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Convergence of iterated Aluthge transform sequence for diagonalizable matrices II: $\lambda$-Aluthge transform
Jorge Antezana, Enrique Pujals, Demetrio Stojanoff
TL;DR
The paper studies convergence and parameter dependence of iterated λ-Aluthge transforms for diagonalizable matrices. Using a dynamical-systems and stable-manifold approach, it proves convergence, establishes regularity of the limiting map, and characterizes important cases where the limit is constant in λ.
Problem
The paper asks whether iterated λ-Aluthge transforms converge for diagonalizable matrices and when their limiting value is independent of λ.
Method
It analyzes λ-Aluthge transforms on similarity orbits using stable-manifold theory and derivative decompositions near the fixed unitary orbit.
Results
The iterates converge for every diagonalizable matrix; the limiting map has stated smoothness properties, and it is constant across S(D) when D has two eigenvalues of equal modulus.
Takeaways & Limitations
The results provide a dynamical-systems framework for understanding λ-Aluthge convergence and identify a broad spectral case with λ-independent limits.
Takeaways & Limitations
For diagonal D with more than two eigenvalues of the same modulus, the constancy question remains unresolved and is supported only by computational experiments and a conjecture.
Abstract
from arXiv · showhide
Let $\lambda \in (0,1)$ and let $T$ be a $r\times r$ complex matrix with polar decomposition $T=U|T|$. Then, the $\la$- Aluthge transform is defined by $$ \Delta_\lambda (T )= |T|^{\lambda} U |T |^{1-\lambda}. $$ Let $\Delta_\lambda^{n}(T)$ denote the n-times iterated Aluthge transform of $T$, $n\in\mathbb{N}$. We prove that the sequence $\{\Delta_\lambda^{n}(T)\}_{n\in\mathbb{N}}$ converges for every $r\times r$ {\bf diagonalizable} matrix $T$. We show regularity results for the two parameter map $(\la, T) \mapsto \alulit{\infty}{T}$, and we study for which matrices the map $(0,1)\ni \lambda \mapsto \Delta_\lambda^{\infty}(T)$ is constant.
1 Introduction.
The paper applies dynamical-systems and stable-manifold methods to λ-Aluthge transforms, proving convergence for diagonalizable matrices and establishing regularity and constancy results for the limiting map.
- Relation to prior work: The paper extends earlier convergence results for λ = 1/2 and complements prior results covering matrices whose eigenvalues have different moduli.It also develops the general λ case using a dynamical-systems approach rather than the approach used by Huajun Huang and Tin-Yau Tam.
- Dynamical-systems framework: The dynamical-systems approach studies the transform on a similarity orbit, where the unitary orbit U(D) is fixed by the Aluthge transform.For invertible T, the λ-Aluthge transform remains in T’s similarity orbit, motivating this restriction.
- Stable-manifold mechanism: Stable-manifold arguments show that points near U(D) converge exponentially to points of the orbit, after sufficiently many iterations enter an open neighborhood covered by stable manifolds.At each N ∈ U(D), the derivative has tangent and transversal invariant directions, with contraction in the transversal direction.
- Convergence results: The iterated λ-Aluthge transforms converge for every r × r diagonalizable matrix T, with limits in the unitary orbit U(D).Here D is a diagonal matrix similar to T; U(D) consists of the normal matrices in the corresponding similarity orbit.
- Regularity: The limiting map (λ, T) ↦ Δ∞_λ(T) is C∞ on (0, 1) × D*_r(C) and has additional regularity on (0, 1) × S(D).D*_r(C) denotes the open dense set of invertible matrices with r different eigenvalues, as described in the cited preceding work.
- Dependence on λ: The map R_T(λ) = Δ∞_λ(T) is generally nonconstant for diagonalizable matrices, but is constant throughout S(D) when D has exactly two eigenvalues of equal modulus.The paper conjectures this is the unique spectral configuration making R_T constant for every T in S(D).
2 Preliminaries.
The preliminaries establish the matrix, manifold, orbit, and λ-Aluthge-transform framework used to analyze iteration through stable-manifold methods. They also record continuity, smoothness, spectral, and diagonalizability properties of the transform.
- Stable manifold theorem: The stable manifold theorem provides invariant transversal manifolds near a pseudo-hyperbolic submanifold, with trajectories asymptotic to those on the invariant set.The setup uses invariant subbundles, contraction in the stable direction, and expansion along the complementary direction.
- Similarity orbit of a diagonal matrix: For a diagonal matrix D, its similarity orbit is a smooth manifold, while its compact unitary orbit consists of the normal elements and admits smooth local cross sections.The tangent space at N in the similarity orbit is described by commutators, with coordinates constrained by repeated diagonal entries.
- λ-Aluthge transforms: The λ-Aluthge transform is defined from the polar decomposition and is continuous on all matrices, while (λ,T) ↦ Δ_λ(T) is smooth on (0,1) × GL_r(C).For invertible T, the fractional power is expressed as |T|^λ=(T* T)^(λ/2)=exp((λ/2) log(T* T)).
- λ-Aluthge transforms: The transform is equivariant under unitary conjugation and respects direct sums, enabling orbitwise and blockwise analysis.These structural properties follow from the transform’s definition and are used alongside its regularity.
- λ-Aluthge transforms: Every limit point of an iterated λ-Aluthge sequence is normal and preserves the spectrum with algebraic multiplicities; fixed points are exactly the normal matrices.Diagonalizability is preserved by the λ-Aluthge transform.
3 Convergence
The convergence analysis reduces diagonalizable matrices to the invertible case and applies a stable-manifold decomposition around the unitary orbit of a diagonal matrix. This yields convergence of all iterates and smooth orbitwise limit maps.
- 3.2 Main Theorem: A smooth invariant pre-lamination of the similarity orbit consists of sheets W_N,λ converging exponentially toward N, with dist(Δ_λ^n(T),N)≤dist(T,N)ρ^n for any k_D<ρ<1.The associated projection p:W(D)→U(D) is C∞ and assigns each point to its limiting unitary-orbit point.
- 3.3 Convergence for fixed λ.: The stable-manifold argument applies the normal-limit result to enter the neighborhood W(D), where the smooth projection controls the eventual dynamics.The paper states that the λ-Aluthge transform yields the same convergence properties as the classical transform Δ_1/2.
- 3.3 Convergence for fixed λ.: For every λ ∈(0,1), the iterates Δ_λ^n(T) converge for every diagonalizable matrix T.The proof first reduces the possibly singular case to an invertible diagonalizable restriction after one iteration.
- 3.3 Convergence for fixed λ.: On the similarity orbit S(D), the iterates converge uniformly on compact sets to a C∞ map Δ_λ^∞:S(D)→U(D).The limit map is a C∞ retraction, and the similarity orbit is partitioned into smooth sheets associated with points of the unitary orbit.
4 Regularity properties of ∆∞
The paper establishes smooth dependence of the iterated λ-Aluthge limit on λ and matrices in a similarity orbit, including a smooth retraction onto the unitary orbit. It also characterizes important cases in which the limit is independent of λ.
- 4.1 On the orbit S(D).: The fixed points of the extended dynamical map are exactly UL(D), and the stable submanifolds vary smoothly with λ.The fibers W+_(λ,N)=(Δ∞)^−1(λ,N) are smooth submanifolds, representing sheets that move smoothly with λ.
- 4.1 On the orbit S(D).: Iterated transforms enter a neighborhood of the fixed-point orbit and converge there exponentially along stable manifolds.For points in the stable sheets, distances contract at least as ρ^n with ρ<1.
- 4.1 On the orbit S(D).: The map Δ∞:(0,1) × S(D) → U(D) is of class C∞ and acts as a smooth retraction onto the unitary orbit.The corresponding extended map on SL(D) is a C∞ retraction onto UL(D).
- 4.1 On the orbit S(D).: The smoothness results extend from invertible diagonal D to noninvertible diagonal matrices by reducing the dynamics to the invertible restriction on ker(T)⊥.The reduction uses smooth kernel projections and unitary conjugation.
- 4.4 The map λ ↦ Δ∞_λ(T) for fixed T.: The limit map RT(λ)=Δ∞_λ(T) is generally not constant, but it is constant for every T∈S(D) when σ(D)={d1,d2} with |d1|=|d2|.If D has two eigenvalues with distinct moduli, at least one matrix in its similarity orbit has a nonconstant RT; the paper conjectures the two-eigenvalue equal-modulus case is the unique universal case.
5 The proof of Theorem 3.2.1.
The proof analyzes the differential of the λ-Aluthge transform on tangent spaces of similarity orbits using Hadamard-product operators and spectral moduli. This yields smooth stable subspaces and the contraction structure needed for the dynamical convergence theorem.
- 5 The proof of Theorem 3.2.1.: The proof represents tangent vectors through Hadamard products and computes the differential using matrices whose entries depend on the eigenvalue moduli and λ.The operator H(λ) is decomposed into real and imaginary parts, with explicit entries involving |di| and |dj|.
- 5 The proof of Theorem 3.2.1.: The contraction estimate follows from bounding the relevant Hadamard operator, giving ∥A1_N(λ)∥<1 and making I−A1_N(λ) invertible.This invertibility permits the construction of E^s_N,λ as a graph over the contracting directions.
- 5 The proof of Theorem 3.2.1.: The tangent-space analysis separates Hermitian and anti-Hermitian components through the projections PRe and PIm.This decomposition is used to express the differential and the stable projection on T_D S(D).
- 5 The proof of Theorem 3.2.1.: The λ-Aluthge transform is C∞ on S(D), and each unitary-orbit point N has a stable subspace E^s_N,λ.The stable subspace is complementary to the tangent space of U(D), and the associated projection depends smoothly on (λ,N).
- 5 The proof of Theorem 3.2.1.: When all eigenvalues of D have the same modulus, the stable projection reduces to Q_N, so the stable subspace distribution is independent of λ.This is the differential-geometric mechanism used later to establish λ-independence in the equal-modulus two-eigenvalue case.