Let $V$ be the set of $n\times n$ complex or real general matrices, Hermitian matrices, symmetric matrices, positive definite (resp. semi-definite) matrices, diagonal matrices, or upper triangular matrices. Fix $k\in \mathbb{Z}\setminus \{0, 1\}$. We characterize linear maps $ψ:V\to V$ that satisfy $ψ(A^k)=ψ(A)^k$ on an open neighborhood $S$ of $I_n$ in $V$. The $k$-power preservers are necessarily $k$-potent preservers, and the case $k=2$ corresponds to Jordan homomorphisms. Applying the results, we characterize maps $ϕ,ψ:V\to V$ that satisfy "$ \operatorname{tr}(ϕ(A)ψ(B)^k)=\operatorname{tr}(AB^k)$ for all $A\in V$, $B\in S$, and $ψ$ is linear" or "$ \operatorname{tr}(ϕ(A)ψ(B)^k)=\operatorname{tr}(AB^k)$ for all $A, B\in S$ and both $ϕ$ and $ψ$ are linear." The characterizations systematically extend existing results in literature, and they have many applications in areas like quantum information theory. Some structural theorems and power series over matrices are widely used in our characterizations.
Let T be be a zero-product preserving bounded linear map between C*-algebras A and B. Here neither A nor B is necessarily unital. In this note, we investigate when T gives rise to a Jordan homomorphism. In particular, we show that A and B are isomorphic as Jordan algebras if T is bijective and sends zero products of self-adjoint elements to zero products. They are isomorphic as C*-algebras if T is bijective and preserves the full zero product structure.
Let $\mathcal{B} (X)$ be the algebra of all bounded linear operators on an infinite-dimensional complex Banach space $X$. For an operator $ T \in \mathcal{B} (X)$, $K(T)$ denotes as usual the analytic core of $T$. We determine the form of surjective maps $ ϕ$ on $ \mathcal{B} (X)$ satisfying $$ K(ϕ(T) ϕ(S) + ϕ(S) ϕ(T)) = K(TS + ST) $$ for all $T, S \in \mathcal{B} (X)$.
Let $n_1,\ldots,n_k $ be integers larger than or equal to 2. We characterize linear maps $ϕ: M_{n_1\cdots n_k}\rightarrow M_{n_1\cdots n_k}$ such that $${\mathrm rank}\,(ϕ(A_1\otimes \cdots \otimes A_k))=1\quad\hbox{whenever}\quad{\mathrm rank}\, (A_1\otimes \cdots \otimes A_k)=1 \quad \hbox{for all}\quad A_i \in M_{n_i},\, i = 1,\dots,k.$$ Applying this result, we extend two recent results on linear maps that preserving the rank of special classes of matrices.
Denote by $M_n$ the set of $n\times n$ complex matrices. Let $f: M_n \rightarrow [0,\infty)$ be a continuous map such that $f(μUAU^*)= f(A)$ for any complex unit $μ$, $A \in M_n$ and unitary $U \in M_n$, $f(X)=0$ if and only if $X=0$ and the induced map $t \mapsto f(tX)$ is monotonic increasing on $[0,\infty)$ for any rank 1 nilpotent $X \in M_n$. Characterizations are given for surjective maps $ϕ$ on $M_n$ satisfying $f(AB-BA) = f(ϕ(A)ϕ(B)-ϕ(B)ϕ(A))$. The general theorem are then used to deduce results on special cases when the function is the pseudo spectrum and the pseudo spectral radius, t…
In a first objective we improve our understanding about surjective and bijective bounded linear operators preserving orthogonality from a JB$^*$-algebra $\mathcal{A}$ into a JB$^*$-triple $E$. Among many other conclusions, it is shown that a bounded linear bijection $T: \mathcal{A}\to E$ is orthogonality preserving if, and only if, it is biorthogonality preserving if, and only if, it preserves zero-triple-products in both directions (i.e., $\{a,b,c\}=0 \Leftrightarrow \{T(a),T(b),T(c)\}=0$). In the second main result we establish a complete characterization of all one-parameter groups of ortho…
Let ${\mathcal B}(H)$ denote the Banach algebra of all bounded linear operators on a complex Hilbert space $H$ with $\dim H\geq 3$, and let $\mathcal A$ and $\mathcal B$ be subsets of ${\mathcal B}(H)$ which contain all rank one operators. Suppose $F(\cdot )$ is a unitary invariant norm, the pseudo spectra, the pseudo spectral radius, the $C$-numerical range, or the $C$-numerical radius for some finite rank operator $C$. The structure is determined for surjective maps $Φ:{\mathcal A}\rightarrow \mathcal B$ satisfying $F(A^*B)=F(Φ(A)^*Φ(B))$ for all $A, B \in {\mathcal A}$. To establish the pro…
Due to the corresponding fact concerning Hilbert spaces, it is natural to ask if the linearity and the orthogonality structure of a Hilbert $C^*$-module determine its $C^*$-algebra-valued inner product. We verify this in the case when the $C^*$-algebra is commutative (or equivalently, we consider a Hilbert bundle over a locally compact Hausdorff space). More precisely, a $\mathbb{C}$-linear map $θ$ (not assumed to be bounded) between two Hilbert $C^*$-modules is said to be "orthogonality preserving" if $\left<θ(x),θ(y)\right> =0$ whenever $\left<x,y\right> =0$. We prove that if $θ$ is an ortho…
In this paper, we describe linear maps between complex Banach algebras that preserve products equal to fixed elements. This generalizes some important special cases where the fixed elements are the zero or identity element. First we show that if such map preserves products equal to a finite-rank operator, then it must also preserve the zero product. In several instances, this is enough to show that a product preserving map must be a scalar multiple of an algebra homomorphism. Second, we explore a more general problem concerning the existence of product preserving maps and the relationship betw…
In this paper we continue the study of groups of trace class and consider in particular the case of semi-direct products. One of the highlights is the theorem saying that the semi-direct product of a semisimple Lie group G and its Lie algebra is a trace class group if and only if G is compact.
We prove the linearity and injectivity of two maps $ϕ_1$ and $ϕ_2$ on certain subsets of $M_n$ that satisfy $\operatorname{tr}(ϕ_1(A)ϕ_2(B))=\operatorname{tr}(AB)$. We apply it to characterize maps $ϕ_i:\mathcal{S}\to \mathcal{S}$ ($i=1, \ldots, m$) satisfying $$\operatorname{tr} (ϕ_1(A_1)\cdots ϕ_m(A_m))=\operatorname{tr} (A_1\cdots A_m)$$ in which $\mathcal{S}$ is the set of $n$-by-$n$ general, Hermitian, or symmetric matrices for $m\ge 3$, or positive definite or diagonal matrices for $m\ge 2$. The real versions are also given.
A Lie group G is called a trace class group if for every irreducible unitary representation R of G and every C-infinity function f with compact support the operator R(f) is of trace class. In this note we prove that the semidirect product of R^n and a real semisimple algebraic subgroup G of GL(n;R) is a trace class group only if G is compact. The converse has been shown elsewhere. We also make a descent start with the study of semidirect products with Heisenberg-type groups.
In this article we study simplicity and traces of reduced $L^p$ operator crossed products $F^p_{\mathrm{r}}(G, A, α)$. Given $p \in (1, \infty)$, let $G$ be a Powers group, and let $α\colon G \to Aut(A)$ be an isometric action of $G$ on a unital $L^p$ operator algebra $A$ such that $A$ is $G$-simple. We prove that the reduced $L^p$ operator crossed product of $A$ by $G$, $F^p_{\mathrm{r}}(G, A, α)$, is simple. Moreover, we show that traces on $F^p_{\mathrm{r}}(G, A, α)$ are in correspondence with $G$-invariant traces on A. Our results generalize the results obtained by de la Harpe for reduced …
This note reports partial results related to the Gaussian product inequality (GPI) conjecture for the joint distribution of traces of Wishart matrices. In particular, several GPI-related results from Wei (2014) and Liu et al. (2015) are extended in two ways: by replacing the power functions with more general classes of functions, and by replacing the usual Gaussian and multivariate gamma distributional assumptions by the more general trace-Wishart distribution assumption. These findings suggest that a Kronecker product form of the GPI holds for diagonal blocks of any Wishart distribution.
This paper explores the long journey from projective tensor products of a pair of Banach spaces, passing through the definition of nuclear operators still on the realm of projective tensor products, to the of notion of trace-class operators on a Hilbert space, and shows how and why these concepts (nuclear and trace-class operators, that is) agree in the end.
The Kerman-Sawyer trace theorem for product Morrey spaces
Graduate School of Information Science and Technology, The University of Osaka（大阪大学信息科学技术研究科）；Center for Mathematics and Data Science, Gunma University（群马大学数学与数据科学中心）；College of Science and Technology, Nihon University（日本大学理工学部）；Research and Support Center on Higher Education for the hearing and Visually Impaired, National University Corporation Tsukuba University of Technology（筑波技术大学国立大学法人听力及视力障碍高等教育研究与支援中心）
We propose a new class of traces motivated by a trace/trace class property discovered by Laurie, Nordgren, Radjavi and Rosenthal concerning products of operators outside the trace class. Spectral traces, traces that depend only on the spectrum and algebraic multiplicities, possess this property and we suspect others do, but we know of no other traces that do. This paper is intended to be part survey. We provide here a brief overview of some facts concerning traces on ideals, especially involving Lidskii formulas and spectral traces. We pose the central question: whenever the relevant products,…
We give criteria for products of Toeplitz and Hankel operators on the Fock (Segal-Bargmann) space to belong to the Dixmier class, and compute their Dixmier trace. At the same time, analogous results for the Weyl pseudodifferential operators are also obtained.
With techniques borrowed from quantum information theory, we develop a method to systematically obtain operator inequalities and identities in several matrix variables. These take the form of trace polynomials: polynomial-like expressions that involve matrix monomials $X_{α_1} \cdots X_{α_r}$ and their traces $\operatorname{tr}(X_{α_1} \cdots X_{α_r})$. Our method rests on translating the action of the symmetric group on tensor product spaces into that of matrix multiplication. As a result, we extend the polarized Cayley-Hamilton identity to an operator inequality on the positive cone, charact…
