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Functions preserving Loewner positivity when applied entrywise to positive semidefinite matrices have been widely studied in the literature. Following the work of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J. 26], and others, it is well-known that functions preserving positivity for matrices of all dimensions are absolutely monotonic (i.e., analytic with nonnegative Taylor coefficients). In this paper, we study functions preserving positivity when applied entrywise to sparse matrices, with zeros encoded by a graph $G$ or a family of graphs $G_n$. Our results generalize Schoenberg and Rudins results to a modern setting, where functions are often applied entrywise to sparse matrices in order to improve their properties (e.g. better conditioning). The only such result known in the literature is for the complete graph $K_2$. We provide the first such characterization result for a large family of non-complete graphs. Specifically, we characterize functions preserving Loewner positivity on matrices with zeros according to a tree. These functions are multiplicatively midpoint-convex and super-additive. Leveraging the underlying sparsity in matrices thus admits the use of functions which are not necessarily analytic nor absolutely monotonic. We further show that analytic functions preserving positivity on matrices with zeros according to trees can contain arbitrarily long sequences of negative coefficients, thus obviating the need for absolute monotonicity in a very strong sense. This result leads to the question of exactly when absolute monotonicity is necessary when preserving positivity for an arbitrary class of graphs. We then provide a stronger condition in terms of the numerical range of all symmetric matrices, such that functions satisfying this condition on matrices with zeros according to any family of graphs with unbounded degrees are necessarily absolutely monotonic.
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Entrywise functions preserving the cone of positive semidefinite matrices have been studied by many authors, most notably by Schoenberg [Duke Math. J. 9, 1942] and Rudin [Duke Math. J. 26, 1959]. Following their work, it is well-known that entrywise functions preserving Loewner positivity in all dimensions are precisely the absolutely monotonic functions. However, there are strong theoretical and practical motivations to study functions preserving positivity in a fixed dimension $n$. Such characterizations for a fixed value of $n$ are difficult to obtain, and in fact are only known in the $2 times 2$ case. In this paper, using a novel and intuitive approach, we study entrywise functions preserving positivity on distinguished submanifolds inside the cone obtained by imposing rank constraints. These rank constraints are prevalent in applications, and provide a natural way to relax the elusive original problem of preserving positivity in a fixed dimension. In our main result, we characterize entrywise functions mapping $n times n$ positive semidefinite matrices of rank at most $l$ into positive semidefinite matrices of rank at most $k$ for $1 leq l leq n$ and $1 leq k < n$. We also demonstrate how an important necessary condition for preserving positivity by Horn and Loewner [Trans. Amer. Math. Soc. 136, 1969] can be significantly generalized by adding rank constraints. Finally, our techniques allow us to obtain an elementary proof of the classical characterization of functions preserving positivity in all dimensions obtained by Schoenberg and Rudin.
We prove that the only entrywise transforms of rectangular matrices which preserve total positivity or total non-negativity are either constant or linear. This follows from an extended classification of preservers of these two properties for matrices of fixed dimension. We also prove that the same assertions hold upon working only with symmetric matrices; for total-positivity preservers our proofs proceed through solving two totally positive completion problems.
Entrywise powers of matrices have been well-studied in the literature, and have recently received renewed attention in the regularization of high-dimensional correlation matrices. In this paper, we study powers of positive semidefinite block matrices $(H_{st})_{s,t=1}^n$ with complex entries. We first characterize the powers $alphainmathbb{R}$ such that the blockwise power map $(H_{st}) mapsto (H_{st}^alpha)$ preserves Loewner positivity. The characterization is obtained by exploiting connections with the theory of matrix monotone functions developed by Loewner. Second, we revisit previous work by Choudhury [Proc. AMS 108] who had provided a lower bound on $alpha$ for preserving positivity when the blocks $H_{st}$ pairwise commute. We completely settle this problem by characterizing the full set of powers preserving positivity in this setting. Our characterizations generalize previous work by FitzGerald-Horn, Bhatia-Elsner, and Hiai from scalars to arbitrary block size, and in particular, generalize the Schur Product Theorem. Finally, a natural and unifying framework for studying the case of diagonalizable blocks consists of replacing real powers by general characters of the complex plane. We thus classify such characters, and generalize our results to this more general setting. In the course of our work, given $betainmathbb{Z}$, we provide lower and upper bounds for the threshold power $alpha >0$ above which the complex characters $re^{itheta}mapsto r^alpha e^{ibetatheta}$ preserve positivity when applied entrywise to positive semidefinite matrices. In particular, we completely resolve the $n=3$ case of a question raised in 2001 by Xingzhi Zhan. As an application, we extend previous work by de Pillis [Duke Math. J. 36] by classifying the characters $K$ of the complex plane for which the map $(H_{st})_{s,t=1}^n mapsto (K({rm tr}(H_{st})))_{s,t=1}^n$ preserves positivity.
We study Helson matrices (also known as multiplicative Hankel matrices), i.e. infinite matrices of the form $M(alpha) = {alpha(nm)}_{n,m=1}^infty$, where $alpha$ is a sequence of complex numbers. Helson matrices are considered as linear operators on $ell^2(mathbb{N})$. By interpreting Helson matrices as Hankel matrices in countably many variables we use the theory of multivariate moment problems to show that $M(alpha)$ is non-negative if and only if $alpha$ is the moment sequence of a measure $mu$ on $mathbb{R}^infty$, assuming that $alpha$ does not grow too fast. We then characterize the non-negative bounded Helson matrices $M(alpha)$ as those where the corresponding moment measures $mu$ are Carleson measures for the Hardy space of countably many variables. Finally, we give a complete description of the Helson matrices of finite rank, in parallel with the classical Kronecker theorem on Hankel matrices.
We characterize positivity preserving, translation invariant, linear operators in $L^p(mathbb{R}^n)^m$, $p in [1,infty)$, $m,n in mathbb{N}$.
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