We study symmetric and antisymmetric tensor products of Hilbert-space operators, focusing on norms and spectra for some well-known classes favored by function-theoretic operator theorists. We pose many open questions that should interest the field.

We improve and expand in two directions the theory of norms on complex matrices induced by random vectors. We first provide a simple proof of the classification of weakly unitarily invariant norms on the Hermitian matrices. We use this to extend the main theorem in Chávez, Garcia, and Hurley (2023, Canadian Mathematical Bulletin 66, 808–826) from exponent $d\geq 2$ to $d \geq 1$. Our proofs are much simpler than the originals: they do not require Lewis’ framework for group invariance in convex matrix analysis. This clarification puts the entire theory on simpler foundations while extending its range of applicability.

We discuss how countable subadditivity of operators can be derived from subadditivity under mild forms of continuity, and provide examples manifesting such circumstances.

We introduce a family of norms on the $n \times n$ complex matrices. These norms arise from a probabilistic framework, and their construction and validation involve probability theory, partition combinatorics, and trace polynomials in noncommuting variables. As a consequence, we obtain a generalization of Hunter’s positivity theorem for the complete homogeneous symmetric polynomials.

We study the classical Hermite–Hadamard inequality in the matrix setting. This leads to a number of interesting matrix inequalities such as the Schatten p-norm estimates

$$ \begin{align*}\left(\|A^q\|_p^p + \|B^q\|_p^p\right)^{1/p} \le \|(xA+(1-x)B)^q\|_p+ \|((1-x)A+xB)^q\|_p, \end{align*} $$

$n\times n$

$A,B$

$0<q,x<1$

$X^*X+Y^*Y=XX^*+YY^*=I$

$$ \begin{align*}(X^*AX+Y^*BY)\oplus (Y^*AY+X^*BX) =\frac{1}{2n}\sum_{k=1}^{2n} U_k (A\oplus B)U_k^*, \end{align*} $$

$2n\times 2n$

$U_k$

We study super weakly compact operators through a quantitative method. We introduce a semi-norm $\sigma (T)$ of an operator $T:X\to Y$ , where X, Y are Banach spaces, the so-called measure of super weak noncompactness, which measures how far T is from the family of super weakly compact operators. We study the equivalence of the measure $\sigma (T)$ and the super weak essential norm of T. We prove that Y has the super weakly compact approximation property if and and only if these two semi-norms are equivalent. As an application, we construct an example to show that the measures of T and its dual $T^*$ are not always equivalent. In addition we give some sequence spaces as examples of Banach spaces having the super weakly compact approximation property.

We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range $W(X)$ of the off-diagonal block $X$, especially the distance $d$ from $0$ to $W(X)$. A special consequence is an estimate,

$$\begin{eqnarray}\text{diam}\,W\left(\left[\begin{array}{@{}cc@{}}A & X\\ X^{\ast } & B\end{array}\right]\right)-\text{diam}\,W\biggl(\frac{A+B}{2}\biggr)\geq 2d,\end{eqnarray}$$

We establish inequalities of Jensen’s and Slater’s type in the general setting of a Hermitian unital Banach $\ast$-algebra, analytic convex functions and positive normalised linear functionals.

New inequalities relating the norm $n(X)$ and the numerical radius $w(X)$ of invertible bounded linear Hilbert space operators were announced by Hosseini and Omidvar [‘Some inequalities for the numerical radius for Hilbert space operators’, Bull. Aust. Math. Soc.94 (2016), 489–496]. For example, they asserted that $w(AB)\leq$$2w(A)w(B)$ for invertible bounded linear Hilbert space operators $A$ and $B$. We identify implicit hypotheses used in their discovery. The inequalities and their proofs can be made good by adding the extra hypotheses which take the form $n(X^{-1})=n(X)^{-1}$. We give counterexamples in the absence of such additional hypotheses. Finally, we show that these hypotheses yield even stronger conclusions, for example, $w(AB)=w(A)w(B)$.

We introduce some new refinements of numerical radius inequalities for Hilbert space invertible operators. More precisely, we prove that if $T\in {\mathcal{B}}({\mathcal{H}})$ is an invertible operator, then $\Vert T\Vert \leq \sqrt{2}\unicode[STIX]{x1D714}(T)$.

An investigation is made of the continuity, the compactness and the spectrum of the Cesàro operator $\mathsf{C}$ when acting on the weighted Banach sequence spaces $\ell _{p}(w)$, $1<p<\infty$, for a positive decreasing weight $w$, thereby extending known results for $\mathsf{C}$ when acting on the classical spaces $\ell _{p}$. New features arise in the weighted setting (for example, existence of eigenvalues, compactness) which are not present in $\ell _{p}$.

We continue the study of the boundedness of the operator

on the set of decreasing functions in Lp(w). This operator was first introduced by Braverman and Lai and also studied by Andersen, and although the weighted weak-type estimate was completely solved, the characterization of the weights w such that is bounded is still open for the case in which p > 1. The solution of this problem will have applications in the study of the boundedness on weighted Lorentz spaces of important operators in harmonic analysis.

The relative projection constant${\it\lambda}(Y,X)$ of normed spaces $Y\subset X$ is ${\it\lambda}(Y,X)=\inf \{\Vert P\Vert :P\in {\mathcal{P}}(X,Y)\}$, where ${\mathcal{P}}(X,Y)$ denotes the set of all continuous projections from $X$ onto $Y$. By the well-known result of Bohnenblust, for every $n$-dimensional normed space $X$ and a subspace $Y\subset X$ of codimension one, ${\it\lambda}(Y,X)\leq 2-2/n$. The main goal of the paper is to study the equality case in the theorem of Bohnenblust. We establish an equivalent condition for the equality ${\it\lambda}(Y,X)=2-2/n$ and present several applications. We prove that every three-dimensional space has a subspace with the projection constant less than $\frac{4}{3}-0.0007$. This gives a nontrivial upper bound in the problem posed by Bosznay and Garay. In the general case, we give an upper bound for the number of ($n-1$)-dimensional subspaces with the maximal relative projection constant in terms of the facets of the unit ball of $X$. As a consequence, every $n$-dimensional normed space $X$ has an ($n-1$)-dimensional subspace $Y$ with ${\it\lambda}(Y,X)<2-2/n$. This contrasts with the separable case in which it is possible that every hyperplane has a maximal possible projection constant.

Extending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra $\mathbb{B}\left( H \right)$ of bounded linear operators acting on a Hilbert space $H$. Among other things, we investigate the relationship between the approximate parallelism and norm of inner derivations on $\mathbb{B}\left( H \right)$. We also characterize the parallel elements of a ${{C}^{*}}$-algebra by using states. Finally we utilize the linking algebra to give some equivalent assertions regarding parallel elements in a Hilbert ${{C}^{*}}$-module.

Under the assumption that $\mu $ is a nondoubling measure, we study certain commutators generated by the Lipschitz function and the Marcinkiewicz integral whose kernel satisfies a Hörmander-type condition. We establish the boundedness of these commutators on the Lebesgue spaces, Lipschitz spaces, and Hardy spaces. Our results are extensions of known theorems in the doubling case.

This paper is inspired by a counter example of J. Kurzweil published in [5], whose intention was to demonstrate that a certain property of linear operators on finite-dimensional spaces need not be preserved in infinite dimension. We obtain a stronger result, which says that no infinite-dimensional Banach space can have the given property. Along the way, we will also derive an interesting proposition related to Dvoretzky's theorem.

Let $H$ be a separable, infinite-dimensional, complex Hilbert space and let $A,\,B\,\in \,\mathcal{L}\left( H \right)$, where $\mathcal{L}(H)$ is the algebra of all bounded linear operators on $H$. Let ${{\delta }_{AB}}\,:\mathcal{L}\left( H \right)\to \mathcal{L}\left( H \right)$ denote the generalized derivation ${{\delta }_{AB}}\left( X \right)\,=\,AX\,-\,XB$. This note will initiate a study on the class of pairs $\left( A,\,B \right)$ such that $\overline{R\left( {{\delta }_{AB}} \right)}\,=\,\overline{R\left( {{\delta }_{A*\,B*}} \right)}$.

Let $\mu $ be a nonnegative Radon measure on ${{\mathbb{R}}^{d}}$ that satisfies the growth condition that there exist constants ${{C}_{0}}\,>\,0$ and $n\,\in \,(0,\,d]$ such that for all $x\,\in \,{{\mathbb{R}}^{d}}$ and $r\,>\,0$, $\mu \left( B\left( x,\,r \right) \right)\,\le \,{{C}_{0}}{{r}^{n}}$, where $B(x,\,r)$ is the open ball centered at $x$ and having radius $r$. In this paper, the authors prove that if $f$ belongs to the $\text{BMO}$-type space $\text{RBMO(}\mu \text{)}$ of Tolsa, then the homogeneous maximal function ${{\dot{\mathcal{M}}}_{s}}\left( f \right)$ (when ${{\mathbb{R}}^{d}}$ is not an initial cube) and the inhomogeneous maximal function ${{\overset{{}}{\mathop{\mathcal{M}}}\,}_{s}}\left( f \right)$ (when ${{\mathbb{R}}^{d}}$ is an initial cube) associated with a given approximation of the identity $S $ of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, ${{\dot{\mathcal{M}}}_{s}}$ and ${{\mathcal{M}}_{s}}$ are bounded from $\text{RBMO(}\mu \text{)}$ to the $\text{BLO}$-type space $\text{RBMO(}\mu \text{)}$. The authors also prove that the inhomogeneous maximal operator ${{\mathcal{M}}_{s}}$ is bounded from the local $\text{BMO}$-type space $\text{rbmo(}\mu \text{)}$ to the local $\text{BLO}$-type space $\text{rblo(}\mu \text{)}$.

Let 𝒳 be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, two weighted estimates related to weights are established for singular integral operators with nonsmooth kernels via a new sharp maximal operator associated with a generalized approximation to the identity. As applications, the weighted Lp(𝒳) and weighted endpoint estimates with general weights are obtained for singular integral operators with nonsmooth kernels, their commutators with BMO (𝒳) functions, and associated maximal operators. Some applications to holomorphic functional calculi of elliptic operators and Schrödinger operators are also presented.

Let $A={{({{a}_{j,k}})}_{j,k\ge 1}}$ be a non-negative matrix. In this paper, we characterize those $A$ for which ${{\left\| A \right\|}_{E,F}}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces: ${{\ell }_{p}}$, $d(w,p)$, and ${{\ell }_{p}}(w)$. The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour.