Let T be a bounded linear operator on a separable Banach space that satisfies geometric properties similar to those of $\ell ^p,\, p>1$. We prove that the smallest and the largest norm of weak cluster points of all maximizing sequences for T can only take the values $0$ or $1$. The three classes of bounded linear operators emerging from the dichotomy of these extremal norm values coincide with the partition, created by considering the norm-attaining property and if the essential norm equals the norm.

We develop several $\ell ^p$-operator norm inequalities for $k\times k$ block matrices defined on the $\ell ^p$-sum of Banach spaces. Using these inequalities, we obtain p-numerical radius and spectral radius bounds for $k\times k$ block matrices. We deduce a p-numerical radius bound for the Kronecker product $A\otimes B$, where $A\in {M}_k(\mathbb {C})$ is a $k\times k$ complex matrix and $B\in \mathcal {L}(\mathbb {H})$ is a bounded linear operator on a complex Hilbert space $\mathbb {H}$. This improves and extends Holbrook’s bound $w(A\otimes B)\leq w(A)\|B\|.$ If $\|A\|_{\ell ^p}$ and $w_p(A)$ denote the $\ell ^p$-operator norm and p-numerical radius of $A\in {M}_k(\mathbb {C})$, respectively, then it is shown that

$$ \begin{align*} \frac{1}{2}\|A\|_p+\mu_p(A) \leq w_p(A), \end{align*} $$

$\mu _p(A)$

$\ell ^p$

$\frac {1}{2}\|A\|_p= w_p(A)$

In the present article, we study the norms for symmetric and antisymmetric tensor products of weighted shift operators. By proving that for $n\geq 2$,

$$ \begin{align*}\|S_{\alpha}^{l_1}\odot\cdots \odot S_{\alpha}^{l_k}\odot S_{\alpha}^{*l_{k+1}}\odot\cdots \odot S_{\alpha}^{*l_{n}}\| =\mathop{\prod}_{i=1}^n\left \| S_{\alpha}^{{l_{i}}}\right\|, \text{ for any} \ (l_1,l_2\ldots l_n)\in\mathbb N^n\end{align*} $$

In this article, motivated by the regularity theory of the solutions of doubly nonlinear parabolic partial differential equations, the authors introduce the off-diagonal two-weight version of the parabolic Muckenhoupt class with time lag. Then the authors introduce the uncentered parabolic fractional maximal operator with time lag and characterize its two-weighted boundedness (including the endpoint case) in terms of these weights under an additional mild assumption (which is not necessary for one-weight case). The most novelty of this article exists in that the authors further introduce a new parabolic shaped domain and its corresponding parabolic fractional integral with time lag and, moreover, applying the aforementioned (two-)weighted boundedness of the parabolic fractional maximal operator with time lag, the authors characterize the (two-)weighted boundedness (including the endpoint case) of these parabolic fractional integrals in terms of the off-diagonal (two-weight) parabolic Muckenhoupt class with time lag; as applications, the authors further establish a parabolic weighted Sobolev embedding and a priori estimate for the solution of the heat equation. The key tools to achieve these include the parabolic Calderón–Zygmund-type decomposition, the chaining argument, and the parabolic Welland inequality, which is obtained by making the utmost of the geometrical relation between the parabolic shaped domain and the parabolic rectangle.

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 $\Vert T\Vert \leq \sqrt{2}\unicode[STIX]{x1D714}(T)$ .

In this paper, the commutators of singular integrals with rough kernels are considered. By the method of block decomposition for kernel function and Fourier transform estimates, some new results about the Lp (ℝn) boundedness for these commutators are obtained.

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.