We show that a regular locally compact quantum group $\mathbb{G}$ is discrete if and only if $\mathcal{L}^{\infty}(\mathbb{G})$ contains non-zero compact operators on $\mathcal{L}^{2}(\mathbb{G})$. As a corollary we classify all discrete quantum groups among regular locally compact quantum groups $\mathbb{G}$ where $\mathcal{L}^{1}(\mathbb{G})$ has the Radon--Nikodym property.

We show that for $p$-operator spaces, there are natural notions of minimal and maximal structures. These are useful for dealing with tensor products.

In the paper, we characterize Jordan $*$-derivations of a $2$-torsion free, finite-dimensional semiprime algebra $R$ with involution $*$. To be precise, we prove the theorem: Let $deltacolon R o R$ be a Jordan $*$-derivation. Then there exists a $*$-algebra decomposition $R=Uoplus V$ such that both $U$ and $V$ are invariant under $delta$. Moreover, $*$ is the identity map of $U$ and $delta,|_U$ is a derivation, and the Jordan $*$-derivation $delta,|_V$ is inner. We also prove the theorem: Let $R$ be a noncommutative, centrally closed prime algebra with involution $*$, $operatorname{char},R e 2$, and let $delta$ be a nonzero Jordan $*$-derivation of $R$. If $delta$ is an elementary operator of $R$, then $operatorname{dim}_CRlt infty$ and $delta$ is inner.

We prove weak-type $(1,1)$ estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $\Delta^*\Psi$ where $\Delta^*$ is Bourgain's maximal multiplier operator and $\Psi$ is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the $L^q$ operator norm when $1 \lt q \lt 2.$ We also consider associated variation-norm estimates.

Let $T$ be a contraction on a complex, separable, infinite dimensional Hilbert space and let $\sigma \left( T\right) $ (resp. $\sigma _{e}\left( T\right) )$ be its spectrum (resp. essential spectrum). We assume that $T$ is an essentially isometric operator, that is $I_{H}-T^{\ast }T$ is compact. We show that if $D\diagdown \sigma \left( T\right) \neq \emptyset ,$ then for every $f$ from the disc-algebra, \begin{equation*} \sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma _{e}\left( T\right) \right) , \end{equation*} where $D$ is the open unit disc. In addition, if $T$ lies in the class $ C_{0\cdot }\cup C_{\cdot 0},$ then \begin{equation*} \sigma _{e}\left( f\left( T\right) \right) =f\left( \sigma \left( T\right) \cap \Gamma \right) , \end{equation*} where $\Gamma $ is the unit circle. Some related problems are also discussed.

Some subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional $\varepsilon$ term. It seems not possible to erase this residual term. However, in case of compact operators we show that the $\varepsilon$ term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also stresses on matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings.

For Banach spaces $X$ and $Y$, we show that if $X^\ast$ and $Y$ are weakly sequentially complete and every weakly compact operator from $X$ to $Y$ is compact then the space of all compact operators from $X$ to $Y$ is weakly sequentially complete. The converse is also true if, in addition, either $X^\ast$ or $Y$ has the bounded compact approximation property.

We prove the non-existence of real hypersurfaces in complex projective space whose structure Jacobi operator is Lie $\mathbb{D}$-parallel and satisfies a further condition.

We give a partial answer to a conjecture of Dostanić on the determination of the norm of a class of integral operators induced by the weighted Bergman projection in the upper half plane.

We give sufficient conditions for the following problem: given a topological space $X$, a metric space $Y$, a subspace $Z$ of $Y$, and a continuous map $f$ from $X$ to $Y$, is it possible, by applying to $f$ an arbitrarily small perturbation, to ensure that $f(X)$ does not meet $Z$? We also give a relative variant: if $f(X')$ does not meet $Z$ for a certain subset $X'\subset X$, then we may keep $f$ unchanged on $X'$. We also develop a variant for continuous sections of fibrations and discuss some applications to matrix perturbation theory.

In this article we study the Cesàro operator $$ \mathcal{C}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\zeta)\,d\zeta, $$ and its companion operator $\mathcal{T}$ on Hardy spaces of the upper half plane. We identify $\mathcal{C}$ and $\mathcal{T}$ as resolvents for appropriate semigroups of composition operators and we find the norm and the spectrum in each case. The relation of $\mathcal{C}$ and $\mathcal{T}$ with the corresponding Ces\`{a}ro operators on Lebesgue spaces $L^p(\mathbb R)$ of the boundary line is also discussed.

A vector measure result is used to study the complementation of the space $K(X,Y)$ of compact operators in the spaces $W(X,Y)$ of weakly compact operators, $CC(X,Y)$ of completely continuous operators, and $U(X,Y)$ of unconditionally converging operators. Results of Kalton and Emmanuele concerning the complementation of $K(X,Y)$ in $L(X,Y)$ and in $W(X,Y)$ are generalized. The containment of $c_0$ and $\ell_\infty$ in spaces of operators is also studied.

In this paper we prove weighted norm inequalities with weights in the $A_p$ classes, for pseudodifferential operators with symbols in the class ${S^{n(\rho -1)}_{\rho, \delta}}$ that fall outside the scope of Calderón-Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy-Littlewood type maximal functions. Our weighted norm inequalities also yield $L^{p}$ boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in $\mathrm{OP}S^{m}_{\rho, \delta}$.

It is shown that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. A different proof is given in the well known special case of an abelian radical. A result relating the number of invariants to the dimension of the Cartan subalgebra is also established.

We study the complementation of the space $W(X,Y)$ of weakly compact operators, the space $K(X,Y)$ of compact operators, the space $U(X,Y)$ of unconditionally converging operators, and the space $CC(X,Y)$ of completely continuous operators in the space $L(X,Y)$ of bounded linear operators from $X$ to $Y$. Feder proved that if $X$ is infinite-dimensional and $c_0 \hookrightarrow Y$, then $K(X,Y)$ is uncomplemented in $L(X,Y)$. Emmanuele and John showed that if $c_0 \hookrightarrow K(X,Y)$, then $K(X,Y)$ is uncomplemented in $L(X,Y)$. Bator and Lewis showed that if $X$ is not a Grothendieck space and $c_0 \hookrightarrow Y$, then $W(X,Y)$ is uncomplemented in $L(X,Y)$. In this paper, classical results of Kalton and separably determined operator ideals with property $(*)$ are used to obtain complementation results that yield these theorems as corollaries.

The Banach spaces $L(X, Y)$, $K(X, Y)$, $L_{w^*}(X^*, Y)$, and $K_{w^*}(X^*, Y)$ are studied to determine when they contain the classical Banach spaces $c_o$ or $\ell_\infty$. The complementation of the Banach space $K(X, Y)$ in $L(X, Y)$ is discussed as well as what impact this complementation has on the embedding of $c_o$ or $\ell_\infty$ in $K(X, Y)$ or $L(X, Y)$. Results of Kalton, Feder, and Emmanuele concerning the complementation of $K(X, Y)$ in $L(X, Y)$ are generalized. Results concerning the complementation of the Banach space $K_{w^*}(X^*, Y)$ in $L_{w^*}(X^*, Y)$ are also explored as well as how that complementation affects the embedding of $c_o$ or $\ell_\infty$ in $K_{w^*}(X^*, Y)$ or $L_{w^*}(X^*, Y)$. The $\ell_p$ spaces for $1 = p < \infty$ are studied to determine when the space of compact operators from one $\ell_p$ space to another contains $c_o$. The paper contains a new result which classifies these spaces of operators. A new result using vector measures is given to provide more efficient proofs of theorems by Kalton, Feder, Emmanuele, Emmanuele and John, and Bator and Lewis.

Following a similar result of Uspenskij on the unitary group of a separable Hilbert space, we show that, with respect to the lower (or Roelcke) uniform structure, the Polish group $G= \operatorname{Aut}(\mu)$ of automorphisms of an atomless standard Borel probability space $(X,\mu)$ is precompact. We identify the corresponding compactification as the space of Markov operators on $L_2(\mu)$ and deduce that the algebra of right and left uniformly continuous functions, the algebra of weakly almost periodic functions, and the algebra of Hilbert functions on $G$, i.e., functions on $G$ arising from unitary representations, all coincide. Again following Uspenskij, we also conclude that $G$ is totally minimal.

In this note we study the convergence of sequences of Monge-Ampère measures $\{(dd^cu_s)^n\}$, where $\{u_s\}$ is a given sequence of plurisubharmonic functions, converging in capacity.

We comment on domain conditions that regulate when the adjoint of the sum or product of two unbounded operators is the sum or product of their adjoints, and related closure issues. The quantum mechanical problem PHP essentially selfadjoint for unbounded Hamiltonians is addressed, with new results.

While there is a large variety of univalently induced closed range composition operators on the Bloch space, we show that the only univalently induced, closed range, composition operators on the Bloch-type spaces $B^{\alpha}$ with $\alpha \ne 1$ are the ones induced by a disc automorphism.

We generalize Löwner's method for proving that matrix monotone functions are operator monotone. The relation $x\leq y$ on bounded operators is our model for a definition of $C^{*}$-relations being residually finite dimensional. Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved and verify a technical condition, then such a theorem will follow from its restriction to matrices. Applications are shown regarding norms of exponentials, the norms of commutators, and "positive" noncommutative $*$-polynomials.

We prove, under some conditions on the domains, that the adjoint of the sum of two unbounded operators is the sum of their adjoints in both Hilbert and Banach space settings. A similar result about the closure of operators is also proved. Some interesting consequences and examples "spice up" the paper.

In this paper we study real hypersurfaces in a non-flat complex space form with $\eta$-parallel shape operator. Several partial characterizations of these real hypersurfaces are obtained.

Let $D$ be a connected bounded domain in $\mathbb{R}^n$. Let $0<\mu_1\leq\mu_2\leq\dots\leq\mu_k\leq\cdots$ be the eigenvalues of the following Dirichlet problem: $$ \begin{cases}\Delta^2u(x)+V(x)u(x)=\mu\rho(x)u(x),\quad x\in D u|_{\partial D}=\frac{\partial u}{\partial n}|_{\partial D}=0, \end{cases} $$ where $V(x)$ is a nonnegative potential, and $\rho(x)\in C(\bar{D})$ is positive. We prove the following inequalities: $$\mu_{k+1}\leq\frac{1}{k}\sum_{i=1}^k\mu_i+\Bigl[\frac{8(n+2)}{n^2}\Bigl(\frac{\rho_{\max}} {\rho_{\min}}\Bigr)^2\Bigr]^{1/2}\times \frac{1}{k}\sum_{i=1}^k[\mu_i(\mu_{k+1}-\mu_i)]^{1/2}, $$ $$\frac{n^2k^2}{8(n+2)}\leq \Bigl(\frac{\rho_{\max}}{\rho_{\min}}\Bigr)^2\Bigl[\sum_{i=1}^k\frac{\mu_i^{1/2}}{\mu_{k+1}-\mu_i}\Bigr] \times\sum_{i=1}^k\mu_i^{1/2}. $$

We give a characterization of contact manifolds in terms of symplectic Lie-Rinehart-Jacobi algebras. We also give a sufficient condition for a Jacobi manifold to be a contact manifold.