We use Berezin-type symbols on the Dirichlet space $\mathcal{D}({\mathbb{B}}_n)$ of the unit ball to investigate solutions of some Riccati operator equations of the form $XAX+XB-CX-D=0$. In particular, we discuss the solvability of the operator equation $XAX=AX$ in the set of all bounded Toeplitz operators acting on the Dirichlet space $\mathcal{D}({\mathbb{B}}_n)$. Some other related questions, such as invariant subspaces, are also discussed.

Many multilinear discrete operators are primed for pointwise decomposition; such decompositions give structural information but also an essentially optimal range of bounds. We study the (continuous) slicing method of Jeong and Lee—which when debuted instantly gave sharp multilinear operator bounds—in the discrete setting. Via several examples, number theoretic connections, pointed commentary, and a unified theory we hope that this useful technique will lead to further applications. This work generalizes, and was inspired by, the author’s work with Palsson on a special case.

Among metrics of constant positive curvature on a punctured compact Riemann surface with conical singularities at the punctures, dihedral monodromy means that the action of the monodromy group $\mathcal{M}\subset SO(3)$ globally preserves a pair of antipodal points. Using recent results about local invariants of quadratic differentials, we give a complete characterization of the set of conical angles realized by some cone spherical metric with dihedral monodromy.

Let D be a domain in the complex plane $\mathbb{C}$. The Hardy number of D, which was first introduced by Hansen, is the maximal number $h(D)$ in $[0,+\mathrm{\infty }]$ such that f belongs to the classical Hardy space $H^p(\mathbb{D})$ whenever $0<p<h(D)$ and f is holomorphic on the unit disk $\mathbb{D}$ with values in D. As an analogue notion to the Hardy number of a domain D in $\mathbb{C}$, we introduce the Bergman number of D, and we denote it by $b(D)$. Our main result is that if D is regular, then $h(D)=b(D)$. This generalizes earlier work by the author and Karamanlis for simply connected domains. The Bergman number $b(D)$ is the maximal number in $[0,+\mathrm{\infty }]$ such that f belongs to the weighted Bergman space $A_{\mathit{\alpha }}^p(\mathbb{D})$ whenever $p>0$ and $\mathit{\alpha }>-1$ satisfy $0<\frac{p}{\mathit{\alpha }+2}<b(D)$ and f is holomorphic on $\mathbb{D}$ with values in D. We also establish several results about Hardy spaces and weighted Bergman spaces, and we give a new characterization of the Hardy number and thus of the Bergman number of a regular domain with respect to the harmonic measure.

Given orientable Riemannian manifolds $M^n$ and ${\bar{M}}^{n+1}$, we study flows $F_t:M^n\to {\bar{M}}^{n+1}$, called Weingarten flows, in which the hypersurfaces $F_t(M)$ evolve in the direction of their normal vectors with speed given by a function W of their principal curvatures, called a Weingarten function, which is homogeneous, monotonic increasing with respect to any of its variables, and positive on the positive cone. We obtain existence results for flows with isoparametric initial data, in which the hypersurfaces $F_t:M^n\to {\bar{M}}^{n+1}$ are all parallel, and ${\bar{M}}^{n+1}$ is either a simply connected space form or a rank-one symmetric space of noncompact type. We prove that the avoidance principle holds for Weingarten flows defined by odd Weingarten functions, and also that such flows are embedding preserving.

In the spirit of the famous Komlós (1967) theorem, every sequence of nonnegative, measurable functions ${\{f_n\}}_{n\in \mathbb{N}}$ on a probability space contains a subsequence which—along with all its subsequences—converges a.e. in Cesàro mean to some measurable $f_{\ast }:\mathrm{\Omega }\to [0,\mathrm{\infty }]$. This result of von Weizsäcker (2004) is proved here using a new methodology and elementary tools; these sharpen also a theorem of Delbaen and Schachermayer (1994), replacing general convex combinations by Cesàro means.

It is an interesting question to ask whether or not the Abelian ratio ergodic theorem of Báez-Duarte type holds for positive power-bounded linear (not contraction) operators on $L_1$. To approach this question, we prove a multiparameter Abelian ratio ergodic theorem for positive power-bounded Lamperti operators on $L_1$ with an additional condition. The continuous parameter analogs are also investigated.

We give the realization of spherical CR-symmetric manifolds $M^{2n+1}$, $n\ge 2$, as real hypersurfaces in Hermitian symmetric spaces of rank 1 or rank 2. In addition, we give a complete classification of CR-symmetric contact 3-manifolds. Using the generalized Tanaka–Webster connection, we prove classification theorems of locally pseudo-Hermitian symmetric CR space forms of higher dimension and of dimension 3.

In this article, we characterize the operadic variety of commutative associative algebras over a field via a (categorical) condition: the associativity of the so-called cosmash product. This condition, which is closely related to commutator theory, is quite strong: for example, groups do not satisfy it. However, in the case of commutative associative algebras, the cosmash product is nothing more than the tensor product; which explains why in this case it is associative. We prove that in the setting of operadic varieties of algebras over a field, it is the only example. Further examples in the nonoperadic case are also discussed.

Let X and Y be irreducible normal projective varieties, of same dimension, defined over an algebraically closed field, and let $f:Y⟶X$ be a finite generically smooth morphism such that the corresponding homomorphism between the étale fundamental groups $f_{\ast }:{\mathrm{\pi }}_1^{\mathrm{et}}(Y)⟶{\mathrm{\pi }}_1^{\mathrm{et}}(X)$ is surjective. Fix a polarization on X and equip Y with the pulled-back polarization. For a point $y_0\in Y$, let $\mathrm{\varpi }(Y,y_0)$ (resp. $\mathrm{\varpi }(X,f(y_0))$) be the affine group scheme given by the neutral Tannakian category defined by the strongly pseudo-stable vector bundles of degree zero on Y (resp. X). We prove that the homomorphism $\mathrm{\varpi }(Y,y_0)⟶\mathrm{\varpi }(X,f(y_0))$ induced by f is surjective. Let E be a pseudo-stable vector bundle on X and $F\subset f^{\ast }E$ a pseudo-stable subbundle with $\mathrm{\mu }(F)=\mathrm{\mu }(f^{\ast }E)$. We prove that $f^{\ast }E$ is pseudo-stable and there is a pseudo-stable subbundle $W\subset E$ such that $f^{\ast }W=F$ as subbundles of $f^{\ast }E$.