Motivated by applications of the discrete random Schrödinger operator, mathematical physicists and analysts began studying more general Anderson-type Hamiltonians; that is, the family of self-adjoint operators $$H_{\omega }=H+V_{\omega }$$ on a separable Hilbert space $\mathcal{H}$, where the perturbation is given by $$V_{\omega }={\sum }_n{\omega }_n(\cdot ,{\phi }_n){\phi }_n$$ with a sequence $\left\{{\phi }_n\right\}\subset \mathcal{H}$ and independent identically distributed random variables ${\omega }_n$. We show that the essential parts of Hamiltonians associated to any two realizations of the random variable are (almost surely) related by a rank-one perturbation. This result connects one of the least trackable perturbation problem (with almost surely noncompact perturbations) with one where the perturbation is “only” of rank-one perturbations. The latter presents a basic application of model theory. We also show that the intersection of the essential spectrum with open sets is almost surely either the empty set, or it has nonzero Lebesgue measure.

Our goal in this article is to derive Abelian theorems for the Kontorovich–Lebedev and Mehler–Fock transforms of general order over distributions of compact support and over certain spaces of generalized functions.

In this paper we consider sectorial operators, or more generally, sectorial relations and their maximal-sectorial extensions in a Hilbert space $\mathfrak{H}$. Our particular interest is in sectorial relations $S$, which can be expressed in the factorized form $$S=T^{\ast }(I+iB)T\text{or}S=T(I+iB)T^{\ast },$$ where $B$ is a bounded self-adjoint operator in a Hilbert space $\mathfrak{K}$ and $T:\mathfrak{H}\to \mathfrak{K}$ (or $T:\mathfrak{K}\to \mathfrak{H}$, respectively) is a linear operator or a linear relation which is not assumed to be closed. Using the specific factorized form of $S$, a description of all the maximal-sectorial extensions of $S$ is given, along with a straightforward construction of the extreme extensions $S_F$, the Friedrichs extension, and $S_K$, the Kreĭn extension of $S$, which uses the above factorized form of $S$. As an application of this construction, we also treat the form sum of maximal-sectorial extensions of two sectorial relations.

Let $\Lambda$ be either a subgroup of the integers $\mathbb{Z}$, a semigroup in $\mathbb{N}$, or $\Lambda =\mathbb{Q}$ (resp., ${\mathbb{Q}}^{+}$). We determine the Bass and topological stable ranks of the algebras ${\mathrm{AP}}_{\Lambda }=\{f\in \mathrm{AP}:\sigma (f)\subseteq \Lambda \}$ of almost periodic functions on the real line and with Bohr spectrum in $\Lambda$. This answers a question in the first part of this series of articles under the same heading, where it was shown that, in contrast to the present situation, these ranks were infinite for each semigroup $\Lambda$ of real numbers for which the $\mathbb{Q}$-vector space generated by $\Lambda$ had infinite dimension.

We generalize the notion of quasidiagonality, for extensions, allowing for the case where the canonical ideal has few projections. We prove that the pointwise-norm limit of generalized quasidiagonal extensions is generalized quasidiagonal. We also provide a $K$-theory sufficient condition for generalized quasidiagonality of certain extensions of simple continuous-scale $C^{\ast }$-algebras, including certain continuous-scale hereditary $C^{\ast }$-subalgebras of the stabilized Jiang–Su algebra.

Lin solved a longstanding problem as follows. For each $ϵ>0$, there is $\delta >0$ such that, if $h$ and $k$ are self-adjoint contractive $n\times n$ matrices and $\Vert hk-kh\Vert <\delta$, then there are commuting self-adjoint matrices $h\text{'}$ and $k\text{'}$ such that $\Vert h\text{'}-h\Vert$, $\Vert k\text{'}-k\Vert <ϵ$. Here $\delta$ depends only on $ϵ$ and not on $n$. Friis and Rørdam greatly simplified Lin’s proof by using a property they called $\mathit{IR}$. They also generalized Lin’s result by showing that the matrix algebras can be replaced by any $C^{\ast }$-algebras satisfying $\mathit{IR}$. The purpose of this paper is to study the property $\mathit{IR}$. One of our results shows how $\mathit{IR}$ behaves for $C^{\ast }$-algebra extensions. Other results concern nonstable $K$-theory. One shows that $\mathit{IR}$ (at least the stable version) implies a cancellation property for projections which is intermediate between the strong cancellation satisfied by $C^{\ast }$-algebras of stable rank $1$ and the weak cancellation defined in a 2014 paper by Pedersen and the author.

For a Hankel operator $H_{\xi }$, $\xi \in L^{\infty }\left({\mathbb{T}}^2\right)$, on the Hardy space $H^2$ over the bidisk, $ker{H}_{\overline{\xi }}$ is an invariant subspace of $H^2$. It is known that there is an invariant subspace $M$ such that $ker{H}_{\overline{\xi }}\ne M$ for every $\xi \in L^{\infty }$. Let $\eta \in H^{\infty }$ be a nonconstant function. It is proved that if $\eta \perp \phi \left(z\right)H^2+\psi \left(w\right)H^2$ for Blaschke products $\phi \left(z\right)$ and $\psi \left(w\right)$, then $ker{H}_{\overline{\eta }}={\theta }_1\left(z\right){\theta }_2\left(w\right)H^2$ for some subproducts ${\theta }_1\left(z\right)$ and ${\theta }_2\left(w\right)$ of $\phi \left(z\right)$ and $\psi \left(w\right)$, respectively. If $\eta$ is $\ast$-cyclic, then it is easy to see that $ker{H}_{\overline{\eta }}=\left\{0\right\}$. We give some examples $\eta$ satisfying $ker{H}_{\overline{\eta }}=\left\{0\right\}$ but $\eta$ is not $\ast$-cyclic.

Let $n$ be any natural number. The $n$-centered operator is introduced for adjointable operators on Hilbert $C^{\ast }$-modules. Based on the characterizations of the polar decomposition for the product of two adjointable operators, $n$-centered operators, centered operators as well as binormal operators are clarified, and some results known for the Hilbert space operators are improved. It is proved that for an adjointable operator $T$, if $T$ is Moore–Penrose invertible and is $n$-centered, then its Moore–Penrose inverse is also $n$-centered. A Hilbert space operator $T$ is constructed such that $T$ is $n$-centered, whereas it fails to be $(n+1)$-centered.

We address questions on the existence and structure of universal functions for classes $L^p[0,1{)}^2$, $p\in (0,1)$, with respect to the double Walsh system. It is shown that there exists a measurable set $E\subset [0,1{)}^2$ with measure arbitrarily close to $1$, such that, by a proper modification of any integrable function $f\in L^1[0,1{)}^2$ outside $E$, we can get an integrable function $\tilde{f}\in L^1[0,1{)}^2$, which is universal for each class $L^p[0,1{)}^2$, $p\in (0,1)$, with respect to the double Walsh system in the sense of signs of Fourier coefficients.

We consider two types of maximal operators. We prove that, under some conditions, each maximal operator is bounded from the classical dyadic martingale Hardy space $H_p$ to the classical Lebesgue space $L_p$ and from the variable dyadic martingale Hardy space $H_{p(\cdot )}$ to the variable Lebesgue space $L_{p(\cdot )}$. Using this, we can prove the boundedness of the Cesàro and Riesz maximal operator from $H_{p(\cdot )}$ to $L_{p(\cdot )}$ and from the variable Hardy–Lorentz space $H_{p(\cdot ),q}$ to the variable Lorentz space $L_{p(\cdot ),q}$. As a consequence, we can prove theorems about almost everywhere and norm convergence.

In this article we study Hardy spaces ${\mathcal{H}}^{p,q}\left({\mathbb{R}}^d\right)$, $0<p,q<\infty$, modeled over amalgam spaces $(L^p,{\ell }^q)\left({\mathbb{R}}^d\right)$. We characterize ${\mathcal{H}}^{p,q}\left({\mathbb{R}}^d\right)$ by using first-order classical Riesz transforms and compositions of first-order Riesz transforms, depending on the values of the exponents $p$ and $q$. Also, we describe the distributions in ${\mathcal{H}}^{p,q}\left({\mathbb{R}}^d\right)$ as the boundary values of solutions of harmonic and caloric Cauchy–Riemann systems. We remark that caloric Cauchy–Riemann systems involve fractional derivatives in the time variable. Finally, we characterize the functions in $L^2\left({\mathbb{R}}^d\right)\cap {\mathcal{H}}^{p,q}\left({\mathbb{R}}^d\right)$ by means of Fourier multipliers $m_{\theta }$ with symbol $\theta (\cdot /|\cdot \left|\right)$, where $\theta \in C^{\infty }\left({\mathbb{S}}^{d-1}\right)$ and ${\mathbb{S}}^{d-1}$ denotes the unit sphere in ${\mathbb{R}}^d$.

We review some significant generalizations and applications of the celebrated Douglas theorem on equivalence of factorization, range inclusion, and majorization of operators. We then apply it to find a characterization of the positivity of $2\times 2$ block matrices of operators on Hilbert spaces. Finally, we describe the nature of such block matrices and provide several ways for showing their positivity.