Geometric Mean in HPD GLT Sequences

• The paper demonstrates that if the GLT symbols commute, the geometric mean of HPD sequences is characterized by the explicit form (κξ)^(1/2).
• It employs regularization and functional calculus within an approximating-class-of-sequences framework to extend classical spectral analysis to structured HPD matrices.
• Numerical experiments confirm the theory in both scalar and block cases, underscoring applications from differential operators to quantum spin systems.

The geometric mean of Hermitian positive definite (HPD) matrix-sequences within the framework of Generalized Locally Toeplitz (GLT) algebras defines a powerful functional construction for analyzing the asymptotic spectral distribution of structured matrices. The operation’s behavior, well-posedness, and spectral properties are determined by the GLT symbols associated to the constituent sequences, with commutativity and degeneracy of the symbols central to the resulting theory. This concept receives a comprehensive treatment in contemporary spectral analysis, addressing classical conjectures and establishing maximal generality in the GLT setting.

1. Generalized Locally Toeplitz (GLT) Sequences and HPD Structure

A $d$-level, $r$-block GLT matrix-sequence $\{A_n\}_n$ consists of matrices $A_n \in \mathbb{C}^{d_n \times d_n}$ (with $d_n \to \infty$) associated to a "symbol" $$\kappa(x,\theta): [0,1]^d \times [-\pi,\pi]^d \to \mathbb{C}^{r\times r}$$ such that

$\{A_n\}_n \sim_{\mathrm{GLT}} \kappa(x,\theta),$

where $\sim_{\mathrm{GLT}}$ denotes convergence in singular-value or eigenvalue distribution. The GLT algebra is the smallest $*$-algebra containing all Toeplitz sequences $T_n(f)$ (generated by $f\in L^1$), diagonal sampling matrices $D_n(a)$ (for Riemann-integrable $a$), and zero-distributed sequences (with vanishing spectral measure). The map $\{A_n\} \mapsto \kappa$ respects addition, multiplication, the adjoint, and (on a.e. invertible symbols) inversion.

If each $A_n$ is Hermitian and positive definite, $\{A_n\}_n$ is an HPD matrix-sequence; its spectrum lies strictly in $(0,\infty)$ for all $n$.

2. Matrix and Sequence Geometric Means: Definitions and Canonical Formulation

For two HPD matrices $A,B\in\mathbb{C}^{m\times m}$, the Kubo–Ando (“ALM”) geometric mean is

$$G(A,B) := A^{1/2}\, (A^{-1/2} B A^{-1/2})^{1/2} A^{1/2}.$$

This operation is Hermitian, HPD, and symmetric: $G(A,B) = G(B,A)$. Extending to sequences, for $\{A_n\}_n$ and $\{B_n\}_n$ (HPD of the same size), one defines $\{G(A_n,B_n)\}_n$. The operation is well-defined due to invertibility of $A_n$ and the continuity of $t\mapsto t^{1/2}$ on the positive real axis.

In the context of GLT sequences, the geometric mean must be compatible with the underlying symbol calculus, especially regarding multiplicative structure when the symbols commute.

3. Main Result: Asymptotic Symbol of the Geometric Mean Sequence

The principal theorem established across (Barbarino et al., 2018, Ilyas et al., 6 May 2025), and (Khan, 9 Nov 2025) asserts that, for $\{A_n\}_n\sim_{\mathrm{GLT}}\kappa$ and $\{B_n\}_n\sim_{\mathrm{GLT}}\xi$ (HPD $d$-level, $r$-block GLT sequences), if the symbols $\kappa$ and $\xi$ commute almost everywhere on $[0,1]^d\times[-\pi,\pi]^d$, then the geometric mean sequence is again GLT with symbol

$$\{G(A_n,B_n)\}_n \sim_{\mathrm{GLT}} (\kappa\xi)^{1/2}.$$

No a.e. invertibility of $\kappa$ or $\xi$ is required under commutativity, and the result holds for scalar ($r=1$) and block ($r>1$) cases, resolving the conjecture of [Garoni–Serra-Capizzano] for all $d\ge 1$.

In the non-commuting case, the general symbolic form is

$$G(\kappa,\xi) := \kappa^{1/2} \bigl(\kappa^{-1/2}\xi\,\kappa^{-1/2}\bigr)^{1/2} \kappa^{1/2},$$

with existence and GLT membership subject to the invertibility of $\kappa$.

4. Proof Mechanisms, GLT Calculus, and Limit Processes

The proof combines core GLT algebraic axioms (GLT 2: inclusion of Toeplitz/diagonal, GLT 3: closure under operations, GLT 6: functional calculus) with an approximating-class-of-sequences (a.c.s.) strategy.

• A shift $\varepsilon > 0$ regularizes potential singularities: one analyses $A_{n,\varepsilon} = A_n+\varepsilon I$ and $\kappa_\varepsilon = \kappa+\varepsilon I$, ensuring a.e. invertibility.
• For each fixed $\varepsilon>0$, $$\{G(A_{n,\varepsilon},B_n)\}_n \sim_{\mathrm{GLT}} (\kappa_\varepsilon\xi)^{1/2}$$ by the invertible-commuting theory.
• GLT’s functional calculus (Axiom GLT 6) allows tracking of matrix functions at the level of symbols, so $$(A_{n,\varepsilon}B_n A_{n,\varepsilon})^{1/4} \sim_{\mathrm{GLT}} (\kappa_\varepsilon^2\xi^2)^{1/4} = (\kappa_\varepsilon\xi)^{1/2}$$, and $$G(A_{n,\varepsilon},B_n)-(A_{n,\varepsilon}B_n A_{n,\varepsilon})^{1/4}$$ is zero-distributed.
• Letting $\varepsilon \to 0$ and using the a.c.s. convergence theorem (GLT 4), one concludes $$\{G(A_n,B_n)\}_n \sim_{\mathrm{GLT}} (\kappa\xi)^{1/2}$$ without requiring invertibility of the original symbols.

This argument establishes that the spectral behavior of the geometric mean is inherited from the pointwise geometric mean of the symbols under commutativity.

5. Degenerate and Non-Commuting Symbol Cases

If either symbol is degenerate (vanishing on a positive-measure set) but both commute, the result persists: the geometric mean sequence has symbol $(\kappa\xi)^{1/2}$, even when zeros are present in $\kappa$ or $\xi$.

For non-commuting symbols, even if both are a.e. invertible, the simple formula $(\kappa\xi)^{1/2}$ is generally invalid; the correct symbolic description is the non-commutative geometric mean $G(\kappa,\xi)$. However, if both symbols are singular a.e. and non-commuting, numerical results suggest that the limiting spectral symbol may still exist but is not in general given by the pointwise operator mean $G(\kappa,\xi)$ and may only be described via an approximation by $\varepsilon$-regularization.

This suggests that the commutativity (and/or invertibility) requirement in the general symbolic result is maximal: closed-form spectral distribution for the geometric mean sequence exists only under these structural constraints.

6. Generalizations: The Karcher Mean and Multivariate Means

The framework extends to $k>2$ HPD GLT sequences $\{A_n^{(j)}\}_n \sim_{\mathrm{GLT}} \kappa_j$, $j=1,\ldots,k$, via the Karcher (Riemannian barycenter) mean: $$G(A_1,\ldots,A_k) = \arg\min_{X>0}\sum_{j=1}^k d^2(X, A_j),$$ where $d(\cdot,\cdot)$ is the Riemannian distance on the SPD manifold. The Karcher mean $X$ solves $\sum_j \log(A_j^{-1} X) = 0$. For commuting symbols, the mean sequence has symbol $(\prod_{j=1}^{k}\kappa_j)^{1/k}$; for general HPD symbols, numeric evidence supports $$\{G(A_n^{(1)},\ldots,A_n^{(k)})\}_n \sim_{\mathrm{GLT}} G(\kappa_1,\ldots,\kappa_k)$$, with suitable iterative approximation preserving GLT structure.

7. Applications and Numerical Evidence

Numerical experiments in both scalar and block settings ($d=1,2$, $r=1,2$, $k=2,3$) confirm theoretical predictions. Eigenvalues of $G(A_n,B_n)$ cluster along $(\kappa\xi)^{1/2}$ or the relevant Karcher mean, with convergence for moderate matrix sizes. Block cases display the expected multiplicity and branch structures. In degenerate cases (e.g., $a(x)f(\theta)$ with zeros), the symbol accurately captures the asymptotic spectral behavior, with eigenvalue decay rates matching orders of vanishing of $(\kappa\xi)^{1/2}$.

An application to mean-field quantum spin systems (Curie–Weiss model) demonstrates the relevance in mathematical physics. The sequence of Hamiltonians (restricted to the symmetric subspace) forms a tridiagonal GLT sequence with explicitly computable symbol: $$h_0^{\mathrm{CW}}(x,\theta) = -\frac{\Gamma}{2}(2x-1)^2 - 2B\cos\theta\sqrt{x(1-x)},$$ allowing precise analysis of spectral phenomena such as double-well splitting and ground-state near-degeneracy for finite system sizes.

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In summary, the geometric mean of HPD GLT sequences preserves GLT structure precisely when the symbols commute, yielding an explicit formula for the limiting spectral distribution. The operation is robust to symbol degeneracy under commutativity, whereas the non-commuting case admits only more nuanced or numerical descriptions. The theoretical apparatus admits generalization to $k$-fold geometric means (Karcher mean) and underpins spectral analysis in wide-ranging contexts, including discretizations of differential operators and quantum many-body Hamiltonians (Barbarino et al., 2018, Ilyas et al., 6 May 2025, Khan, 9 Nov 2025).