arXiv subjects

Mohammad Sal Moslehian

Publications and source records attributed to Mohammad Sal Moslehian.

At least 19 recordsLinked to original sources

arXiv2026

Operator Schmidt Rank and the Geometry of Schmidt Profiles

The main result of the paper states that if $0\neq v\in\mathbb C^m\otimes\mathbb C^n$, $d=\min\{m,n\}$, and $p=SR(v)$, then for every $1\le q\le d$, \begin{equation*} \min\left\{OSR(\mathbf U): \begin{array}{l} \mathbf U\in\mathbb M_m\otimes\mathbb M_n\text{ is unitary}, SR(\mathbf Uv)=q \end{array}\right\} = \left\lceil\max\left\{\frac pq,\frac qp\right\}\right\rceil . \end{equation*} Here $OSR(\mathbf U)$ denotes the operator Schmidt rank of $\mathbf U$. Furthermore, if $\mathbf U\in\mathbb M_m\otimes\mathbb M_n$ is unitary and $OSR(\mathbf U)\le k$, then, for every unit vector $x\in\mathbb C^m\otimes\mathbb C^n$ and every $1\le r\le d$, \begin{equation*} F_{\min\{kr,d\}}(\mathbf Ux)\ge F_r(x) \quad\text{and}\quad F_{\min\{kr,d\}}(x)\ge F_r(\mathbf Ux), \end{equation*} where $F_r(x)$ denotes the sum of the $r$ largest squared Schmidt coefficients of $x$. This result is stronger than rank reachability as it controls the entire cumulative Schmidt profile. We also establish a majorization relation and use it to obtain sharp bounds on Schatten-norm distortion and on the change of Rényi entropies under unitaries with bounded operator Schmidt rank.

math-ph↗

arXiv2026

On the Commutant of Invertible Positive Operators

An important result due to Radjabalipour asserts that if ${A\in\BB(\H)}$ is an invertible positive operator and ${T\in\BB(\H)}$ is arbitrary, then the operator sequence $\{A^{-n}TA^n\}_{n\in\ZZ}$ is norm-bounded if and only if ${TA=AT}.$ Using the concept of spectral gaps, we show that if $A$ possesses a spectral gap and the sequence $\{A^{-n}TA^n\}_{n\in\ZZ}$ is norm-bounded, then $T$ must be block diagonal with respect to a certain orthogonal decomposition of $\H$. We also provide an alternative proof of Radjabalipour's theorem. We also extend the theorem to some rich classes of operators, including invertible normal operators and invertible weighted sums of projections. Moreover, we provide simple proofs for the fact that a positive definite matrix ${A\in\MM_N}$ is a positive multiple of the identity if and only if the sequence $\{A^{-n}TA^n\}_{n\in\ZZ}$ is norm-bounded for every~${T\in\MM_N}.$

math.FA↗

arXiv2026

Conjugations on Hilbert $C^*$-modules

This work initiates a systematic development of conjugation theory within the framework of Hilbert $C^*$-modules, extending classical complex symmetric operator theory beyond the Hilbert space setting. We first explore the foundational structure of $*$-conjugate-automorphisms $\sharp$ on $C^*$-algebras and characterize their existence through isomorphisms with opposite algebras. We show that the set $\mathbb{L}_c(\mathscr{E})$ of all adjointable conjugate-linear operators on a Hilbert $C^*$-module $\mathscr{E}$ possesses the structure of a Hilbert $\mathbb{L}(\mathscr{E})$-module. We also introduce the key notion of $\sharp$-conjugation on a Hilbert $C^*$-module and fully characterize such modules in terms of real forms and real $C^*$-algebras. Moreover, we investigate the geometric properties of $\sharp$-conjugate duals. A central achievement is the development of polar decomposition theory for semiregular linear (conjugate-linear) operators acting on a Hilbert $C^*$-module equipped with a $\sharp$-conjugation $C$, thereby extending the celebrated Garcia--Putinar theorem [Trans. Amer. Math. Soc. 359 (2007), 3913--3931]. As a consequence, we show that any $C$-symmetric unitary operator factors as a product of two $\sharp$-conjugations. A comparative analysis of alternative definitions shows that requiring the naive adjoint condition $\langle Cx,y\rangle = \langle x,Cy\rangle^*$ would force the underlying $C^*$-algebra to be essentially commutative, thus justifying the adopted approach. We supply various examples and counterexamples that extensively illustrate our results.

math.OA↗

arXiv2026

Schatten norms on Hilbert $C^*$-modules via pure states

Let $(\mathscr{E}, \langle \cdot, \cdot\rangle)$ be a Hilbert $C^*$-module over a $C^*$-algebra $\mathfrak{A}$. The space of adjointable operators on $\mathscr{E}$ is denoted by $\mathcal{L}\left(\mathscr{E}\right)$. The sets of all states and pure states on $\mathfrak{A}$ are denoted by $\mathcal{S}\left(\mathfrak{A}\right)$ and $\mathcal{P}\left( \mathfrak{A}\right)$, respectively. For $τ\in\mathcal{S}\left( \mathfrak{A}\right)$, let us define $\mathcal{N}^{\mathscr{E}}_τ:=\left\lbrace x\in\mathscr{E}:τ\left( \left\langle x,x\right\rangle\right)=0 \right\rbrace$. The Hilbert completion of ${\mathscr{E}}/{\mathcal{N}^{\mathscr{E}}_τ}$ is denoted by $\mathscr{E}_τ$. For $T\in\mathcal{L}(\mathscr{E})$, the operator $T_{\mathscr{E}_τ}\in\mathbb{B}\left( \mathscr{E}_τ\right)$, is defined by $T_{\mathscr{E}_τ}\left(x+\mathcal{N}^{\mathscr{E}}_τ\right)=Tx+\mathcal{N}^{\mathscr{E}}_τ$. In this paper, we show that $\mathscr{E}_τ={\mathscr{E}}/{\mathcal{N}^{\mathscr{E}}_τ}$ when $\mathfrak{A}$ either is a $C^*$-algebra of compact operators or is commutative. We introduce a quantity in the context of Hilbert $C^*$-modules, denoted by $π^{\mathscr{E}}_k(\cdot)$ for $k\geq1$. We prove that $π^{\mathscr{E}}_k(T)\leq\sup_{τ\in\mathcal{P}\left( \mathfrak{A}\right)}\left\|T_{\mathscr{E}_τ}\right\|_{\left(k\right)}\leqπ^{\mathscr{E}^{\sharp}}_k(T_{\mathscr{E}^{\sharp}})$ for every $T\in\mathcal{L}\left(\mathscr{E}\right)$, where the space $\mathscr{E}^{\sharp}$ is constructed as the extension of $\mathscr{E}$ by the embedding of $\mathfrak{A}$ into its enveloping von Neumann algebra $\mathfrak{A}^{**}$.

math.OA↗

arXiv2026

Convergence of Random Products of Projections Under Infinite-Periodic Selections

Let $\{P_j\}$ be a sequence of projections onto the closed subspaces $\mathcal{M}_j$ of a Hilbert space $\mathscr{H}$. Consider an infinite sequence $P_{i_1}, P_{i_2}, \ldots$ with each $P_{i_n} \in \{P_1, P_2, \ldots\}$, possibly repeating in some order or randomly. The question is: Under what conditions does the sequence $\{P_{i_n} \cdots P_{i_2} P_{i_1} x\}_{n=1}^{\infty}$ converge strongly or weakly to $Px$ for every $x \in \mathscr{H}$, where $P$ is the projection onto the intersection $\mathcal{M} = \bigcap_{i=1}^{\infty} \mathcal{M}_i$? In this paper, we present some results concerning random products of countably infinitely many projections $\{P_j\}_{j=1}^{\infty}$ that incorporates the notion of an infinite-periodic function. More precisely, we introduce a new class of functions $σ\colon \mathbb{N} \to \mathbb{N}$, called infinite-periodic functions, and rigorously show that the sequence $\{T_n x\}$ defined by \[ T_1 := P_{σ(1)}\quad \mbox{and} \quad T_n := P_{σ(n)} T_{n-1} \quad \text{for all } n \geq 2, \] for $x \in \mathscr{H}$, converges weakly to $Px$, where $P$ is the projection onto $\bigcap_{j=1}^{\infty} \mathcal{R}(P_j)$. We also provide some technical examples to illustrate our results.

math.FA↗

arXiv2026

Spectral geometric mean and trace characterizations

We use nearly parallel pure states to characterize positive linear functionals $ϕ$ on $\mathbb{M}_n$ as positive multiples of the trace if and only if $ϕ(A \natural B) \leq \sqrt{ϕ(A) ϕ(B)}$ for all positive definite matrices $A$ and $B$. Here $A \natural B = (A^{-1} \# B)^{1/2} A (A^{-1} \# B)^{1/2}$ represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality $ϕ(A \natural B) \leq ϕ((A+B)/2)$ for all positive definite matrices $A$ and $B$. We also present a trace inequality related to quantum fidelity that applies to all positive definite matrices, and demonstrate that it does not characterize the trace.

quant-ph↗

arXiv2025

Convergence of Random Products of Countably Infinitely Many Projections

Let $r \in \mathbb{N}\cup\{\infty\}$ be a fixed number and let $P_j\,\, (1 \leq j\leq r )$ be the projection onto the closed subspace $\mathcal{M}_j$ of $\mathscr{H}$. We are interested in studying the sequence $P_{i_1}, P_{i_2}, \ldots \in\{P_1, \ldots, P_r\}$. A significant problem is to demonstrate conditions under which the sequence $\{P_{i_n}\cdots P_{i_2}P_{i_1}x\}_{n=1}^\infty$ converges strongly or weakly to $Px$ for any $x\in\mathscr{H}$, where $P$ is the projection onto the intersection $\mathcal{M}=\mathcal{M}_1\cap \ldots \cap \mathcal{M}_r$. Several mathematicians have presented their insights on this matter since von Neumann established his result in the case of $r=2$. In this paper, we give an affirmative answer to a question posed by M. Sakai. We present a result concerning random products of countably infinitely many projections (the case $r=\infty$) incorporating the notion of pseudo-periodic function.

math.FA↗

arXiv2025

A new estimation of the quantum Chernoff bound

Relating to finding possible upper bounds for the probability of error for discriminating between two quantum states, it is well-known that \begin{align*} \mathrm{tr}(A+B) - \mathrm{tr}|A-B|\leq 2\, \mathrm{tr}\big(f(A)g(B)\big) \end{align*} holds for every positive-valued matrix monotone function $f$, where $g(x)=x/f(x)$, and all positive definite matrices $A$ and $B$. In this paper, we introduce a new class of functions that satisfy the above inequality. As a consequence, we derive a novel estimation of the quantum Chernoff bound. Additionally, we characterize matrix decreasing functions and establish matrix Powers-Störmer type inequalities for perspective functions.

quant-ph↗

arXiv2024

A separation theorem for Hilbert $W^*$-modules

Let $\mathscr E$ be a Hilbert $\mathscr A$-module over a $C^*$-algebra $\mathscr A$. For each positive linear functional $ω$ on $\mathscr A$, we consider the localization $\mathscr E_ω$ of $\mathscr E$, which is the completion of the quotient space $\mathscr E/\mathscr {N}_ω$, where $\mathscr N_ω=\{x\in \mathscr E:ω\langle x,x\rangle=0\}$. Let $\mathscr H$ and $\mathscr K$ be closed submodules of $\mathscr E$ such that $\mathscr H\cap \mathscr K$ is orthogonally complemented, and let $ω=\sum_{j=1}^{\infty}λ_jω_j$, where $λ_j>0$, $\sum_{j=1}^{\infty}λ_j=1$, and $ω_j$'s are positive linear functionals on $\mathscr A$. We prove that if $(\mathscr H\cap \mathscr K)_{ω_j}=\mathscr H_{ω_j}\cap \mathscr K_{ω_j}$ for each $j$, then \[ (\mathscr H\cap \mathscr K)_ω=\mathscr H_ω\cap \mathscr K_ω\,. \] Furthermore, let $\mathscr L$ be a closed submodule of a Hilbert $\mathscr A$-module $\mathscr E$ over a $W^*$-algebra $\mathscr A$. We pose the following separation problem: ``Does there exist a normal state $ω$ such that $ι_ω(\mathscr L)$ is not dense in $\mathscr E_ω$?'' In this paper, among other results, we give an affirmative answer to this problem, when $\mathscr E$ is a self-dual Hilbert $C^*$-module over a $W^*$-algebra $\mathscr A$ such that $\mathscr E\backslash \mathscr L$ has a nonempty interior with respect to the weak$^*$-topology. This is a step toward answering the above problem.

math.OA↗

arXiv2024

Around Strongly Operator Convex Functions

In this paper, we obtain the subadditivity inequality of strongly operator convex functions on $(0, \infty)$ and $(-\infty,0)$. Applying the properties of operator convex functions, we deduce the subadditivity property of operator monotone functions on $(0, \infty)$. We give new operator inequalities involving strongly operator convex functions on an interval and the weighted operator means. We also investigate relations between strongly operator convex functions and Kwong functions on $(0, \infty)$. Moreover, we study the strongly operator convex functions on $(a, \infty)$ with $a>-\infty$ and also on left half line $(-\infty,b)$ with $b< \infty$. We show that any non-constant strongly operator convex function on $(a, \infty)$ is strictly operator decreasing, and any non-constant strongly operator convex function on $(-\infty,b)$ is strictly operator monotone. Consequently, if $g$ is a strongly operator convex function on $(a, \infty)$ or $(-\infty,b)$, we estimate lower bounds of $|g(A)-g(B)|$ whenever $A-B>0$.

math.FA↗

arXiv2024

Operator mean inequalities and Kwong functions

In this paper, we study operator mean inequalities for the weighted arithmetic, geometric and harmonic means. We give a slight modification of Audenaert's result to show the relation between Kwong functions and operator monotone functions. Operator mean inequalities provide some analogs of the geometric concavity property for Kwong functions, operator convex, and operator monotone functions. Moreover, we give our points across by way of some examples which show the usage of our main results.

math.FA↗

arXiv2024

Extensions of the Hilbert-multi-norm in Hilbert $C^*$-modules

Dales and Polyakov introduced a multi-norm $\left( \left\|\cdot\right\|_n^{(2,2)}:n\in\mathbb{N}\right)$ based on a Banach space $\mathscr{X}$ and showed that it is equal with the Hilbert-multi-norm $\left( \left\|\cdot\right\|_n^{\mathscr{H}}:n\in\mathbb{N}\right)$ based on an infinite-dimensional Hilbert space $\mathscr{H}$. We enrich the theory and present three extensions of the Hilbert-multi-norm for a Hilbert $C^*$-module $\mathscr{X}$. We denote these multi-norms by $\left( \left\|\cdot\right\|_n^{\mathscr{X}}:n\in\mathbb{N}\right)$, $\left( \left\|\cdot\right\|_n^{*}:n\in\mathbb{N}\right)$, and $\left( \left\|\cdot\right\|_n^{\mathcal{P}\left(\mathfrak{A} \right) }:n\in\mathbb{N}\right)$. We show that $\left\|x\right\|_n^{\mathcal{P}\left(\mathfrak{A} \right) }\geq\left\|x\right\|_n^{\mathscr{X}}\leq \left\|x\right\|_n^{*}$ for each $x\in\mathscr{X}^n$. In the case when $\mathscr{X}$ is a Hilbert $\mathbb{K}\left(\mathscr{H}\right)$-module, for each $x\in\mathscr{X}^n$, we observe that $\left\|\cdot\right\|_n^{\mathcal{P}\left(\mathfrak{A} \right)}=\left\|\cdot\right\|_n^{\mathscr{X}}$. Furthermore, if $\mathscr{H}$ is separable and $\mathscr{X}$ is infinite-dimensional, we prove that $\left\|x\right\|_n^{\mathscr{X}}=\left\|x\right\|_n^{*}$. Among other things, we show that small and orthogonal decompositions with respect to these multi-norms are equivalent. Several examples are given to support the new findings.

math.FA↗

arXiv2024

Decomposition of tracial positive maps and applications in quantum information

Every positive multilinear map between $C^*$-algebras is separately weak$^*$-continuous. We show that the joint weak$^*$-continuity is equivalent to the joint weak$^*$-continuity of the multiplications of $C^*$-algebras under consideration. We study the behavior of general tracial positive maps on properly infinite von Neumann algebras and by applying the Aron--Berner extension of multilinear maps, we establish that under some mild conditions every tracial positive multilinear map between general $C^*$-algebras enjoys a decomposition $Φ=φ_2 \circ φ_1$, in which $φ_1$ is a tracial positive linear map with the commutative range and $φ_2$ is a tracial completely positive map with the commutative domain. As an immediate consequence, tracial positive multilinear maps are completely positive. Furthermore, we prove that if the domain of a general tracial completely positive map $Φ$ between $C^*$-algebra is a von Neumann algebra, then $Φ$ has a similar decomposition. As an application, we investigate the generalized variance and covariance in quantum mechanics via arbitrary positive maps. Among others, an uncertainty relation inequality for commuting observables in a composite physical system is presented.

math.OA↗

arXiv2024

Power-norms based on Hilbert $C^*$-modules

Suppose that $\mathscr{E}$ and $\mathscr{F}$ are Hilbert $C^*$-modules. We present a power-norm $\left(\left\|\cdot\right\|^{\mathscr{E}}_n:n\in\mathbb{N}\right)$ based on $\mathscr{E}$ and obtain some of its fundamental properties. We introduce a new definition of the absolutely $(2,2)$-summing operators from $\mathscr{E}$ to $\mathscr{F}$, and denote the set of such operators by $\tildeΠ_2(\mathscr{E},\mathscr{F})$ with the convention $\tildeΠ_2(\mathscr{E})=\tildeΠ_2(\mathscr{E},\mathscr{E})$. It is known that the class of all Hilbert--Schmidt operators on a Hilbert space $\mathscr{H}$ is the same as the space $\tildeΠ_2(\mathscr{H})$. We show that the class of Hilbert--Schmidt operators introduced by Frank and Larson coincides with the space $\tildeΠ_2(\mathscr{E})$ for a countably generated Hilbert $C^*$-module $\mathscr{E}$ over a unital commutative $C^*$-algebra. These results motivate us to investigate the properties of the space $\tildeΠ_2(\mathscr{E},\mathscr{F})$.

math.OA↗

arXiv2024

Matrix Inequalities between $f(A)σf(B)$ and $AσB$

Let $A$ and $ B$ be $n\times n$ positive definite complex matrices, let $σ$ be a matrix mean, and let $f : [0,\infty)\to [0,\infty)$ be a differentiable convex function with $f(0)=0$. We prove that $$f^{\prime}(0)(A σB)\leq \frac{f(m)}{m}(AσB)\leq f(A)σf(B)\leq \frac{f(M)}{M}(AσB)\leq f^{\prime}(M)(AσB),$$ where $m$ represents the smallest eigenvalues of $A$ and $B$ and $M$ represents the largest eigenvalues of $A$ and $B$. If $f$ is differentiable and concave, then the reverse inequalities hold. We use our result to improve some known subadditivity inequalities involving unitarily invariant norms under certain mild conditions. In particular, if $f(x)/x$ is increasing, then $$|||f(A)+f(B)|||\leq\frac{f(M)}{M} |||A+B|||\leq |||f(A+B)|||$$ holds for all $A$ and $B$ with $M\leq A+B$. Furthermore, we apply our results to explore some related inequalities. As an application, we present a generalization of Minkowski's determinant inequality.equality.

math.FA↗

arXiv2021

Improved inequalities for the numerical radius via Cartesian decomposition

We develop various lower bounds for the numerical radius $w(A)$ of a bounded linear operator $A$ defined on a complex Hilbert space, which improve the existing inequality $w^2(A)\geq \frac{1}{4}\|A^*A+AA^*\|$. In particular, for $r\geq 1$, we show that \begin{eqnarray*}\frac{1}{4}\|A^*A+AA^*\| \leq\frac{1}{2} \left( \frac{1}{2}\|\Re(A)+\Im(A)\|^{2r}+\frac{1}{2}\|\Re(A)-\Im(A)\|^{2r}\right)^{\frac{1}{r}} \leq w^{2}(A),\end{eqnarray*} where $\Re(A)$ and $\Im(A)$ are the real and imaginary parts of $A$, respectively. Furthermore, we obtain upper bounds for $w^2(A)$ refining the well-known upper bound $w^2(A)\leq \frac{1}{2} \left(w(A^2)+\|A\|^2\right)$. Separate complete characterizations for $w(A)=\frac{\|A\|}{2}$ and $w(A)=\frac{1}{2}\sqrt{\|A^*A+AA^*\|}$ are also given.

math.FA↗

arXiv2020

B-spline interpolation problem in Hilbert C*-modules

We introduce the $B$-spline interpolation problem corresponding to a $C^*$-valued sesquilinear form on a Hilbert $C^*$-module and study its basic properties as well as the uniqueness of solution. We first study the problem in the case when the Hilbert $C^*$-module is self-dual. Extending a bounded $C^*$-valued sesquilinear form on a Hilbert $C^*$-module to a sesquilinear form on its second dual, we then provide some necessary and sufficient conditions for the $B$-spline interpolation problem to have a solution. Passing to the setting of Hilbert $W^*$-modules, we present our main result by characterizing when the spline interpolation problem for the extended $C^*$-valued sesquilinear to the dual $\mathscr{X}'$ of the Hilbert $W^*$-module $\mathscr{X}$ has a solution. As a consequence, we give a sufficient condition that for an orthogonally complemented submodule of a self-dual Hilbert $W^*$-module $\mathscr{X}$ is orthogonally complemented with respect to another $C^*$-inner product on $\mathscr{X}$. Finally, solutions of the $B$-spline interpolation problem for Hilbert $C^*$-modules over $C^*$-ideals of $W^*$-algebras are extensively discussed. Several examples are provided to illustrate the existence or lack of a solution for the problem.

math.OA↗

arXiv2019

Douglas factorization theorem revisited

Inspired by the Douglas lemma, we investigate the solvability of the operator equation $AX=C$ in the framework of Hilbert C*-modules. Utilizing partial isometries, we present its general solution when $A$ is a semi-regular operator. For such an operator $A$, we show that the equation $AX=C$ has a positive solution if and only if the range inclusion ${\mathcal R}(C) \subseteq {\mathcal R}(A)$ holds and $CC^*\le t\, CA^*$ for some $t>0$. In addition, we deal with the solvability of the operator equation $(P+Q)^{1/2}X=P$, where $P$ and $Q$ are projections. We provide a counterexample to show that there exists a $C^*$-algebra $\mathfrak{A}$, a Hilbert $\mathfrak{A}$-module $\mathscr{H}$ and projections $P$ and $Q$ on $\mathscr{H}$ such that the operator equation $(P+Q)^{1/2}X=P$ has no solution. Moreover, we give a perturbation result related to the latter equation.

math.OA↗