We provide extension procedures for nonlinear expectations to the space of all bounded measurable functions. We first discuss a maximal extension for convex expectations which have a representation in terms of finitely additive measures. One of the main results of this article is an extension procedure for convex expectations which are continuous from above and therefore admit a representation in terms of countably additive measures. This can be seen as a nonlinear version of the Daniell–Stone theorem. From this, we deduce a robust Kolmogorov extension theorem which is then used to extend nonlinear kernels to an infinite-dimensional path space. We then apply this theorem to construct nonlinear Markov processes with a given family of nonlinear transition kernels.

In this article, we investigate the complex interpolation of predual spaces of general local Morrey-type spaces. By showing that these spaces are equal to the associate space of general local Morrey-type spaces, we prove that predual spaces of general local Morrey-type spaces behave well under the first complex interpolation.

Given a higher-rank graph $\Lambda$, we investigate the relationship between the cohomology of $\Lambda$ and the cohomology of the associated groupoid ${\mathcal{G}}_{\Lambda }$. We define an exact functor between the Abelian category of right modules over a higher-rank graph $\Lambda$ and the category of ${\mathcal{G}}_{\Lambda }$-sheaves, where ${\mathcal{G}}_{\Lambda }$ is the path groupoid of $\Lambda$. We use this functor to construct compatible homomorphisms from both the cohomology of $\Lambda$ with coefficients in a right $\Lambda$-module, and the continuous cocycle cohomology of ${\mathcal{G}}_{\Lambda }$ with values in the corresponding ${\mathcal{G}}_{\Lambda }$-sheaf, into the sheaf cohomology of ${\mathcal{G}}_{\Lambda }$.

Ornstein and Sucheston first proved that for a given positive contraction $T:L_1\to L_1$ there exists $m\in \mathbb{N}$ such that if $\Vert T^{m+1}-T^m\Vert <2$, then ${lim\hspace{0.17em}}_{n\to \infty }\Vert T^{n+1}-T^n\Vert =0$. This result was referred to as the zero-two law. In the present article, we prove a generalized uniform zero-two law for the multiparametric family of positive contractions of noncommutative $L_1$-spaces. Moreover, we also establish a vector-valued analogue of the uniform zero-two law for positive contractions of $L_1(M,\Phi )$—the noncommutative $L_1$-spaces associated with center-valued traces.

Extending the corresponding notion for matrices or bounded linear operators on a Hilbert space, we define a generalized Schur complement for a nonnegative linear operator mapping a linear space into its dual, and we derive some of its properties.

We investigate pointwise convergence of entangled ergodic averages of Dunford–Schwartz operators $T_0,T_1,\dots ,T_m$ on a Borel probability space. These averages take the form

$$\frac{1}{N^k}{\sum }_{1\le n_1,\dots ,n_k\le N}{T}_m^{n_{\alpha \left(m\right)}}{A}_{m-1}{T}_{m-1}^{n_{\alpha (m-1)}}\cdots A_2{T}_2^{n_{\alpha \left(2\right)}}{A}_1{T}_1^{n_{\alpha \left(1\right)}}f,$$ where $f\in L^p(X,\mu )$ for some $1\le p<\infty$, and $\alpha :\{1,\dots ,m\}\to \{1,\dots ,k\}$ encodes the entanglement. We prove that, under some joint boundedness and twisted compactness conditions on the pairs $(A_i,T_i)$, convergence holds almost everywhere for all $f\in L^p$. We also present an extension to polynomial powers in the case $p=2$, in addition to a continuous version concerning Dunford–Schwartz $C_0$-semigroups.

In this paper we first develop the rotation theorem of the Gaussian paths on Wiener space. We next analyze the generalized analytic Fourier–Feynman transform. As an application of our rotation theorem, we represent the multiple generalized analytic Fourier–Feynman transform as a single generalized Fourier–Feynman transform.

We derive sharp weighted norm estimates for positive kernel operators on spaces of homogeneous type. Similar problems are studied for one-sided fractional integrals. Bounds of weighted norms are of mixed type. The problems are studied using the two-weight theory of positive kernel operators. As special cases, we derive sharp weighted estimates in terms of Muckenhoupt characteristics.

We describe a novel method of handling general sharp square function inequalities for monotone bases and contractive projections in $L^1$. The technique rests on the construction of an appropriate special function enjoying certain size and convexity-type properties. As an illustration, we establish a strong $L^1\to L^1$ and a weak-type $L^1\to L^{1,\infty }$ estimate for square functions.

In this article, based on some geometric properties of Banach spaces and one feature of the metric projection, we introduce a new class of bounded linear operators satisfying the so-called $(\alpha ,\beta )$-USU (uniformly strong uniqueness) property. This new convenient property allows us to take the study of the stability problem of the Moore–Penrose metric generalized inverse a step further. As a result, we obtain various perturbation bounds of the Moore–Penrose metric generalized inverse of the perturbed operator. They offer the advantage that we do not need the quasiadditivity assumption, and the results obtained appear to be the most general case found to date. Closely connected to the main perturbation results, one application, the error estimate for projecting a point onto a linear manifold problem, is also investigated.

In this article, we investigate disjointness-preserving orthogonally additive operators in the setting of vector lattices. First, we present a formula for the band projection onto the band generated by a single positive, disjointness-preserving, order-bounded, orthogonally additive operator. Then we prove a Radon–Nikodým theorem for a positive, disjointness-preserving, order-bounded, orthogonally additive operator defined on a vector lattice $E$, taking values in a Dedekind-complete vector lattice $F$. We conclude by obtaining an analytical representation for a nonlinear lattice homomorphism between order ideals of spaces of measurable almost everywhere finite functions.

For a set $\mathcal{M}$ of operators on a complex Banach space $\mathcal{X}$, the reflexive cover of $\mathcal{M}$ is the set $Ref\left(\mathcal{M}\right)$ of all those operators $T$ satisfying $Tx\in \overline{\mathcal{M}x}$ for every $x\in \mathcal{X}$. Set $\mathcal{M}$ is reflexive if $Ref\left(\mathcal{M}\right)=\mathcal{M}$. The notion is well known, especially for Banach algebras or closed spaces of operators, because it is related to the problem of invariant subspaces. We study reflexivity for general sets of operators. We are interested in how the reflexive cover behaves towards basic operations between sets of operators. It is easily seen that the intersection of an arbitrary family of reflexive sets is reflexive, as well. However this does not hold for unions, since the union of two reflexive sets of operators is not necessarily a reflexive set. We give some sufficient conditions under which the union of reflexive sets is reflexive. We explore how the reflexive cover of the sum (resp., the product) of two sets is related to the reflexive covers of summands (resp., factors). We also study the relation between reflexivity and convexity, with special interest in the question: under which conditions is the convex hull of a reflexive set reflexive?