric isometry; for cone-like singular spin manifolds with scalar curvature at least $L^2(\mathbb{R}_+;E_0)$37, the corresponding map is a smooth Riemannian isometry onto a punctured sphere (Cecchini et al., 20 May 2025).

A distinct geometric usage appears in projective geometry, where the cone operator $L^2(\mathbb{R}_+;E_0)$38 is a degree $L^2(\mathbb{R}_+;E_0)$39 linear operator on currents or forms obtained by double coning to linear centers in projective space. After averaging, it is cohomological and induces on cohomology the inverse Lefschetz operator:

$L^2(\mathbb{R}_+;E_0)$40

Equivalently, it satisfies the Lefschetz commutator relations on cohomology, so it realizes the lowering operator in the $L^2(\mathbb{R}_+;E_0)$41-structure determined by the Kähler form (Wang, 2018).

Taken together, these literatures show that abstract cone operators serve as a transfer mechanism from cone geometry to operator theory. Depending on context, the cone may encode radial singularity, matrix positivity, boundary asymptotics, logical resource structure, or cohomological lowering. The resulting operators classify LMIs, control asymptotic domains, determine selfadjoint extensions, furnish monoidal closed structures, implement residuation, or support index-theoretic rigidity theorems. The diversity of the term is therefore substantive rather than accidental: each usage isolates a different way in which a cone can be made analytically or categorically operative.