We present a characterization for the continuous, isotropic, and positive definite kernels on a product of spheres along the lines of a classical result of Schoenberg on positive definiteness on a single sphere. We also discuss a few issues regarding the characterization, including topics for future investigation.

Note: An incorrect version of this article was posted from August 31, 2016, through September 7, 2016. The PDF is now correct.

This paper is concerned with the study of geometric structures in spaces of polynomials. More precisely, we discuss for $E$ and $F$ Banach spaces, whether the class of $n$-homogeneous polynomials, ${\mathcal{P}}_w{(}^{n}E,F)$, which are weakly continuous on bounded sets, is an HB-subspace or an $M(1,C)$-ideal in the space of continuous $n$-homogeneous polynomials, $\mathcal{P}{(}^{n}E,F)$. We establish sufficient conditions under which the problem can be positively solved. Some examples are given. We also study when some ideal structures pass from ${\mathcal{P}}_w{(}^{n}E,F)$ as an ideal in $\mathcal{P}{(}^{n}E,F)$ to the range space $F$ as an ideal in its bidual $F^{\ast \ast }$.

Let $\mathcal{A}$ be a maximal subdiagonal algebra of semifinite von Neumann algebra $\mathcal{M}$. For $0<p\le \infty$, we define the noncommutative Hardy–Lorentz spaces $H^{p,\omega }\left(\mathcal{A}\right)$, and give some properties of these spaces. We obtain that the Herglotz maps are bounded linear maps from ${\Lambda }_{\omega }^p\left(\mathcal{M}\right)$ into ${\Lambda }_{\omega }^p\left(\mathcal{M}\right)$, and with this result we characterize the dual spaces of $H^{p,\omega }\left(\mathcal{A}\right)$ for $1<p<\infty$. We also present the Hartman–Wintner spectral inclusion theorem of Toeplitz operators and Pisier’s interpolation theorem for this case.

Let

$$Lf\left(x\right)=-\frac{1}{\omega \left(x\right)}{\sum }_{i,j}{\partial }_i\left(a_{ij}\right(\cdot \left){\partial }_{j}f\right)\left(x\right)+V\left(x\right)f\left(x\right)$$ be the degenerate Schrödinger operator, where $\omega$ is a weight from the Muckenhoupt class $A_2$ and $V$ is a nonnegative potential that belongs to a certain reverse Hölder class with respect to the measure $\omega \left(x\right)dx$. Based on some smoothness estimates of the Poisson semigroup $e^{-t\sqrt{L}}$, we introduce the area function $S_P^L$ associated with $e^{-t\sqrt{L}}$ to characterize the Hardy space associated with $L$.

In this paper, we introduce Hardy spaces with variable exponents defined on a probability space and develop the martingale theory of variable Hardy spaces. We prove the weak-type and strong-type inequalities on Doob’s maximal operator, and we get a $(1,p(\cdot ),\infty )$-atomic decomposition for Hardy martingale spaces associated with conditional square functions. As applications, we obtain a dual theorem and the John–Nirenberg inequalities in the frame of variable exponents. The key ingredient is that we find a condition with a probabilistic characterization of $p(\cdot )$ to replace the so-called log-Hölder continuity condition in ${\mathbb{R}}^n$.

We study when diameter $2$ properties can be inherited by subspaces. We obtain that the slice diameter $2$ property (resp., the diameter $2$ property, strong diameter $2$ property) passes from a Banach space $X$ to a subspace $Y$ whenever $X/Y$ is finite-dimensional and $Y$ is complemented by a norm $1$ projection (resp., the quotient $X/Y$ is finite-dimensional and strongly regular). Also, we study the same problem for the dual properties of diameter $2$ properties, such as having octahedral, weakly octahedral, or $2$-rough norm.

Let $X$ be a completely regular Hausdorff space, let $\mathcal{D}$ be a cover of $X$, and let $\pi :\mathcal{E}\to X$ be a bundle of Banach spaces (algebras). Let $\Gamma \left(\pi \right)$ be the space of sections of $\pi$, and let ${\Gamma }_b(\pi ,\mathcal{D})$ be the subspace of $\Gamma \left(\pi \right)$ consisting of sections which are bounded on each $D\in \mathcal{D}$. We study the subspace (ideal) and quotient structures of some spaces of vector-valued functions which arise from endowing ${\Gamma }_b(\pi ,\mathcal{D})$ with the cover-strict topology.

K. M. R. Audenaert (2010), R. A. Horn and F. Zhang (2010), Z. Huang (2011), A. R. Schep (2011), A. Peperko (2012), and D. Chen and Y. Zhang (2015) have proved inequalities on the spectral radius and the operator norm of Hadamard products and ordinary matrix products of finite and infinite nonnegative matrices that define operators on sequence spaces. In the present article, we extend and refine several of these results, and we also prove some analogues for the numerical radius.

Let $\nu$ be a $\sigma$-finite Banach-space-valued measure defined on a $\delta$-ring. We find a wide class of measures $\nu$ for which interpolation with a parameter function of couples of Banach lattices of $p$-integrable and weakly $p$-integrable functions with respect to $\nu$ produces a Lorentz-type space. Moreover, we prove that if we interpolate between sums and intersections of them, then they still yield another Lorentz-type space closely related with the first one.

For two unknown functions $f$, $g$, the equation

$$\langle f\left(x\right)\left|g\right(y)\rangle =\langle x\mid y\rangle$$ and its stability as well as the approximate orthogonality-preserving property $$x\perp y⟹fx{\perp }^{\epsilon }gy$$ are considered.

This work is inspired by the study of wandering vectors and frame vectors for unitary systems. We investigate the structure and properties of complete wandering subspaces for unitary systems, and, in particular, we consider the unitary systems with a structure similar to wavelet systems. Given a unitary system with a complete wandering subspace, a necessary and sufficient condition for a closed subspace to be a Parseval fusion frame generator is obtained. Moreover, we study the dilation property for Parseval fusion frame generators for unitary groups.

Let $n$ and $k$ be nonnegative integers such that $1\le k\le n+1$. The convex cone ${\mathcal{F}}_{+}^{k:n}$ of all functions $f$ on an arbitrary interval $I\subseteq \mathbb{R}$ whose derivatives $f^{\left(j\right)}$ of orders $j=k-1,\dots ,n$ are nondecreasing is characterized. A simple description of the convex cone dual to ${\mathcal{F}}_{+}^{k:n}$ is given. In particular, these results are useful in, and were motivated by, applications in probability. In fact, the results are obtained in a more general setting with certain generalized derivatives of $f$ of the $j$th order in place of $f^{\left(j\right)}$. Somewhat similar results were previously obtained, in terms of Tchebycheff–Markov systems, in the case when the left endpoint of the interval $I$ is finite, with certain additional integrability conditions; such conditions fail to hold in the mentioned applications. Development of substantially new methods was needed to overcome the difficulties.