We study the classical Hermite–Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such as the Schatten p-norm estimates

$$ \begin{align*}\left(\|A^q\|_p^p + \|B^q\|_p^p\right)^{1/p} \le \|(xA+(1-x)B)^q\|_p+ \|((1-x)A+xB)^q\|_p, \end{align*} $$

$n\times n$

$A,B$

$0<q,x<1$

$X^*X+Y^*Y=XX^*+YY^*=I$

$$ \begin{align*}(X^*AX+Y^*BY)\oplus (Y^*AY+X^*BX) =\frac{1}{2n}\sum_{k=1}^{2n} U_k (A\oplus B)U_k^*, \end{align*} $$

$2n\times 2n$

$U_k$

We provide a direct proof of the Medvedev–Scanlon’s conjecture from Medvedev and Scanlon (Ann. Math. Second Series 179(2014), 81–177) regarding Zariski dense orbits under the action of regular self-maps on split semiabelian varieties defined over a field of characteristic $0$ . Besides obtaining significantly easier proofs than the ones previously obtained in Ghioca and Scanlon (Trans. Am. Math. Soc. 369(2017), 447–466; for the case of abelian varieties) and Ghioca and Satriano (Trans. Am. Math. Soc. 371(2019), 6341–6358; for the case of semiabelian varieties), our method allows us to exhibit numerous starting points with Zariski dense orbits, which the methods from Ghioca and Scanlon (Trans. Am. Math. Soc. 369(2017), 447–466) and Ghioca and Satriano (Trans. Am. Math. Soc. 371(2019), 6341–6358) could not provide.

Motivated by the recent result in Samei and Wiersma (2020, Advances in Mathematics 359, 106897) that quasi-Hermitian groups are amenable, we consider a generalization of this property on discrete groups associated to certain Roe-type algebras; we call it uniformly quasi-Hermitian. We show that the class of uniformly quasi-Hermitian groups is contained in the class of supramenable groups and includes all subexponential groups. We also show that they are invariant under quasi-isometry.

Let $\Phi $ be a correspondence from a normed vector space X into itself, let $u: X\to \mathbf {R}$ be a function, and let $\mathcal {I}$ be an ideal on $\mathbf {N}$ . In addition, assume that the restriction of u on the fixed points of $\Phi $ has a unique maximizer $\eta ^\star $ . Then, we consider feasible paths $(x_0,x_1,\ldots )$ with values in X such that $x_{n+1} \in \Phi (x_n)$ , for all $n\ge 0$ . Under certain additional conditions, we prove the following turnpike result: every feasible path $(x_0,x_1,\ldots )$ which maximizes the smallest $\mathcal {I}$ -cluster point of the sequence $(u(x_0),u(x_1),\ldots )$ is necessarily $\mathcal {I}$ -convergent to $\eta ^\star $ .

We provide examples that, on the one hand, justify the hypotheses of our result and, on the other hand, prove that we are including new cases which were previously not considered in the related literature.

In this paper, we consider an equivalence relation on the space $AP(\mathbb {R},X)$ of almost periodic functions with values in a prefixed Banach space X. In this context, it is known that the normality or Bochner-type property, which characterizes these functions, is based on the relative compactness of the family of translates. Now, we prove that every equivalence class is sequentially compact and the family of translates of a function belonging to this subspace is dense in its own class, i.e., the condition of almost periodicity of a function $f\in AP(\mathbb {R},X)$ yields that every sequence of translates of f has a subsequence that converges to a function equivalent to f. This extends previous work by the same authors on the case of numerical almost periodic functions.

We provide a description of the spectrum and compute the eigenvalues distribution of circulant Hankel matrices obtained as symmetrization of classical Toeplitz circulant matrices. Other types of circulant matrices such as simple and Cesàro circulant matrices are also considered.

In this paper, we will show that the unitary equivalence of two multiplication operators on the Bergman spaces on polygons depends on the geometry of the polygon.

Thurston norms are invariants of 3-manifolds defined on their second homology and understanding the shape of their dual unit balls is a widely open problem. In this article, we provide a large family of polytopes in $\mathbb {R}^{2g}$ that appear like dual unit balls of Thurston norms, generalizing Thurston’s construction for polygons in $\mathbb {R}^2$ .

À mes enseignants, premiers

In this note, we give a new property of de Branges–Rovnyak kernels. As the main theorem, it is shown that the exponential of de Branges–Rovnyak kernel is strictly positive definite if the corresponding Schur class function is nontrivial.

For $0<\lambda ,p<1$ , the Dirichlet–Morrey space $\mathcal {D}_p^{\lambda } $ is the space of all analytic function on the unit disc such that the measure $ |f'(z)|^2(1-|z|^2)^pdA(z)$ is a $p\lambda $ -Carleson measure. In this paper, we show that the corona theorem and the Wolff theorem hold for the multiplier algebra of Dirichlet–Morrey spaces.

We compute the leading coefficient in the asymptotic expansion of the eigenvalue counting function for the Kohn Laplacian on the spheres. We express the coefficient as an infinite sum and as an integral.

We show that the Tits index $E_{8,1}^{133}$ cannot be obtained by means of the Tits construction over a field with no odd degree extensions. The proof uses a general method coming from the theory of symmetric spaces. We construct two cohomological invariants, in degrees $6$ and $8$ , of the Tits construction and the more symmetric Allison–Faulkner construction of Lie algebras of type $E_8$ and show that these invariants can be used to detect the isotropy rank.

A sequence $\left \{g_k\right \}_{k=1}^{\infty }$ in a Hilbert space ${\cal H}$ has the expansion property if each $f\in \overline {\text {span}} \left \{g_k\right \}_{k=1}^{\infty }$ has a representation $f=\sum _{k=1}^{\infty } c_k g_k$ for some scalar coefficients $c_k.$ In this paper, we analyze the question whether there exist small norm-perturbations of $\left \{g_k\right \}_{k=1}^{\infty }$ which allow to represent all $f\in {\cal H};$ the answer turns out to be yes for frame sequences and Riesz sequences, but no for general basic sequences. The insight gained from the analysis is used to address a somewhat dual question, namely, whether it is possible to remove redundancy from a sequence with the expansion property via small norm-perturbations; we prove that the answer is yes for frames $\left \{g_k\right \}_{k=1}^{\infty }$ such that $g_k\to 0$ as $k\to \infty ,$ as well as for frames with finite excess. This particular question is motivated by recent progress in dynamical sampling.

We investigate a method of construction of central deformations of associative algebras, which we call centrification. We prove some general results in the case of Hopf algebras and provide several examples.

Let $j_n$ be the modular function obtained by applying the nth Hecke operator on the classical j-invariant. For $n>m\ge 2$ , we prove that between any two zeros of $j_m$ on the unit circle of the fundamental domain, there is a zero of $j_n$ .

We compare the Pontryagin duals of fine Selmer groups of two congruent p-adic Galois representations over admissible pro-p, p-adic Lie extensions $K_\infty $ of number fields K. We prove that in several natural settings the $\pi $ -primary submodules of the Pontryagin duals are pseudo-isomorphic over the Iwasawa algebra; if the coranks of the fine Selmer groups are not equal, then we can still prove inequalities between the $\mu $ -invariants. In the special case of a $\mathbb {Z}_p$ -extension $K_\infty /K$ , we also compare the Iwasawa $\lambda $ -invariants of the fine Selmer groups, even in situations where the $\mu $ -invariants are nonzero. Finally, we prove similar results for certain abelian non-p-extensions.

We generalize Radó’s extension theorem from the complex plane to reduced complex spaces.

Let $\Omega $ be a bounded Reinhardt domain in $\mathbb {C}^n$ and $\phi _1,\ldots ,\phi _m$ be finite sums of bounded quasi-homogeneous functions. We show that if the product of Toeplitz operators $T_{\phi _m}\cdots T_{\phi _1}=0$ on the Bergman space on $\Omega $ , then $\phi _j=0$ for some j.

In this note, we show that in a complete $\operatorname {\mathrm {CAT}}(0)$ space pointwise convergence of proximal mappings under a certain normalization condition implies Mosco convergence.

We show that any weakly separated Bessel system of model spaces in the Hardy space on the unit disc is a Riesz system and we highlight some applications to interpolating sequences of matrices. This will be done without using the recent solution of the Feichtinger conjecture, whose natural generalization to multidimensional model subspaces of ${\mathrm {H}}^2$ turns out to be false.