Let $A$ be a $C^{\ast }$-algebra, and let $A^{\ast \ast }$ be its enveloping von Neumann algebra. Akemann suggested a kind of noncommutative topology in which certain projections in $A^{\ast \ast }$ play the role of open sets, and he used two operator inequalities in connection with compactness. Both of these inequalities are equivalent to compactness for a closed projection in $A^{\ast \ast }$, but only one is equivalent to relative compactness for a general projection. A third operator inequality, also related to compactness, was used by the author. The study of all three inequalities can be unified by considering a numerical invariant which is equivalent to the distance of a projection from the set of relatively compact projections. Tomita’s concept of regularity of projections seems relevant, and so we give some results and examples on regularity. We also include a few related results on semicontinuity.

We consider the Calderón–Lozanovskii construction $\phi (X_0,X_1)$ in the context of quasi-Banach lattices, and we provide an extension of a result by Ovchinnikov concerning the associated interpolation methods ${\phi }^c$ and ${\phi }^0$. Our approach is based on the interpolation properties of $(\infty ,1)$-regular operators between quasi-Banach lattices.

Using the local approach to the global structure of a symmetric space $E$, we establish a relationship between strict $K$-monotonicity, lower (resp., upper) local uniform $K$-monotonicity, order continuity, and the Kadec–Klee property for global convergence in measure. We also answer the question: Under which condition does upper local uniform $K$-monotonicity coincide with upper local uniform monotonicity? Finally, we present a correlation between $K$-order continuity and lower local uniform $K$-monotonicity in a symmetric space $E$ under some additional assumptions on $E$.

We study totally bounded sets in variable Lebesgue spaces. The full characterization of this kind of sets is given for the case of variable Lebesgue space on metric measure spaces. Furthermore, the sufficient conditions for compactness are shown without assuming $log$-Hölder continuity of the exponent.

We use the duality between functional and differential equations to solve several classes of abstract Cauchy problems related to special functions. As a general framework, we investigate operator functions which are multiplicative with respect to convolution of a hypergroup. This setting contains all representations of (hyper)groups, and properties of continuity are shown; examples are provided by translation operator functions on homogeneous Banach spaces and weakly stationary processes indexed by hypergroups. Then we show that the concept of a multiplicative operator function can be used to solve a variety of abstract Cauchy problems, containing discrete, compact, and noncompact problems, including $C_0$-groups and cosine operator functions, and more generally, Sturm–Liouville operator functions.

If $\mu$ is a positive Borel measure on the interval $[0,1)$, we let ${\mathcal{H}}_{\mu }$ be the Hankel matrix ${\mathcal{H}}_{\mu }=({\mu }_{n,k}{)}_{n,k\ge 0}$ with entries ${\mu }_{n,k}={\mu }_{n+k}$, where, for $n=0,1,2,\dots$, ${\mu }_n$ denotes the moment of order $n$ of $\mu$. This matrix formally induces the operator

$${\mathcal{H}}_{\mu }\left(f\right)\left(z\right)={\sum }_{n=0}^{\infty }\left({\sum }_{k=0}^{\infty }{\mu }_{n,k}{a}_k\right)z^n$$ on the space of all analytic functions $f\left(z\right)={\sum }_{k=0}^{\infty }{a}_k{z}^k$, in the unit disk $\mathbb{D}$. This is a natural generalization of the classical Hilbert operator. The action of the operators $H_{\mu }$ on Hardy spaces has been recently studied. This article is devoted to a study of the operators $H_{\mu }$ acting on certain conformally invariant spaces of analytic functions on the disk such as the Bloch space, the space BMOA, the analytic Besov spaces, and the $Q_s$-spaces.

In this article the authors present a systematic study of several new $L^p$-boundedness properties and Parseval-type relations concerning the operators with complex Gaussian kernels over the spaces $L^p(\mathbb{R},cosh(\alpha x\left)dx\right)$ and $L^p(\mathbb{R},cosh(\alpha x^2\left)dx\right)$, $1\le p\le \infty$, $\alpha \in \mathbb{R}$. Relevant connections with various earlier related results are also pointed out.

In this paper we prove that, in the case of some unbounded Vilenkin groups, the Riesz logarithmic means converges in the norm of the spaces $X\left(G\right)$ for every $f\in X\left(G\right)$, where by $X\left(G\right)$ we denote either the class of continuous functions with supremum norm or the class of integrable functions.

In this article, we consider some algebraic properties of Toeplitz operators on weighted pluriharmonic Bergman space on the unit ball. We characterize the commutants of Toeplitz operators whose symbols are certain separately radial functions or holomorphic monomials, and then give a partial answer to the finite-rank product problem of Toeplitz operators.

In this paper, we give a uniform characterization for the reducing subspaces for $T_{\phi }$ with the symbol $\phi \left(z\right)=z^k+{\overline{z}}^l$ ($k,l\in {\mathbb{Z}}_{+}^2$) on the Bergman spaces over the bidisk, including the known cases that $\phi (z_1,z_2)=z_1^N{z}_2^M$ and $\phi (z_1,z_2)=z_1^N+{\overline{z}}_2^M$ with $N,M\in {\mathbb{Z}}_{+}$. Meanwhile, the reducing subspaces for $T_{z^N+{\overline{z}}^M}$ ($N,M\in {\mathbb{Z}}_{+}$) on the Bergman space over the unit disk are also described. Finally, we state these results in terms of the commutant algebra ${\mathcal{V}}^{\ast }\left(\phi \right)$.

We extend the notion of completely rank-nonincreasing (CRNI) linear maps to include the multilinear maps. We show that a bilinear map on a finite-dimensional vector space on any field is CRNI if and only if it is a skew-compression bilinear map. We also characterize CRNI continuous bilinear maps defined on the set of compact operators.

The operator ideal of $(p,q)$-factorable operators can be characterized as the class of operators that factors through the embedding $L^{q\text{'}}\left(\mu \right)\hookrightarrow L^p\left(\mu \right)$ for a finite measure $\mu$, where $p,q\in [1,\infty )$ are such that $1/p+1/q\ge 1$. We prove that this operator ideal is included into a Banach operator ideal characterized by means of factorizations through $r$th and $s$th power factorable operators, for suitable $r,s\in [1,\infty )$. Thus, they also factor through a positive map $L^s(m_1{)}^{*}\to L^r(m_2)$, where $m_1$ and $m_2$ are vector measures. We use the properties of the spaces of $u$-integrable functions with respect to a vector measure and the $u$th power factorable operators to obtain a characterization of $(p,q)$-factorable operators and conditions under which a $(p,q)$-factorable operator is $r$-summing for $r\in [1,p]$.