Let $\mathcal{L}=-\Delta +\mu$ be the generalized Schrödinger operator on ${\mathbb{R}}^n,n\ge 3$, where $\Delta$ is the Laplacian and $\mu \not\equiv 0$ is a nonnegative Radon measure on ${\mathbb{R}}^n$. In this article, we introduce two families of Carleson measures $\left\{d{\nu }_{h,k}\right\}$ and $\left\{d{\nu }_{P,k}\right\}$ generated by the heat semigroup $\left\{e^{-t\mathcal{L}}\right\}$ and the Poisson semigroup $\left\{e^{-t\sqrt{\mathcal{L}}}\right\}$, respectively. By the regularities of semigroups, we establish the Carleson measure characterizations of BMO-type spaces ${\mathrm{BMO}}_{\mathcal{L}}\left({\mathbb{R}}^n\right)$ associated with the generalized Schrödinger operators.

Let $G$ be a compact Abelian group, let $\mu$ be the corresponding Haar measure, and let $\hat{G}$ be the Pontryagin dual of $G$. Furthermore, let ${\mathcal{C}}_p$ denote the Schatten class of operators on some separable infinite-dimensional Hilbert space, and let $L^p(G;{\mathcal{C}}_p)$ denote the corresponding Bochner space. If $G\ni \theta \mapsto A_{\theta }$ is the mapping belonging to $L^p(G;{\mathcal{C}}_p)$, then If $G$ is a finite group, then the previous equations comprise several generalizations of Clarkson–McCarthy inequalities obtained earlier (e.g., $G={\mathbf{Z}}_n$ or $G={\mathbf{Z}}_2^n$), as well as the original inequalities, for $G={\mathbf{Z}}_2$. We also obtain other related inequalities.

Let $\phi :{\mathbb{R}}^n\times [0,\infty )\to [0,\infty )$ satisfy that $\phi (x,\cdot )$, for any given $x\in {\mathbb{R}}^n$, is an Orlicz function and that $\phi (\cdot ,t)$ is a Muckenhoupt $A_{\infty }$ weight uniformly in $t\in (0,\infty )$. The weak Musielak–Orlicz Hardy space ${\mathit{WH}}^{\phi }\left({\mathbb{R}}^n\right)$ is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak–Orlicz space ${\mathit{WL}}^{\phi }\left({\mathbb{R}}^n\right)$. For parameter $\rho \in (0,\infty )$ and measurable function $f$ on ${\mathbb{R}}^n$, the parametric Marcinkiewicz integral ${\mu }_{\Omega }^{\rho }$ related to the Littlewood–Paley $g$-function is defined by setting, for all $x\in {\mathbb{R}}^n$,

$${\mu }_{\Omega }^{\rho }\left(f\right)\left(x\right):=\left({\int }_0^{\infty }\right|{\int }_{|x-y|\le t}\frac{\Omega (x-y)}{|x-y{|}^{n-\rho }}f\left(y\right)dy{|}^2\frac{dt}{t^{2\rho +1}}{)}^{1/2},$$ where $\Omega$ is homogeneous of degree zero satisfying the cancellation condition.

In this article, we discuss the boundedness of the parametric Marcinkiewicz integral ${\mu }_{\Omega }^{\rho }$ with rough kernel from weak Musielak–Orlicz Hardy space ${\mathit{WH}}^{\phi }\left({\mathbb{R}}^n\right)$ to weak Musielak–Orlicz space ${\mathit{WL}}^{\phi }\left({\mathbb{R}}^n\right)$. These results are new even for the classical weighted weak Hardy space of Quek and Yang, and probably new for the classical weak Hardy space of Fefferman and Soria.

The main purpose of this article is to investigate Weyl-almost periodic solutions and asymptotically Weyl-almost periodic solutions of abstract Volterra integro-differential equations and inclusions. The class of asymptotically Weyl-almost periodic functions that we introduce here seems not to have been considered elsewhere, even in the scalar-valued case. We analyze the Weyl-almost periodic and asymptotically Weyl-almost periodic properties of convolution products and various types of degenerate solution operator families subgenerated by multivalued linear operators.

Given a $C^{\ast }$-algebra $A$, let $S\left(A^{+}\right)$ denote the set of positive elements in the unit sphere of $A$. Let $H_1$, $H_2$, $H_3$, and $H_4$ be complex Hilbert spaces, where $H_3$ and $H_4$ are infinite-dimensional and separable. In this article, we prove a variant of Tingley’s problem by showing that every surjective isometry $\Delta :S\left(B\right(H_1{)}^{+})\to S(B\left(H_2{)}^{+}\right)$ (resp., $\Delta :S\left(K\right(H_3{)}^{+})\to S(K\left(H_4{)}^{+}\right)$) admits a unique extension to a surjective complex linear isometry from $B\left(H_1\right)$ onto $B\left(H_2\right)$ (resp., from $K\left(H_3\right)$ onto $K\left(H_4\right)$). This provides a positive answer to a conjecture recently posed by Nagy.

We prove that every evolution algebra $A$ is a normed algebra, for an $l_1$-norm defined in terms of a fixed natural basis. We further show that a normed evolution algebra $A$ is a Banach algebra if and only if $A=A_1\oplus A_0$, where $A_1$ is finite-dimensional and $A_0$ is a zero-product algebra. In particular, every nondegenerate Banach evolution algebra must be finite-dimensional and the completion of a normed evolution algebra is therefore not, in general, an evolution algebra. We establish a sufficient condition for continuity of the evolution operator $L_B$ of $A$ with respect to a natural basis $B$, and we show that $L_B$ need not be continuous. Moreover, if $A$ is finite-dimensional and $B=\{e_1,\dots ,e_n\}$, then $L_B$ is given by $L_e$, where $e={\sum }_i{e}_i$ and $L_a$ is the multiplication operator $L_a\left(b\right)=ab$, for $b\in A$. We establish necessary and sufficient conditions for convergence of $\left(L_a^n\right(b){)}_n$, for all $b\in A$, in terms of the multiplicative spectrum ${\sigma }_m\left(a\right)$ of $a$. Namely, $\left(L_a^n\right(b){)}_n$ converges, for all $b\in A$, if and only if ${\sigma }_m\left(a\right)\subseteq \Delta \cup \left\{1\right\}$ and $\nu (1,a)\le 1$, where $\nu (1,a)$ denotes the index of $1$ in the spectrum of $L_a$.

Let $T_1$, $T_2$ be two Calderón–Zygmund operators, and let $T_{1,b}$ be the commutator of $T_1$ with symbol $b\in BMO\left({\mathbb{R}}^n\right)$. In this article, we establish the quantitative weighted bounds on $L^p({\mathbb{R}}^n,w)$ with $w\in A_p\left({\mathbb{R}}^n\right)$ for the composite operator $T_{1,b}{T}_2$.

For a pure bounded rationally cyclic subnormal operator $S$ on a separable complex Hilbert space $\mathcal{H}$, Conway and Elias showed that $clos\left(\sigma \right(S)\setminus {\sigma }_e(S\left)\right)=clos(Int(\sigma \left(S\right)\left)\right)$. This article examines the property for rationally multicyclic ($N$-cyclic) subnormal operators. We show that there exists a $2$-cyclic irreducible subnormal operator $S$ with $clos\left(\sigma \right(S)\setminus {\sigma }_e(S\left)\right)\ne clos(Int(\sigma \left(S\right)\left)\right)$. We also show the following. For a pure rationally $N$-cyclic subnormal operator $S$ on $\mathcal{H}$ with the minimal normal extension $M$ on $\mathcal{K}\supset \mathcal{H}$, let ${\mathcal{K}}_m=clos(span\{(M^{\ast }{)}^{k}x:x\in \mathcal{H},0\le k\le m\}$. Suppose that $M{|}_{{\mathcal{K}}_{N-1}}$ is pure. Then $clos\left(\sigma \right(S)\setminus {\sigma }_e(S\left)\right)=clos(Int(\sigma \left(S\right)\left)\right)$.

In this article, we study some geometric properties like parallelism, orthogonality, and semirotundity in the space of bounded linear operators. We completely characterize parallelism of two compact linear operators between normed linear spaces $\mathbb{X}$ and $\mathbb{Y}$, assuming $\mathbb{X}$ to be reflexive. We also characterize parallelism of two bounded linear operators between normed linear spaces $\mathbb{X}$ and $\mathbb{Y}$. We investigate parallelism and approximate parallelism in the space of bounded linear operators defined on a Hilbert space. Using the characterization of operator parallelism, we study Birkhoff–James orthogonality in the space of compact linear operators as well as bounded linear operators. Finally, we introduce the concept of semirotund points (semirotund spaces) which generalizes the notion of exposed points (strictly convex spaces). We further study semirotund operators and prove that $\mathbb{B}(\mathbb{X},\mathbb{Y})$ is a semirotund space which is not strictly convex if $\mathbb{X},\mathbb{Y}$ are finite-dimensional Banach spaces and $\mathbb{Y}$ is strictly convex.

In this article, we establish the Cameron–Martin translation theorems for the analytic Fourier–Feynman transform of functionals on the product function space $C_{a,b}^2[0,T]$. The function space $C_{a,b}[0,T]$ is induced by the generalized Brownian motion process associated with continuous functions $a(t)$ and $b(t)$ on the time interval $[0,T]$. The process used here is nonstationary in time and is subject to a drift $a(t)$. To study our translation theorem, we introduce a Fresnel-type class ${\mathcal{F}}_{A_1,A_2}^{a,b}$ of functionals on $C_{a,b}^2[0,T]$, which is a generalization of the Kallianpur and Bromley–Fresnel class ${\mathcal{F}}_{A_1,A_2}$. We then proceed to establish the translation theorems for the functionals in ${\mathcal{F}}_{A_1,A_2}^{a,b}$.

Our aim in this article is to establish weighted Hardy-type inequalities in a broad family of means. In other words, for a fixed vector of weights $({\lambda }_n{)}_{n=1}^{\infty }$ and a weighted mean $\mathcal{M}$, we search for the smallest extended real number $C$ such that

$${\sum }_{n=1}^{\infty }{\lambda }_n\mathcal{M}((x_1,\dots ,x_n),({\lambda }_1,\dots ,{\lambda }_n))\le C{\sum }_{n=1}^{\infty }{\lambda }_n{x}_n\text{for all}x\in {\ell }_1(\lambda ).$$ The main results provide a complete answer in the case when $\mathcal{M}$ is monotone and satisfies the weighted counterpart of the Kedlaya inequality. In particular, this is the case if $\mathcal{M}$ is symmetric, concave, and the sequence $(\frac{{\lambda }_n}{{\lambda }_1+\cdots +{\lambda }_n}{)}_{n=1}^{\infty }$ is nonincreasing. In addition, we prove that if $\mathcal{M}$ is a symmetric and monotone mean, then the biggest possible weighted Hardy constant is achieved if $\lambda$ is the constant vector.

We characterize those bounded multilinear operators that factor through a Hilbert space in terms of its behavior in finite sequences. This extends a result, essentially due to Kwapień, from the linear to the multilinear setting. We prove that Hilbert–Schmidt and Lipschitz $2$-summing multilinear operators naturally factor through a Hilbert space. We also prove that the class $\Gamma$ of all multilinear operators that factor through a Hilbert space is a maximal multi-ideal; moreover, we give an explicit formulation of a finitely generated tensor norm $\gamma$ which is in duality with $\Gamma$.