Let $m,n\ge 2$ be integers. Denote by $M_n$ the set of $n\times n$ complex matrices and $\|\cdot \|_{(p,k)}$ the $(p,k)$ norm on $M_{mn}$ with a positive integer $k\leq mn$ and a real number $p>2$. We show that a linear map $\phi :M_{mn}\rightarrow M_{mn}$ satisfies

$$ \begin{align*}\|\phi(A\otimes B)\|_{(p,k)}=\|A\otimes B\|_{(p,k)} \mathrm{\quad for~ all\quad}A\in M_m\ \mathrm{and}\ B\in M_n\end{align*} $$

if and only if there exist unitary matrices $U,V\in M_{mn}$ such that

$$ \begin{align*}\phi(A\otimes B)=U(\varphi_1(A)\otimes \varphi_2(B))V \mathrm{\quad for~ all\quad}A\in M_m\ \mathrm{and}\ B\in M_n,\end{align*} $$

where $\varphi _s$ is the identity map or the transposition map $X\mapsto X^T$ for $s=1,2$. The result is also extended to multipartite systems.

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.

The existence of isometric embedding of $S_q^m$ into $S_p^n$, where $1\leq p\neq q\leq \infty$ and $m,n\geq 2$, has been recently studied in [6]. In this article, we extend the study of isometric embeddability beyond the above-mentioned range of $p$ and $q$. More precisely, we show that there is no isometric embedding of the commutative quasi-Banach space $\ell _q^m(\mathbb {R})$ into $\ell _p^n(\mathbb {R})$, where $(q,p)\in (0,\infty )\times (0,1)$ and $p\neq q$. As non-commutative quasi-Banach spaces, we show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in (0,2)\setminus \{1\}\times (0,1)$ $\cup \, \{1\}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$ $\cup \, \{\infty \}\times (0,1)\setminus \left \{\!\frac {1}{n}:n\in \mathbb {N}\right \}$ and $p\neq q$. Moreover, in some restrictive cases, we also show that there is no isometric embedding of $S_q^m$ into $S_p^n$, where $(q,p)\in [2, \infty )\times (0,1)$. A new tool in our paper is the non-commutative Clarkson's inequality for Schatten class operators. Other tools involved are the Kato–Rellich theorem and multiple operator integrals in perturbation theory, followed by intricate computations involving power-series analysis.

In 2005, N. Nikolski proved among other things that for any $r\in (0,1)$ and any $K\geq 1$, the condition number $CN(T)=\Vert T\Vert \cdot \Vert T^{-1}\Vert $ of any invertible n-dimensional complex Banach space operators T satisfying the Kreiss condition, with spectrum contained in $\left \{ r\leq |z|<1\right \}$, satisfies the inequality $CN(T)\leq CK(T)\Vert T \Vert n/r^{n}$ where $K(T)$ denotes the Kreiss constant of T and $C>0$ is an absolute constant. He also proved that for $r\ll 1/n,$ the latter bound is asymptotically sharp as $n\rightarrow \infty $. In this note, we prove that this bound is actually achieved by a family of explicit $n\times n$ Toeplitz matrices with arbitrary singleton spectrum $\{\lambda \}\subset \mathbb {D}\setminus \{0\}$ and uniformly bounded Kreiss constant. Independently, we exhibit a sequence of Jordan blocks with Kreiss constants tending to $\infty $ showing that Nikolski’s inequality is still asymptotically sharp as K and n go to $\infty $.

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 investigate a class of generalised stochastic complex matrices constructed from the class of all doubly stochastic matrices and a special class of circulant matrices. We determine the exact values of the structured singular values of all matrices in the class in terms of the constant row (column) sum.

We continue our investigation of the real space H of Hermitian matrices in $${M_n}(\mathbb{C})$$ with respect to norms on $${\mathbb{C}^n}$$. We complete the commutative case by showing that any proper real subspace of the real diagonal matrices on $${\mathbb{C}^n}$$ can appear as H. For the non-commutative case, we give a complete solution when n=3 and we provide various illustrative examples for n ≥ 4. We end with a short list of problems.

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 investigate the real space H of Hermitian matrices in $M_n(\mathbb{C})$ with respect to norms on $\mathbb{C}^n$ . For absolute norms, the general form of Hermitian matrices was essentially established by Schneider and Turner [Schneider and Turner, Linear and Multilinear Algebra (1973), 9–31]. Here, we offer a much shorter proof. For non-absolute norms, we begin an investigation of H by means of a series of examples, with particular reference to dimension and commutativity.

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.

In this paper we generalize the notion of the C-numerical range of a matrix to operators in arbitrary tracial von Neumann algebras. For each self-adjoint operator C, the C-numerical range of such an operator is defined; it is a compact, convex subset of ℂ. We explicitly describe the C-numerical ranges of several operators and classes of operators.

In this paper we establish operator quasilinearity properties of some functionals associated with Davis–Choi–Jensen’s inequality for positive maps and operator convex or concave functions. Applications for the power function and the logarithm are provided.

Let $\mathbf{A}$ be a density matrix in ${{\mathbb{M}}_{m}}\,\otimes \,{{\mathbb{M}}_{n}}$. Audenaert [J. Math. Phys. 48(2007) 083507] proved an inequality for Schatten $p$-norms:

$$1\,+\,||\mathbf{A}|{{|}_{p}}\,\ge \,{{\left\| \text{T}{{\text{r}}_{1}}\,\mathbf{A} \right\|}_{p}}\,+\,||\text{T}{{\text{r}}_{2}}\,\mathbf{A}|{{|}_{p}},$$

where $\text{T}{{\text{r}}_{1}}$ and $\text{T}{{\text{r}}_{2}}$ stand for the first and second partial trace, respectively. As an analogue of his result, we prove a determinantal inequality

$$1\,+\,\det \,\mathbf{A}\,\ge \,\det {{\left( \text{T}{{\text{r}}_{1}}\mathbf{A} \right)}^{m}}\,+\,\det {{\left( \text{T}{{\text{r}}_{2}}\mathbf{A} \right)}^{2}}.$$

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.

We prove a normalized version of the restricted invertibility principle obtained by Spielman and Srivastava in [An elementary proof of the restricted invertibility theorem. Israel J. Math. 190 (2012), 83–91]. Applying this result, we get a new proof of the proportional Dvoretzky–Rogers factorization theorem recovering the best current estimate in the symmetric setting while we improve the best known result in the non-symmetric case. As a consequence, we slightly improve the estimate for the Banach–Mazur distance to the cube: the distance of every $n$-dimensional normed space from ${ \ell }_{\infty }^{n} $ is at most $\mathop{(2n)}\nolimits ^{5/ 6} $. Finally, using tools from the work of Batson et al in [Twice-Ramanujan sparsifiers. In STOC’09 – Proceedings of the 2009 ACM International Symposium on Theory of Computing, ACM (New York, 2009), 255–262], we give a new proof for a theorem of Kashin and Tzafriri [Some remarks on the restriction of operators to coordinate subspaces. Preprint, 1993] on the norm of restricted matrices.

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.

The tracial numerical range of operators on a 2-dimensional Krein space is investigated. Results in the vein of those obtained in the context of Hilbert spaces are obtained.

Let Mn, be the algebra of all n × n matrices over a field F, where n ≧ 2. Let S be a subset of Mn containing all rank one matrices. We study mappings φ: S → Mn, such that F(φ (A)φ (B)) = F(A B) for various families of functions F including all the unitary similarity invariant functions on real or complex matrices. Very often, these mappings have the form A ↦ μ(A)S(σ (aij))S-1 for all A= (aij) ∈ S for some invertible S ∈ Mn, field monomorphism σ of F, and an F*-valued mapping μ defined on S. For real matrices, σ is often the identity map; for complex matrices, σ is often the identity map or the conjugation map: z ↦ z. A key idea in our study is reducing the problem to the special case when F:Mn → {0, 1} is defined by F(X) = 0, if X = 0, and F(X) = 1 otherwise. In such a case, one needs to characterize φ: S → Mn such that φ(A) φ (B) = 0 if and only if AB = 0. We show that such a map has the standard form described above on rank one matrices in S.

As an attempt to understand linear isometries between C*-algebras without the surjectivity assumption, we study linear isometries between matrix algebras. Denote by Mm the algebra of m × m complex matrices. If k ≥ n and φ: Mn → Mk has the form X ↦ U[X ⊕ f(X)] V or X ↦ U[X1 ⊕ f(X)]V for some unitary U, V ∈ Mk and contractive linear map f: Mn → Mk, then ║φ(X)║ = ║X║ for all X ∈ Mn. We prove that the converse is true if k ≤ 2n - 1, and the converse may fail if k ≥ 2n. Related results and questions involving positive linear maps and the numerical range are discussed.