Generalized Sectorial Decomposition

• Generalized sectorial decomposition is a framework that partitions operators, matrices, flows, and integrals into sectors, making hidden spectral and asymptotic structures explicit.
• It employs techniques such as logarithmic functional calculus, congruence and Cholesky factorizations, and dyadic decompositions to handle non-selfadjoint and non-normal characteristics.
• This approach offers a unified method to simplify complex analyses in operator theory, matrix analysis, harmonic analysis, geometric topology, and dynamical systems.

Generalized sectorial decomposition denotes a family of constructions in which an operator, matrix, flow, symplectic manifold, or integral is partitioned by rays, sectors, or sectorial hypersurfaces so that spectral, asymptotic, or microlocal structure becomes explicit. In operator theory this language is tied to sectorial projections and logarithmic functional calculus; in matrix analysis it is tied to congruence decompositions and phase data; in harmonic analysis it is tied to Paley–Littlewood and Triebel–Lizorkin decompositions attached to sectorial operators; and in geometric and dynamical settings it appears in Liouville sectorial coverings and in higher-dimensional analogues of Poincaré’s planar sectorial decomposition (Grubb, 2011, Qiu et al., 2022, Kriegler et al., 2014, Dai, 20 Nov 2025, Alonso-González et al., 2023).

1. Elliptic operators, logarithms, and sectorial projections

For a classical elliptic pseudodifferential operator $P$ of order $m>0$ on a closed manifold, sectorial decomposition begins with two rays $e^{i\theta}\mathbb R_+$ and $e^{i\varphi}\mathbb R_+$ that separate the eigenvalues of the principal symbol $p_m(x,\xi)$. Under the hypotheses that these rays are rays of minimal growth and contain no eigenvalues of $P$, the sectorial projection is defined by the contour integral

$$\Pi_{\theta,\varphi}(P) = \frac{i}{2\pi}\int_{\Gamma_{\theta,\varphi}} \lambda^{-1}P(P-\lambda)^{-1}\,d\lambda.$$

The decisive structural identity is

$$\Pi_{\theta,\varphi}(P)=\frac{i}{2\pi}\bigl(\log_\theta P-\log_\varphi P\bigr),$$

which converts the projection problem into a comparison of two logarithms with different branch cuts (Grubb, 2011).

This formula resolves a flaw in earlier direct proofs. Booss–Bavnbek, Chen, Lesch, and Zhu had observed that one cannot in general modify the principal symbol $p_m(x,\xi)$ so that $(p_m(x,\xi)-\lambda)^{-1}$ is uniformly well defined along both rays for all $m>0$0. The logarithmic approach avoids that obstruction because in the construction of $m>0$1 one only modifies $m>0$2 near a single ray, as in Seeley’s theory of complex powers. The resulting symbol has the form

$m>0$3

so the logarithmic terms cancel in the difference, and $m>0$4 becomes a classical pseudodifferential operator of order $m>0$5 (Grubb, 2011).

At symbol level, the principal symbol of the projection is the matrix Riesz projection of $m>0$6 onto the eigenvalues in the open sector $m>0$7: $m>0$8 This gives the decomposition a microlocal meaning: the operator splits phase space according to sectorially separated branches of the principal symbol. The same framework formally extends to several rays $m>0$9, yielding projections associated with adjacent sectors and a decomposition of the identity into sectorial pieces. This suggests a generalized sectorial decomposition in the non-selfadjoint elliptic calculus, with logarithms replacing direct resolvent integration as the robust construction.

2. Matrix decompositions, phases, and semi-sectorial normal forms

In matrix analysis, sectorial decomposition is first a congruence normal form. If $e^{i\theta}\mathbb R_+$0 is sectorial, there exist a nonsingular matrix $e^{i\theta}\mathbb R_+$1 and a diagonal unitary matrix $e^{i\theta}\mathbb R_+$2 such that

$e^{i\theta}\mathbb R_+$3

The diagonal entries of $e^{i\theta}\mathbb R_+$4 determine the phases $e^{i\theta}\mathbb R_+$5, which coincide with the canonical angles of $e^{i\theta}\mathbb R_+$6. The unitary factor is unique up to permutation, and two canonical refinements of this decomposition are the symmetric polar decomposition

$e^{i\theta}\mathbb R_+$7

with $e^{i\theta}\mathbb R_+$8 and $e^{i\theta}\mathbb R_+$9 unitary, and the generalized Cholesky factorization

$e^{i\varphi}\mathbb R_+$0

with $e^{i\varphi}\mathbb R_+$1 upper triangular with positive diagonal and $e^{i\varphi}\mathbb R_+$2 unitary; both are unique (Wang et al., 2019).

The semi-sectorial theory extends this picture to singular matrices and to numerical ranges that may touch the origin. For a quasi-sectorial matrix $e^{i\varphi}\mathbb R_+$3 of rank $e^{i\varphi}\mathbb R_+$4,

$e^{i\varphi}\mathbb R_+$5

where $e^{i\varphi}\mathbb R_+$6 is sectorial. In the generic semi-sectorial case with nonempty interior, one has the generalized sectorial decomposition

$e^{i\varphi}\mathbb R_+$7

where $e^{i\varphi}\mathbb R_+$8 and $e^{i\varphi}\mathbb R_+$9 is a block diagonal matrix built from rotated Jordan blocks

$p_m(x,\xi)$0

The phases of $p_m(x,\xi)$1 are the angles from the $p_m(x,\xi)$2-block together with copies of the extremal boundary angles $p_m(x,\xi)$3. This is the canonical extension of sectorial phase data to semi-sectorial matrices (Qiu et al., 2022).

This decomposition is stable under several fundamental operations. For a semi-sectorial matrix $p_m(x,\xi)$4 of rank $p_m(x,\xi)$5, the phases of the Moore–Penrose inverse satisfy

$p_m(x,\xi)$6

Compressions and generalized Schur complements satisfy phase interlacing inequalities, and for quasi-sectorial $p_m(x,\xi)$7 and semi-sectorial $p_m(x,\xi)$8 there exists an isometry $p_m(x,\xi)$9 such that

$P$0

which yields the generalized matrix small phase theorem: $P$1 if and only if

$P$2

The theory also introduces essential phases via positive diagonal similarity and proves that for a strongly connected directed graph, the Laplacian becomes quasi-sectorial after the canonical scaling $P$3, where $P$4 is the positive left eigenvector satisfying $P$5 (Qiu et al., 2022).

A related inequality-oriented viewpoint treats a sectorial matrix as a positive real part plus a bounded-angle phase. Using the decomposition $P$6, one obtains sharp control of singular values, determinants, inverses, and operator means in terms of $P$7 and the sector angle $P$8 (Nasiri et al., 2020). Numerical-radius refinements use the same geometry: if $P$9 and $$\Pi_{\theta,\varphi}(P) = \frac{i}{2\pi}\int_{\Gamma_{\theta,\varphi}} \lambda^{-1}P(P-\lambda)^{-1}\,d\lambda.$$0, then

$$\Pi_{\theta,\varphi}(P) = \frac{i}{2\pi}\int_{\Gamma_{\theta,\varphi}} \lambda^{-1}P(P-\lambda)^{-1}\,d\lambda.$$1

and if $$\Pi_{\theta,\varphi}(P) = \frac{i}{2\pi}\int_{\Gamma_{\theta,\varphi}} \lambda^{-1}P(P-\lambda)^{-1}\,d\lambda.$$2 are double commuting sectorial matrices with sectorial index $$\Pi_{\theta,\varphi}(P) = \frac{i}{2\pi}\int_{\Gamma_{\theta,\varphi}} \lambda^{-1}P(P-\lambda)^{-1}\,d\lambda.$$3, then

$$\Pi_{\theta,\varphi}(P) = \frac{i}{2\pi}\int_{\Gamma_{\theta,\varphi}} \lambda^{-1}P(P-\lambda)^{-1}\,d\lambda.$$4

(Bhunia et al., 2022). These estimates quantify how decomposition by sector or phase controls non-normality.

3. Functional-calculus decompositions and abstract smoothness scales

For Banach-function-space operators, generalized sectorial decomposition takes the form of scale decompositions generated by holomorphic functional calculus. If $$\Pi_{\theta,\varphi}(P) = \frac{i}{2\pi}\int_{\Gamma_{\theta,\varphi}} \lambda^{-1}P(P-\lambda)^{-1}\,d\lambda.$$5 is $$\Pi_{\theta,\varphi}(P) = \frac{i}{2\pi}\int_{\Gamma_{\theta,\varphi}} \lambda^{-1}P(P-\lambda)^{-1}\,d\lambda.$$6-sectorial and $$\Pi_{\theta,\varphi}(P) = \frac{i}{2\pi}\int_{\Gamma_{\theta,\varphi}} \lambda^{-1}P(P-\lambda)^{-1}\,d\lambda.$$7, the generalized Triebel–Lizorkin norm is

$$\Pi_{\theta,\varphi}(P) = \frac{i}{2\pi}\int_{\Gamma_{\theta,\varphi}} \lambda^{-1}P(P-\lambda)^{-1}\,d\lambda.$$8

with discrete variants

$$\Pi_{\theta,\varphi}(P) = \frac{i}{2\pi}\int_{\Gamma_{\theta,\varphi}} \lambda^{-1}P(P-\lambda)^{-1}\,d\lambda.$$9

The resulting homogeneous and inhomogeneous spaces $$\Pi_{\theta,\varphi}(P)=\frac{i}{2\pi}\bigl(\log_\theta P-\log_\varphi P\bigr),$$0 are independent of the auxiliary cutoff $$\Pi_{\theta,\varphi}(P)=\frac{i}{2\pi}\bigl(\log_\theta P-\log_\varphi P\bigr),$$1, interpolate well, and the part of $$\Pi_{\theta,\varphi}(P)=\frac{i}{2\pi}\bigl(\log_\theta P-\log_\varphi P\bigr),$$2 in these spaces has a bounded $$\Pi_{\theta,\varphi}(P)=\frac{i}{2\pi}\bigl(\log_\theta P-\log_\varphi P\bigr),$$3-functional calculus (Kunstmann et al., 2012).

When $$\Pi_{\theta,\varphi}(P)=\frac{i}{2\pi}\bigl(\log_\theta P-\log_\varphi P\bigr),$$4 on $$\Pi_{\theta,\varphi}(P)=\frac{i}{2\pi}\bigl(\log_\theta P-\log_\varphi P\bigr),$$5, this abstract construction reproduces the classical spaces: $$\Pi_{\theta,\varphi}(P)=\frac{i}{2\pi}\bigl(\log_\theta P-\log_\varphi P\bigr),$$6 In this sense, sectorial calculus replaces Fourier decomposition by a holomorphic spectral decomposition adapted to $$\Pi_{\theta,\varphi}(P)=\frac{i}{2\pi}\bigl(\log_\theta P-\log_\varphi P\bigr),$$7 (Kunstmann et al., 2012).

A complementary dyadic version is the Paley–Littlewood decomposition for $$\Pi_{\theta,\varphi}(P)=\frac{i}{2\pi}\bigl(\log_\theta P-\log_\varphi P\bigr),$$8-sectorial operators with bounded $$\Pi_{\theta,\varphi}(P)=\frac{i}{2\pi}\bigl(\log_\theta P-\log_\varphi P\bigr),$$9-calculus. For a homogeneous dyadic partition $p_m(x,\xi)$0, the series

$p_m(x,\xi)$1

converges unconditionally, and the fractional-domain norms satisfy

$p_m(x,\xi)$2

Real interpolation of these fractional-domain scales yields abstract Besov spaces $p_m(x,\xi)$3 and $p_m(x,\xi)$4, while variants for bisectorial operators and strip-type generators extend the same principle beyond single sectors (Kriegler et al., 2014).

The unifying point is that the dyadic pieces $p_m(x,\xi)$5 or the continuous pieces $p_m(x,\xi)$6 function as sectorially localized frequency bands. This is a genuine decomposition of regularity by the spectrum of a sectorial operator rather than by Euclidean frequency.

4. L-systems, non-selfadjoint spectral expansions, and fractional evolution

In the theory of L-systems, sectorial decomposition is encoded by the relation between a dissipative or accumulative main operator $p_m(x,\xi)$7, a state-space $p_m(x,\xi)$8-extension $p_m(x,\xi)$9, and the impedance function

$(p_m(x,\xi)-\lambda)^{-1}$0

For minimal L-systems with nonnegative symmetric part, the classes $(p_m(x,\xi)-\lambda)^{-1}$1, $(p_m(x,\xi)-\lambda)^{-1}$2, $(p_m(x,\xi)-\lambda)^{-1}$3, and $(p_m(x,\xi)-\lambda)^{-1}$4 of sectorial Stieltjes and inverse Stieltjes functions govern the exact angle of sectoriality of $(p_m(x,\xi)-\lambda)^{-1}$5 and of $(p_m(x,\xi)-\lambda)^{-1}$6 or the associated operator $(p_m(x,\xi)-\lambda)^{-1}$7. In particular, if $(p_m(x,\xi)-\lambda)^{-1}$8 or $(p_m(x,\xi)-\lambda)^{-1}$9, then the main and state-space operators have the same exact angle $m>0$00; in the general two-parameter case,

$m>0$01

For one-dimensional Schrödinger operators on the half-line, the theory makes these relations explicit in terms of the Weyl–Titchmarsh function $m>0$02 and the boundary parameter $m>0$03 (Belyi et al., 2017).

A different spectral use of sectoriality appears in the monograph on fractional calculus and Abel–Lidskii theory. There the central object is a sectorial operator $m>0$04 with compact resolvent and compact inverse $m>0$05. The monograph formulates sufficient conditions for the Abel–Lidskii basis property, strengthens the conditions regarding the semi-angle of the sector, weakens the conditions regarding the involved parameters, introduces the Schatten-von Neumann class of the convergence exponent, and develops a method of contour integration based on a sequence of contours of the power type (Kukushkin, 2024).

The outcome is a generalized root-vector decomposition for non-selfadjoint sectorial operators. The Hilbert space is expanded in Jordan chains of root vectors and biorthogonal vectors, and contour integrals defining $m>0$06 produce Abel–Lidskii expansions of the form

$m>0$07

These expansions are then used for evolution equations in abstract Hilbert space and for operators arising from fractional derivatives, including Kipriyanov-type constructions (Kukushkin, 2024). In this setting, generalized sectorial decomposition is not a projection calculus but a regularized decomposition of the space along root subspaces of a non-normal sectorial operator.

5. Liouville sectors and symmetric products of surfaces

In symplectic topology, sectorial decomposition refers to covers by Liouville sectors whose boundaries form a sectorial collection of hypersurfaces. For a Riemann surface $m>0$08 with a quadratic Stein structure, cutting $m>0$09 along stable arcs of saddle points produces a decomposition that lifts to the second symmetric product. The central result is that there exists a collection of smooth hypersurfaces $m>0$10 and a decomposition

$m>0$11

where each $m>0$12 is a Liouville sector with corners, obtained as the closure of the stable manifold of the critical point $m>0$13 of a Stein potential $m>0$14 on $m>0$15 (Dai, 20 Nov 2025).

Locally, the model is built on $m>0$16, where stable manifolds of asymptotic regions define hypersurfaces $m>0$17 and $m>0$18, and one gets a sectorial decomposition

$m>0$19

Globally, the functions $m>0$20 required in the definition of sectorial coverings are produced by transporting $m>0$21 along the Liouville flow; the resulting hypersurfaces satisfy the orthogonality and Poisson-commutation conditions that define a sectorial collection (Dai, 20 Nov 2025).

This construction has categorical consequences. Sectorial descent reconstructs the wrapped Fukaya category of $m>0$22 from the categories of the pieces $m>0$23 and their overlaps. Applied to $m>0$24, the decomposition of $m>0$25 into $m>0$26, $m>0$27, and $m>0$28 yields a new geometric proof of Homological Mirror Symmetry for the complex two-dimensional pair of pants (Dai, 20 Nov 2025). Here generalized sectorial decomposition is a geometric gluing formalism rather than a spectral splitting.

6. Dynamical stratifications and adjacent algorithmic usages

For analytic vector fields in $m>0$29 with an isolated singularity at the origin, a reduction of singularities $m>0$30 produces a three-dimensional analogue of Poincaré’s planar sectorial decomposition. Under the hypotheses that the induced foliated manifold $m>0$31 is spherical, non-dicritical, hyperbolic, acyclic, non s-resonant, and of Morse–Smale type, there exists a fundamental system of neighborhoods $m>0$32 of the divisor $m>0$33 such that each $m>0$34 admits an analytic $m>0$35-stratification into elementary dynamical pieces. These pieces are classified as elliptic, hyperbolic, parabolic attracting, or parabolic repelling according to their $m>0$36- and $m>0$37-limit sets, and they occur in dimensions $m>0$38 and $m>0$39 (Alonso-González et al., 2023).

The construction uses a graph $m>0$40 on the divisor, fitting domains with carefully controlled frontier types, and two classes of invariant two-dimensional surfaces: fixed separating surfaces and mobile separating surfaces. Their mutual disjointness and their traces on $m>0$41 provide the walls of the stratification, while the three-dimensional complementary components play the role of higher-dimensional sectors. The result is explicitly presented as a generalization to dimension three of Poincaré’s decomposition of planar flows into parabolic, elliptic, and hyperbolic invariant sectors (Alonso-González et al., 2023).

A distinct but terminologically adjacent usage occurs in perturbative quantum field theory. There, sector decomposition means subdivision of the Feynman-parameter domain so that infrared singularities factor into monomials. A non-iterative version recasts the factorization step as a problem in computational geometry: exponent vectors of the relevant polynomials define convex polyhedral cones

$m>0$42

which are then triangulated to produce a finite set of sectors with monomial changes of variables. This method is deterministic, avoids infinite loops, and in the examples reported produces fewer sectors than several iterative strategies (Kaneko et al., 2010). Although this is sector decomposition rather than sectoriality in the operator-theoretic sense, it belongs to the broader history of decompositions organized by asymptotic cones and sectors.

Across these settings, generalized sectorial decomposition has no single invariant definition. Its stable content is structural: a complicated object is cut into sectorially organized pieces—by logarithmic branch cuts, congruence phases, dyadic functional-calculus bands, Liouville hypersurfaces, invariant separating surfaces, or asymptotic cones—so that non-selfadjoint, non-normal, or singular behavior becomes tractable.