Entrywise Positivity Preservers on Green Matrices

Abstract: We classify the entrywise functions that preserve positive semidefiniteness on discrete Green matrices (G(p,q)=(p_{\min(i,j)}q_{\max(i,j)})) with positive parameters, without requiring the resulting matrix to retain Green structure. For matrices of all orders, the preservers are the zero function and the functions (f(t)=\int_{[0,\infty)}t^α\,dμ(α)), where (μ) is a nonzero finite positive measure and the integral is finite for every (t>0). Requiring the resulting matrix to be totally nonnegative reduces the nonzero preservers to (f(t)=ct^α), where (c>0) and (α\ge0). These power functions also preserve positive semidefinite Green structure, while strict Green structure is preserved precisely when (α>0). No regularity assumption is needed for these classifications. We also characterize continuously differentiable functions that are entrywise Loewner monotone on every fixed-(q) Green family: this holds precisely when (f') is a positive mixture of nonnegative real powers, with the zero measure allowed.

• The paper proves that PSD-preserving functions are exactly zero or finite positive mixtures of nonnegative powers, f(t)=∫t^αdμ(α), with rank-one and 2×2 tests sufficient to force the classification.
• It shows that total-nonnegativity preservation is far more rigid than PSD preservation, allowing only monomials ct^α, while strict Green inputs produce positive-definite outputs exactly when μ assigns positive mass to (0,∞).
• It classifies fixed-q Loewner preservers through integrated power mixtures and characterizes Green-structure preservers as c exp(φ(log t)), which reduces to ct^α under mild regularity assumptions.

Overview and setting

This paper classifies the functions $f:(0,\infty)\to\mathbb{R}$ that act entrywise, $f[A] = (f(a_{ij}))$, on the cone of positive-parameter discrete Green matrices $G(p,q)$ with entries $g_{ij} = p_{\min(i,j)}q_{\max(i,j)}$, $p_i, q_i > 0$. Green matrices of this form arise as discrete Green kernels of second-order boundary-value problems and as inverses of irreducible tridiagonal $Z$-matrices. The paper distinguishes sharply between several output requirements: positive semidefiniteness (PSD) alone, total nonnegativity (TN), positive definiteness on strict Green inputs, Loewner monotonicity on fixed-parameter families, and preservation of Green structure itself. Except for the Loewner-monotonicity problem, no regularity is assumed on $f$ a priori; positivity, monotonicity, and continuity are derived as consequences of the preservation property.

The key structural tool is the factorization $$G(p,q) = D_q L\,\mathrm{diag}(\Delta_1,\dots,\Delta_n)\,L^T D_q$$, where $D_q$ is diagonal, $L$ is the lower-triangular summation matrix, and $f[A] = (f(a_{ij}))$0 are the increments of the ratio sequence $f[A] = (f(a_{ij}))$1 with $f[A] = (f(a_{ij}))$2. This yields the criteria $f[A] = (f(a_{ij}))$3 and $f[A] = (f(a_{ij}))$4; combined with Pinkus's criterion, PSD and TN coincide on the input Green class.

PSD preservers: a Laplace-transform representation

The central theorem states that $f[A] = (f(a_{ij}))$5 preserves PSD on the Green class at all orders if and only if either $f[A] = (f(a_{ij}))$6 or

$f[A] = (f(a_{ij}))$7

for a nonzero finite positive Borel measure $f[A] = (f(a_{ij}))$8 with finite integral at every $f[A] = (f(a_{ij}))$9. The representing measure is unique. The proof proceeds in two steps. First, testing on positive rank-one matrices $G(p,q)$0 (which are Green with $G(p,q)$1) forces $G(p,q)$2 to be a continuous PSD Hankel kernel on $G(p,q)$3, so the Hankel kernel representation of Belton–Guillot–Khare–Putinar gives a bilateral Laplace representation $G(p,q)$4 with $G(p,q)$5 finite. Second, a two-by-two test matrix of the form $G(p,q)$6 with $G(p,q)$7 forces $G(p,q)$8 to be nondecreasing; a support lemma then shows that a finite measure whose bilateral Laplace transform is nondecreasing must be supported on $G(p,q)$9. Conversely, any such measure works because $g_{ij} = p_{\min(i,j)}q_{\max(i,j)}$0 is PSD for $g_{ij} = p_{\min(i,j)}q_{\max(i,j)}$1 and $g_{ij} = p_{\min(i,j)}q_{\max(i,j)}$2 for $g_{ij} = p_{\min(i,j)}q_{\max(i,j)}$3, so $g_{ij} = p_{\min(i,j)}q_{\max(i,j)}$4 is an integral of nonnegative quadratic forms.

A notable refinement is a reduced test family: the full classification is already forced by (i) all positive rank-one matrices and (ii) the two-by-two matrices above. Thus the entire preserver class is determined by tests of order at most two, and every nonzero preserver is automatically positive, nondecreasing, continuous, and multiplicatively log-convex, $g_{ij} = p_{\min(i,j)}q_{\max(i,j)}$5.

Loewner monotonicity on fixed-$g_{ij} = p_{\min(i,j)}q_{\max(i,j)}$6 fibers

Fixing $g_{ij} = p_{\min(i,j)}q_{\max(i,j)}$7, the family $g_{ij} = p_{\min(i,j)}q_{\max(i,j)}$8 is convex — in contrast, the paper shows $g_{ij} = p_{\min(i,j)}q_{\max(i,j)}$9 itself is not convex, via an explicit sum of two rank-one Green matrices that violates the Green multiplicative identity $p_i, q_i > 0$0. For $p_i, q_i > 0$1, Loewner monotonicity $p_i, q_i > 0$2 within every fixed-$p_i, q_i > 0$3 family holds if and only if

$p_i, q_i > 0$4

with $p_i, q_i > 0$5 a finite positive Borel measure, the zero measure allowed. Necessity follows by differentiating along the direction $p_i, q_i > 0$6 and applying the PSD classification to $p_i, q_i > 0$7; sufficiency follows from convexity of the fiber and a Schur-product integral argument. The paper leaves open whether the same classification holds when $p_i, q_i > 0$8 and $p_i, q_i > 0$9 have different $Z$0-sequences.

Rigidity of totally nonnegative output

Requiring TN output collapses the preservers dramatically: the only possibilities are $Z$1 and $Z$2 with $Z$3, $Z$4. The proof is a clean rigidity argument: TN of a specific nonprincipal minor of an order-three strict Green test matrix yields the reverse Cauchy–Schwarz inequality $Z$5 for the normalized Laplace transform $Z$6, while Cauchy–Schwarz itself gives $Z$7; equality forces the representing measure to be a Dirac mass. The paper emphasizes the contrast: although PSD and TN coincide on Green inputs, the output problems differ — the function $Z$8 (measure $Z$9) preserves PSD on the Green class but maps the strict Green matrix $f$0 to a matrix with a negative minor. This is a concrete disanalogy with the full-PSD-cone setting.

Positive definite output and structure preservation

For strict Green inputs (strictly increasing ratios, hence positive definite), PSD output upgrades to positive definite output precisely when the representing measure has positive mass on $f$1; measures concentrated at $f$2 give rank-one images $f$3.

For structure preservation, the paper first gives an intrinsic characterization: a symmetric positive-entry matrix is Green if and only if $f$4 for all $f$5. For positive-entry Green matrices with no PSD requirement, the preservers are exactly $f$6 with $f$7 an arbitrary additive function — possibly discontinuous, since no regularity is imposed. Under any mild regularity (continuity, measurability, local boundedness, or monotonicity on an interval), this reduces to $f$8 with $f$9. Requiring PSD or strict Green output forces monotonicity of $$G(p,q) = D_q L\,\mathrm{diag}(\Delta_1,\dots,\Delta_n)\,L^T D_q$$0, hence $$G(p,q) = D_q L\,\mathrm{diag}(\Delta_1,\dots,\Delta_n)\,L^T D_q$$1 or $$G(p,q) = D_q L\,\mathrm{diag}(\Delta_1,\dots,\Delta_n)\,L^T D_q$$2 respectively.

A sharpness result shows that order-three tests cannot be replaced by order two: $$G(p,q) = D_q L\,\mathrm{diag}(\Delta_1,\dots,\Delta_n)\,L^T D_q$$3 maps every PSD Green matrix of order two to a PSD Green matrix (via the AM–GM inequality $$G(p,q) = D_q L\,\mathrm{diag}(\Delta_1,\dots,\Delta_n)\,L^T D_q$$4) and every strict Green matrix of order two to a strict one, yet is not of the form $$G(p,q) = D_q L\,\mathrm{diag}(\Delta_1,\dots,\Delta_n)\,L^T D_q$$5. Since every symmetric positive-entry $$G(p,q) = D_q L\,\mathrm{diag}(\Delta_1,\dots,\Delta_n)\,L^T D_q$$6 matrix is Green, order two carries no information beyond positive definiteness in this setting.

Limitations and open questions

The classifications concern all matrix orders and functions on $$G(p,q) = D_q L\,\mathrm{diag}(\Delta_1,\dots,\Delta_n)\,L^T D_q$$7; preservers for Green matrices admitting zero entries or parameters, and preserver classes fixed to a single dimension, are explicitly left unstudied. The Loewner-monotonicity classification is restricted to $$G(p,q) = D_q L\,\mathrm{diag}(\Delta_1,\dots,\Delta_n)\,L^T D_q$$8 functions and to fixed-$$G(p,q) = D_q L\,\mathrm{diag}(\Delta_1,\dots,\Delta_n)\,L^T D_q$$9 fibers, and the cross-$D_q$0 version is open. The TN rigidity argument relies on a specific order-three test family; whether weaker test families (e.g., only principal minors) suffice is not addressed.

Conclusion

The paper provides a complete, regularity-free classification of entrywise positivity preservers on discrete Green matrices, organized by output requirement: Laplace-transform mixtures of nonnegative real powers for PSD output, monomials for TN output and for Green-structure preservation (with $D_q$1 for strict structure), and additive-perturbation forms for Loewner monotonicity on fixed-$D_q$2 fibers. The proofs are elementary, relying on rank-one and low-order test matrices together with the Hankel kernel representation, and the paper identifies precisely which test orders are necessary.