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Register a new userOn Helson matrices: moment problems, non-negativity, boundedness, and finite rank
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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.
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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.
This paper is devoted to give several characterizations on a more general level for the boundedness of $tau$-Wigner distributions acting from weighted modulation spaces to weighted modulation and Wiener amalgam spaces. As applications, sharp exponents are obtained for the boundedness of $tau$-Wigner distributions on modulation spaces with power weights. We also recapture the main theorems of Wigner distribution obtained in cite{CorderoNicola2018IMRNI,Cordero2020a}. As consequences, the characterizations of the boundedness on weighted modulation spaces of several types of pseudodifferential operators are established. In particular, we give the sharp exponents for the boundedness of pseudodifferential operators with symbols in Sj{o}strands class and the corresponding Wiener amalgam spaces.
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.
In 1975, P.R. Chernoff used iterates of the Laplacian on $mathbb{R}^n$ to prove an $L^2$ version of the Denjoy-Carleman theorem which provides a sufficient condition for a smooth function on $mathbb{R}^n$ to be quasi-analytic. In this paper, we prove an exact analogue of Chernoffs theorem for all rank one Riemannian symmetric spaces (of noncompact and compact types) using iterates of the associated Laplace-Beltrami operators.
The higher rank numerical ranges of generic matrices are described in terms of the components of their Kippenhahn curves. Cases of tridiagonal (in particular, reciprocal) 2-periodic matrices are treated in more detail.
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