Dorin Ervin Dutkay
We present a unitary approach to the construction of representations and intertwining operators. We apply it to the $C^*$-algebras, groups, Gabor type unitary systems and wavelets. We give an application of our method to the theory of frames, and we prove a general dilation theorem which is in turn applied to specific cases, and we obtain in this way a dilation theorem for wavelets.
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We study various convex functions on $R^n$ associated with positive definite matrices. This yiels some exotic Holder matrix inequalities.
Given a function $f:(0,\infty)\rightarrow\RR$ and a positive semidefinite $n\times n$ matrix $P$, one may define a trace functional on positive definite $n\times n$ matrices as $A\mapsto \Tr(Pf(A))$. For differentiable functions $f$, the function $A\mapsto \Tr(Pf(A))$ is differentiable at all positive definite matrices $A$. Under certain continuity conditions on~$f$, this function may be extended to certain non-positive-definite matrices $A$, and the \emph{directional} derivatives of $\Tr(Pf(A)$ may be computed there. This note presents conditions for these directional derivatives to exist and…
In this paper we determine those bijective maps of the set of all positive definite $n\times n$ complex matrices which preserve a given Bregman divergence corresponding to a differentiable convex function that satisfies certain conditions. We cover the cases of the most important Bregman divergences and present the precise structure of the mentioned transformations. Similar results concerning Jensen divergences and their preservers are also given.
We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least two-dimensional complex Hilbert space which preserve the quantum $χ_α^2$-divergence for some $α\in [0,1]$. We prove that any such transformation is necessarily implemented by either a unitary or an antiunitary operator. Similar results concerning maps on the cone of positive semidefinite operators as well as on the set of all density operators are also derived.
Positive definiteness and Fell bundles over discrete groups
本文为离散群上 Fell 丛之间的丛映射引入正定性概念，证明其在全截面和约化截面 C*-代数间诱导完全正映射，并借此定义推广 Exel 逼近性质的新逼近性质，该性质仍蕴含弱包含性质。
We introduce bi-parametric fractional integrals of the Erdelyi-Kober type that generalize known Garding-Gindikin constructions associated to the cone of positive definite matrices. It is proved that the Radon transform, which maps a zonal function on the Grassmann manifold $G_{n,m}$ of $m$-dimensional linear subspaces of $R^n$ into a function on the similar manifold $G_{n,k}$, $ 1\leq m<k \leq n-1$, is represented as analytic continuation of the corresponding Erdelyi-Kober integral. This result shows that different Grinberg-Rubin's formulas for such transforms [GR] have, in fact, a common stru…
Injectivity, stability, and positive definiteness of max filtering
本文研究了最大滤波的注入性、稳定性和正定性，给出了将轨道空间嵌入欧氏空间的近乎尖锐条件，并估计了有限群下的度量畸变。
A definition of frames in Krein spaces is proposed which extends the concept of $J$-frames defined by J.I. Giribet et al., J. Math. Anal. Appl. ${\textbf{393}}$ (2012), 122-137. The principal difference consists in the fact that a $J$-frame is related to maximal definite subspaces $\mathcal{M}_\pm$ which are not assumed to be uniformly definite. The latter allows one to extend the collection of $J$-frames. In particular, some complete $J$-orthogonal sequences and $J$-orthogonal Schauder bases can be interpreted as $J$-frames.
Suppose a map $ϕ$ on the set of positive definite matrices satisfies $\det(A+B)=\det(ϕ(A)+ϕ(B))$. Then we have $${\rm tr}(AB^{-1}) = {\rm tr}(ϕ(A){ϕ(B)}^{-1}).$$ Through this viewpoint, we show that $ϕ$ is of the form $ϕ(A)= M^*AM$ or $ϕ(A)= M^*A^tM$ for some invertible matrix $M$ with $\det (M^*M)=1$. We also characterize the map $ϕ: \mathcal{S} \rightarrow \mathcal{S}$ preserving the determinant of convex combinations in $\mathcal{S}$ by using similar method. Here $\mathcal{S}$ can be the set of complex matrices, positive definite matrices, symmetric matrices, and upper triangular matrices.
In this paper we deal with two quantum relative entropy preserver problems on the cones of positive (either positive definite or positive semidefinite) operators. The first one is related to a quantum Rényi relative entropy like quantity which plays an important role in classical-quantum channel decoding. The second one is connected to the so-called maximal $f$-divergences introduced by D. Petz and M. B. Ruskai who considered this quantity as a generalization of the usual Belavkin-Staszewski relative entropy. We emphasize in advance that all the results are obtained for finite dimensional Hilb…
A very general KSGNS type dilation theorem in the context of right (not necessarily Hilbert) modules over $C^*$-algebras is presented. The proof uses Kolmogorov type decompositions for positive-definite kernels with values in spaces of sesquilinear maps. More specific functional analytic applications are obtained by adding assumptions.
An operator convex function on (0,\infty) which satisfies the symmetry condition k(1/x) = x k(x) can be used to define a type of non-commutative multiplication by a positive definite matrix (or its inverse) using the primitive concepts of left and right multiplication and the functional calculus. The operators for the inverse can be used to define quadratic forms associated with Riemannian metrics which contract under the action of completely positive trace-preserving maps. We study the question of when these operators define maps which are also completely positive (CP). Although A --> D^{-1/2…
Non-linear characterization of Jordan $*$-isomorphisms via maps on positive cones of $C^*$-algebras
本文研究含单位元$C^*$-代数正锥间的保持映射，刻画三类保持性质的满射与Jordan $*$-同构的关系，推广相关推论并解决Molnár提出的问题。
Strict Positivity and $D$-Majorization
该研究受量子热力学启发，探究严格正性的判定与性质，并将其用于推广$D$-优化排序，给出二维情形下该排序的迹范数刻画及序性质。
In this paper we consider power means of positive Hilbert space operators both in the conventional and in the Kubo-Ando senses. We describe the corresponding isomorphisms (bijective transformations respecting those means as binary operations) on positive definite cones and on positive semidefinite cones in operator algebras. We also investigate the question when those two sorts of power means can be transformed into each other.
In this paper, we generalize some matrix inequalities involving matrix power and Karcher means of positive definite matrices. Among other inequalities, it is shown that if ${\mathbb A}=(A_{1},...,A_{n})$ is a $n$-tuple of positive definite matrices such that $0<m\leq A_{i}\leq M\, (i=1,\cdots,n)$ for some scalars $m< M$ and $ω=(w_{1},\cdots,w_{n})$ is a weight vector with $w_{i}\geq0$ and $\sum_{i=1}^{n}w_{i}=1$, then \begin{align*} Φ^{p}\Big(\sum_{i=1}^{n}w_{i}A_{i}\Big)\leq α^{p}Φ^{p}(P_{t}(ω; {\mathbb A})) \end{align*} and \begin{align*} Φ^{p}\Big(\sum_{i=1}^{n}w_{i}A_{i}\Big)\leq α^{p}Φ^{p…
Given a positive definite matrix $A\in \mathbb{C}^{n\times n}$ and a Hermitian matrix $D\in \mathbb{C}^{m\times m}$, we characterize under which conditions there exists a strictly contractive matrix $K\in \mathbb{C}^{n\times m}$ such that the non-Hermitian block-matrix \[ \left[ \begin{array}{cc} A & -AK \\ K^*A & D \end{array} \right] \] has a positive definite Schur complement with respect to its submatrix~$A$. Additionally, we show that~$K$ can be chosen such that diagonalizability of the block-matrix is guaranteed and we compute its spectrum. Moreover, we show a connection to the recently …
A classical theorem of Herglotz states that a function $n\mapsto r(n)$ from $\mathbb Z$ into $\mathbb C^{s\times s}$ is positive definite if and only there exists a $\mathbb C^{s\times s}$-valued positive measure $dμ$ on $[0,2π]$ such that $r(n)=\int_0^{2π}e^{int}dμ(t)$for $n\in \mathbb Z$. We prove a quaternionic analogue of this result when the function is allowed to have a number of negative squares. A key tool in the argument is the theory of slice hyperholomorphic functions, and the representation of such functions which have a positive real part in the unit ball of the quaternions. We st…
Let $A$ be a positive definite operator on a Hilbert space $H$, and $|||.|||$ be a unitarily invariant norm on $B(H)$. We show that if $f$ is an operator monotone function on $(0,\infty)$ and $n\in \mathbb{N}$, then $|||D^n f(A)|||\leq\|f^{(n)}(A)\|$ and $\|f^{(n)}(\cdot)\|$ is a quasi-convex function on the set of all positive definite operators in $B(H)$. We establish some estimates of the right hand side of some Hermite-Hadamard type inequalities in which differentiable functions are involved, and norms of the maps induced by them on the set of self adjoint operators are convex, quasi-conve…
The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function $ϕ$ in the form $K_D^ϕ(H,K)=\sum_{i,j}ϕ(λ_i,λ_j)^{-1} Tr P_iHP_jK$ when $\sum_iλ_iP_i$ is the spectral decomposition of the foot point $D$ and the Hermitian matrices $H,K$ are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping $D\mapsto G(D)$ is a kernel metric as well. Several Riemannian geometries of the literature are particular cases, for example, the Fisher-Rao metric for multivariate Gaussian distribu…
Let $f$ be an operator convex function on $(0,\infty)$, and $Φ$ be a unital positive linear maps on $B(H)$. we give a complementary inequality to Davis-Choi-Jensen's inequality as follows \begin{equation*} f(Φ(A))\geq \frac{4R(A,B)}{(1+R(A,B))^2}Φ(f(A)), \end{equation*} where $R(A, B)=\max\{r(A^{-1}B) ,r(B^{-1}A)\}$ and $r(A)$ is the spectral radius of $A$. We investigate the complementary inequalities related to the operator power means and the Karcher means via unital positive linear maps, and obtain the following result: If $A_{1}$, $A_{2}$,\dots, $A_{n}$, are positive definite operators in…
The notion of positive-definite functions over locally compact quantum groups was recently introduced and studied by Daws and Salmi. Based on this work, we generalize various well-known results about positive-definite functions over groups to the quantum framework. Among these are theorems on "square roots" of positive-definite functions, comparison of various topologies, positive-definite measures and characterizations of amenability, and the separation property with respect to compact quantum subgroups.
A Kernel Approach to the Stinespring and Kraus Representations
本文通过标量正定核推导了完全正映射的Stinespring和Kraus结构定理。
Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if $g(t)=\sum_{k=0}^m a_kt^k$ is a polynomial of degree $m$ with non-negative coefficients, then, for all positive operators $A,\,B$ and all symmetric norms, $\|g(A+B)\|^{1/m} \le \|g(A)\|^{1/m} + \|g(B)\|^{1/m}$. To give parallel superadditivity results, we investigate anti-norms, a class of functionals containing the Schatten $q$-norms for $q\in(0,1]$ and $q<0$. The results are extensions of the Minkowski determinantal inequality. A few estimates for block-matrices are derived. For instance…
Complete discrete Schoenberg-Delsarte theory for homogeneous spaces
本文提出了一种关于同质空间的完整离散Schoenberg-Delsarte理论，扩展了Schoenberg的经典特征化，应用于受限排列问题和Delsarte理论，并证明了特定正定函数必须来自多项式。
Factorization of positive definite kernels. Correspondences: $C^{*}$-algebraic and operator valued context vs scalar valued kernels
本文引入一类广义正定核，通过标量值核和RKHS方法，给出其Stinespring型分解，并刻画核支配关系，恢复纯态与不可约表示的经典对应。
The main purpose of our paper is a new approach to design of algorithms of Kaczmarz type in the framework of operators in Hilbert space. Our applications include a diverse list of optimization problems, new Karhunen-Loève transforms, and Principal Component Analysis (PCA) for digital images. A key feature of our algorithms is our use of recursive systems of projection operators. Specifically, we apply our recursive projection algorithms for new computations of PCA probabilities and of variance data. For this we also make use of specific reproducing kernel Hilbert spaces, factorization for kern…
Recent results from real algebraic geometry and the theory of polynomial optimization are related in a new framework to the existence question of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle by Ron and Shen are interpreted in terms of hermitian sums of squares of certain nonnegative trigonometric polynomials and in terms of semi-definite programming. The latter together with the recent results in algebraic geometry and semi-definite programming allow us to answer affir…
With view to applications in stochastic analysis and geometry, we introduce a new correspondence for positive definite kernels (p.d.) $K$ and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of factorizations: (i) Probabilistic: Starting with a positive definite kernel $K$ we analyze associated Gaussian processes $V$. Properties of the Gaussian processes will be derived from certain factorizations of $K$, arising as a covariance kernel of $V$. (ii) Geometric analysis: We discuss families of measure spaces arising as boundaries for $K$. Our results entail an …
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}…
Let G be a locally compact group, M(G) denote its measure algebra and L^1(G) denote its group algebra. Also, let pi:G->U(H) be a strongly continuous unitary representation, and let CB^{sigma}(B(H)) be the space of normal completely bounded maps on B(H). We study the range of the map Gamma_pi:M(G)->CB^sigma(B(H)), Gamma_pi(mu)= int_G pi(s)\otimes pi(s)^*dmu(s) where we identify CB^sigma(B(H)) with the extended Haagerup tensor product B(H)\otimes^{eh}B(H)$. We use the fact that the C*-algebra generated by integrating pi to L^1(G) is unital exactly when pi is norm continuous to show that Gamma_pi…
From the Choi Formalism in Infinite Dimensions to Unique Decompositions of Generators of Completely Positive Dynamical Semigroups
本文在可分无穷维Hilbert空间中，利用修改的Choi形式证明了完全正动力学半群生成元可唯一分解为有界算子K与完全正映射Φ之和，并刻画了该对应的单射与满射条件，同时说明结果不可推广到不可分情形。
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