Mohammad Sal Moslehian

Ferdowsi University Of Mashhad | FUM · Department of Pure Mathematics

Let $A$ and $ B$ be $n\times n$ positive definite complex matrices, let $\sigma$ be a matrix mean, and let $f : [0,\infty)\to [0,\infty)$ be a differentiable convex function with $f(0)=0$. We prove that $$f^{\prime}(0)(A \sigma B)\leq \frac{f(m)}{m}(A\sigma B)\leq f(A)\sigma f(B)\leq \frac{f(M)}{M}(A\sigma B)\leq f^{\prime}(M)(A\sigma B),$$ where $...

Regarding finding possible upper bounds for the probability of error for discriminating between two quantum states, it is known that $$ \mathrm{tr}(A+B) - \mathrm{tr}|A-B|\leq 2\, \mathrm{tr}\big(f(A)g(B)\big) $$ holds for every positive matrix monotone function $f$, where $g(x)=x/f(x)$, and all positive matrices $A$ and $B$. We show that the class...

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 \(\mathscr {H}\)...

One of the most fundamental and widely used inequalities in mathematics is the celebrated Cauchy–Schwarz inequality, also known as the Cauchy–Bunyakovsky–Schwarz inequality. The elementary form of Cauchy–Schwarz inequality states that if and are real numbers, then Its general form in an inner-product space is (1.2). The Cauchy–Schwarz inequality wa...

The spectral mapping theorem ensures that for a bounded linear operator A on a complex Hilbert space , where f is an analytic function on a domain containing Unfortunately, there is no such relation for the numerical range of a bounded linear operator, that is, for .

We introduce and study an extension of the Roberts orthogonality, in the setting of \(C^*\)-algebras. More precisely, in a \(C^*\)-algebra \(\mathscr {A},\) for \(a,b \in \mathscr {A}\) and a nonempty subset of \(\mathscr {A}\), say \(\mathscr {B}\), a is called \(\mathscr {B}\)-Roberts orthogonal to b, denoted by \(a \perp ^{\mathscr {B}}_R b\), i...

In this chapter, we collect some basic facts needed to study the numerical range and numerical radius of a bounded linear operator defined on a Hilbert space and fix our notation.

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 \({\math...

Suppose that \({\mathfrak {N}}\) is a von Neumann algebra and that \(\varphi \) is a normal semifinite trace on \({\mathfrak {N}}\). We study bands, projection bands, and band projections in the setting of noncommutative Banach function spaces. Let \(S({\mathfrak {N}}, \varphi )\) be the set of all \(\varphi \)-measurable operators. A subspace \({\...

We generalize a two-parameter extended Wigner–Yanase skew information given by Z. Zhang [J Math Anal Appl. 2021;497(1): 124851] to any general operator monotone function. We prove several fundamental properties including the joint concavity and an uncertainty relation for the proposed generalized Wigner–Yanase skew information.

We introduce some measures of the dependence such as the strong mixing and uniform mixing coefficients in von Neumann algebras and then define the noncommutative strong and uniform mixing sequences. We establish some notable nonncommutative mixing inequalities such as Ibragimov inequality. Moreover, we extend the notion of mixingale sequence to the...

The Johnson--Kadison--Ringrose theorem states that the Aron--Berner extension of a multilinear map from $C^*$-algebras to second dual spaces is unique and separately weak$^*$-weak$^*$-continuous. We show that every positive multilinear map between $C^*$-algebras is jointly weak$^*$-weak$^*$-continuous. Utilizing this fact, we obtain joint weak$^*$-...

The aim of this paper is to present a unified framework in the setting of Hilbert \(C^*\)-modules for the scalar- and vector-valued reproducing kernel Hilbert spaces and \(C^*\)-valued reproducing kernel spaces. We investigate conditionally negative definite kernels with values in the \(C^*\)-algebra of adjointable operators acting on a Hilbert \(C...

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. Passing to the setting of Hilbert W∗-modules, we present our main result by charac...

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...

For a selfadjoint involutive matrix J we equip Cn with the indefinite inner product [x,y]=〈Jx,y〉, where 〈⋅,⋅〉 denotes the usual inner product on Cn and endow the matrix algebra Mn with a partially ordered relation ≤J. We define the notion of J-mean σf(⋅,⋅) of J-selfadjoint matrices with spectra in (0,∞) for any normal matrix monotone function f:(0,...

We give a modified definition of a reproducing kernel Hilbert C⁎-module (shortly, RKHC⁎M) without using the condition of self-duality and discuss some related aspects; in particular, an interpolation theorem is presented. We investigate the exterior tensor product of RKHC⁎Ms and find their reproducing kernel. In addition, we deal with left multipli...

For a probability measure of compact support μ on the set Pn of all positive definite matrices and t∈(0,1], let Pt(μ) be the unique positive solution of X=∫PnX♯tZdμ(Z). In this paper, we show that if μ and ν are probability measures on Pn and Pm, respectively, and Φ:Mn×Mm→Mk is a unital positive bilinear map, then Φ(Pt(μ),Pt(ν))≤Pt(λ) for all t∈[−1...

In this paper, the arithmetic-geometric mean inequalities of indefinite type are discussed. We show that for a J-selfadjoint matrix A satisfying \(I \ge ^J A\) and \({\mathrm{sp}}(A) \subseteq [1, \infty ),\) the inequality $$\begin{aligned} \frac{I + A}{2} \le ^J \sqrt{A} \end{aligned}$$holds, and the reverse does for A with \(I \ge ^J A\) and \({...

We present an expression for the generalized numerical radius associated with a norm on the algebra of bounded linear operators on a Hilbert space and then apply it to obtain upper and lower bounds for the generalized numerical radius. We also establish some generalized numerical radius inequalities involving the product of two operators. Applicati...

We deal with the Roberts numerical radius orthogonality. In the case of 2 × 2 complex matrices, we give some necessary and sufficient conditions for the numerical range to be symmetric by employing the Roberts orthogonality with respect to the numerical radius. In addition, we present an interrelation between the Roberts orthogonality and the Birkh...

Let Φ be a unital positive linear map and let A be a positive invertible operator. We prove that there exist partial isometries U and V such that | Φ ( f ( A ) ) Φ ( A ) Φ ( g ( A ) ) | ≤ U ∗ Φ ( f ( A ) A g ( A ) ) U and Φ f ( A ) − r Φ ( A ) r Φ g ( A ) − r ≤ V ∗ Φ f ( A ) − r A r g ( A ) − r V hold under some mild operator convex conditions and...

Let p be a real number and let \(\varepsilon >0\). An operator \(T\in \mathbb {B}(\mathscr {H})\) is called a \((p,\varepsilon )\)-approximate n-idempotent if $$\begin{aligned} \Vert T^nx- Tx\Vert \le \varepsilon \Vert x\Vert ^p\qquad (x\in \mathscr {H})\,. \end{aligned}$$In this note, we remark that if \(p\ne 1\), then T is an n-idempotent. If \(p...

For an n -tuple of positive invertible operators on a Hilbert space, we present some variants of Ando–Hiai type inequalities for deformed means from an n -variable operator mean by an operator mean, which is related to the information monotonicity of a certain unital positive linear map. As an application, we investigate the monotonicity of the pow...

We investigate the inverse q-numerical range of \(2\times 2\) matrices. A quartic curve is formulated that generates unit vectors for the inverse q-numerical range according to Tsing’s circle formula. Properties of this quartic curve are discussed and corresponding examples are described.

We introduce the notion of a tractable pair of operators as well as that of the generalized parallel sum in the setting of adjointable operators on Hilbert C ∗ -modules. Some significant results about the parallel sum known for matrices and Hilbert space operators are extended to the case of the generalized parallel sum. In particular, a factorizat...

In this paper, we provide a biography of Professor Rajendra Bhatia and discuss some of his influential mathematical works as one of the leading researchers in matrix analysis and linear algebra.

We introduce the notion of acceptable noncommutative random variables and investigate their essential properties. More precisely, we provide several efficient estimation of tail probabilities of sums of noncommutative random variables under some mild conditions. Moreover, we investigate the complete convergence of a sequence of the form [Formula: s...

We introduce the $B$-spline interpolation problem corresponding to a $C^*$-valued sesquilinear form on a Hilbert $C^*$-module and study its basic properties. 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 se...

We present three versions of the Lax–Milgram theorem in the framework of Hilbert \(C^*\)-modules, two for self-dual ones over \(W^*\)-algebras and one for those over \(C^*\)-algebras of compact operators. It is remarkable that while the Riesz theorem is not valid for certain Hilbert \(C^*\)-modules over \(C^*\)-algebras of compact operators, howeve...

In this paper, we aim to replace in the definitions of covariance and correlation the usual trace Tr by a tracial positive map between unital \(C^*\)-algebras and to replace the functions \(x^{\alpha }\) and \(x^{1- \alpha }\) by functions f and g satisfying some mild conditions. These allow us to define the generalized covariance, the generalized...

We introduce the relation ρλ-orthogonality in the setting of normed spaces as an extension of some orthogonality relations based on norm derivatives and present some of its essential properties. Among other things, we give a characterization of inner product spaces via the functional ρλ. Moreover, we consider a class of linear mappings preserving t...

In this paper, we aim to replace in the definitions of covariance and correlation the usual trace {\rm Tr} by a tracial positive map between unital $C^*$-algebras and to throw $x^{\alpha}$ and $x^{1-\alpha}$ in two functions $f$ and $g$ satisfying some mild conditions. These allow us to define the generalized covariance, the generalized variance, t...

It is known that the noncommutative Hardy spaces H1(M) and H1max(M) do not coincide, in general. In fact, it may happen that H1(M)⊈H1max(M). It is an interesting question whether the reverse inclusion holds or not. In this note, motivated by this question, we prove that the validity of inequality ‖x‖H1C(M)≤c‖x‖H1max(M) for some c and all martingale...

We present three versions of the Lax-Milgram theorem in the framework of Hilbert C *-modules, two for those over W *-algebras and one for those over C *-algebras of compact operators. It is remarkable that while the Riesz theorem is not valid for certain Hilbert C *-modules over C *-algebras of compact operators, our Lax-Milgram theorem turns out t...

Halmos' two projections theorem for Hilbert space operators is one of the fundamental results in operator theory. In this paper, we introduce the term of two harmonious projections in the context of adjointable operators on Hilbert $C^*$-modules, extend Halmos' two projections theorem to the case of two harmonious projections. We also give some new...

We review some of the significant generalizations and applications of the celebrated Douglas theorem on the equivalence of factorization, range inclusion, and majorization of operators. We then apply it to find a characterization of the positivity of $2\times 2$ block matrices of operators in Hilbert spaces and finally describe the nature of such b...

Halmos’ two projections theorem for Hilbert space operators is one of the fundamental results in operator theory. In this paper, we introduce the term of two harmonious projections in the context of adjointable operators on Hilbert C ⁎ -modules, extend Halmos’ two projections theorem to the case of two harmonious projections. We also give some new...

Let $\mathbb{B}_J(\mathcal H)$ denote the set of self-adjoint operators acting on a Hilbert space $\mathcal{H}$ with spectra contained in an open interval $J$. A map $\Phi\colon\mathbb{B}_J(\mathcal H)\to {\mathbb B}(\mathcal H)_\text{sa} $ is said to be of Jensen-type if \[ \Phi(C^*AC+D^*BD)\le C^*\Phi(A)C+D^*\Phi(B)D \] for all $ A, B \in B_J(\ma...

Necessary and sufficient conditions are given for the operator system $A_1X=C_1$, $XA_2=C_2$, $A_3XA^*_3=C_3$, and $A_4XA^*_4=C_4$ to have a common positive solution, where $A_i$'s and $C_i$'s are adjointable operators on Hilbert $C^*$-modules. This corrects a published result by removing some gaps in its proof. Finally, a technical example is give...

We present a necessary and sufficient condition for the norm-parallelism of bounded linear operators on a Hilbert space. We also give a characterization of the Birkhoff–James orthogonality for Hilbert space operators. Moreover, we discuss the connection between norm-parallelism to the identity operator and an equality condition for the Davis–Wielan...

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...

The main purpose of this paper is to establish a noncommutative analogue of the Efron-Stein inequality, which bounds the variance of a general function of some independent random variables. Moreover, we state an operator version including random matrices, which extends a result of D. Paulin et al., [Ann. Probab. 44(5) (2016), 3431–3473]. Further, w...

The non-commutative Stein inequality asks whether there exists a constant $C_{p,q}$ depending only on $p, q$ such that \begin{equation*} \left\| \left(\sum_{n} |\mathcal{E}_{n} (x_n) |^{q}\right)^{\frac{1}{q}} \right\|_p \leq C_{p,q} \left\| \left(\sum_{n} | x_n |^q \right)^{\frac{1}{q}}\right \|_p\qquad \qquad (S_{p,q}), \end{equation*} for (posit...

It is proved that for adjointable operators A and B between Hilbert C*-modules, certain majorization conditions are always equivalent without any assumptions on (Formula presented.), where A* denotes the adjoint operator of A and (Formula presented.) is the norm closure of the range of A*. In the case that (Formula presented.) is not orthogonally c...

In this paper, we present some characterizations of linear mappings, which preserve vectors at a specific angle. We introduce the concept of $(\varepsilon, c)$-angle preserving mappings for $|c|<1$ and $0\leq \varepsilon < 1 + |c|$. We show that for a general approximate similarity $S$, if a linear mapping $T$ satisfies $\|T - S\| \leq \theta \|S\|...

We investigate the orthogonality preserving property for pairs of mappings on inner product $C^*$-modules extending existing results for a single orthogonality-preserving mapping. Guided by the point of view that the $C^*$-valued inner product structure of a Hilbert $C^*$-module is determined essentially by the module structure and by the orthogona...

In this paper, we show that for a positive operator A on a Hilbert \(C^*\)-module \( \mathscr {E} \), the range \( \mathscr {R}(A) \) of A is closed if and only if \( \mathscr {R}(A^\alpha ) \) is closed for all \(\alpha \in (0,1)\cup (1,+\,\infty )\), and this occurs if and only if \( \mathscr {R}(A)=\mathscr {R}(A^\alpha ) \) for all \(\alpha \in...

The main goal of this article is to present several quadratic refinements and reverses of the well known Heinz inequality, for numbers and matrices, where the refining term is a quadratic function in the mean parameters. The proposed idea introduces a new approach to these inequalities, where polynomial interpolation of the Heinz function plays a m...

In this article, we present several inequalities treating operator means and the Cauchy–Schwarz inequality. In particular, we present some new comparisons between operator Heron and Heinz means, several generalizations of the difference version of the Heinz means and further refinements of the Cauchy–Schwarz inequality. The techniques used to accom...

Based on a maximal inequality type result of Cuculescu, we establish some noncommutative maximal inequalities such as Haj\'ek--Penyi inequality and Etemadi inequality. In addition, we present a noncommutative Kolmogorov type inequality by showing that if $x_1, x_2, \ldots, x_n$ are successively independent self-adjoint random variables in a noncomm...

In remembrance of Professor Uffe Valentin Haagerup (1949--2015), as a brilliant mathematician, we review some aspects of his life, and his outstanding mathematical accomplishments.

Let A and B be two positive operators with for positive real numbers be an operator mean and be the adjoint mean of If and is a positive unital linear map, then where and is the Kantorovich constant. In addition, for

In this paper, we introduce the concept of completely positive matrix of linear maps on Hilbert A-modules over locally C*-algebras and prove an analogue of Stinespring theorem for it. We show that any two minimal Stinespring representations for such matrices are unitarily equivalent. Finally, we prove an analogue of the Radon-Nikodym theorem for th...

The aim of this paper is to present some results concerning the $\rho_*$-orthogonality in real normed spaces and its preservation by linear operators. Among other things, we prove that if $T\,: X \longrightarrow Y$ is a nonzero linear $(I, \rho_*)$-orthogonality preserving mapping between real normed spaces, then $$\frac{1}{3}\|T\|\|x\|\leq\|Tx\|\l...

We establish a noncommutative Blackwell--Ross inequality for supermartingales under a suitable condition which generalize Khan's works to the noncommutative setting. We then employ it to deduce an Azuma-type inequality.

The aim of this paper is to present a comprehensive study of operator m-convex functions. Let m∈[0,1], and J=[0,b] for some b∈ℝ or J=[0,∞). A continuous function φ:J→ℝ is called operator m-convex if for any t∈[0,1] and any self-adjoint operators A,B∈𝔹⁢(ℋ), whose spectra are contained in J, we have φ⁢(t⁢A+m⁢(1-t)⁢B)≤t⁢φ⁢(A)+m⁢(1-t)⁢φ⁢(B). We first g...

We introduce a notion of approximate orthogonality preserving mappings between Hilbert $C^*$-modules. We define the concept of $(\delta, \varepsilon)$-orthogonality preserving mapping and give some sufficient conditions for a linear mapping to be $(\delta, \varepsilon)$-orthogonality preserving. In particular, if $\mathscr{E}$ is a full Hilbert $\m...

In this paper, motivated by perturbation theory of operators, we present some upper bounds for vertical bar vertical bar vertical bar f (A)Xg(B) + X vertical bar vertical bar vertical bar in terms of vertical bar vertical bar vertical bar vertical bar AXB vertical bar + vertical bar X vertical bar vertical bar vertical bar vertical bar and vertical...

We extend an operator Pólya–Szegö type inequality involving the operator geometric mean to any arbitrary operator mean under some mild conditions. Utilizing the Mond–Pečaric method, we present some other related operator inequalities as well.

In this paper, we study the relationship between operator space norm and operator space numerical radius on the matrix space $\mathcal{M}_n(X)$, when $X$ is a numerical radius operator space. Moreover, we establish several inequalities for operator space numerical radius and the maximal numerical radius norm of $2\times 2$ operator matrices and the...

We show that if $T=H+iK$ is the Cartesian decomposition of $T\in \mathbb{B(\mathscr{H})}$, then for $\alpha ,\beta \in \mathbb{R}$, $\sup_{\alpha ^{2}+\beta ^{2}=1}\Vert \alpha H+\beta K\Vert =w(T)$. We then apply it to prove that if $A,B,X\in \mathbb{B(\mathscr{H})}$ and $0\leq mI\leq X$, then \begin{align*} m\Vert \mbox{Re}(A)-\mbox{Re}(B)\Vert &...

The symmetric doubly stochastic inverse spectral problem is the problem of determining necessary and sufficient conditions for a real -tuple to be the spectrum of an nxn symmetric doubly stochastic matrix. For n>3 , this problem remains open though many partial results are known. In this note, we present a new family of necessary conditions for thi...

We study the operator Q-class functions, present some Hermite-Hadamard inequalities for operator Q-class functions and give some Kantorovich and Jensen type operator inequalities involving Q-class functions. (C) 2015 Mathematical Institute Slovak Academy of Sciences

To extend the Euclidean operator radius, we define $w_p$ for an $n$-tuples of operators $(T_1,\ldots, T_n)$ in $\mathbb{B}(\mathscr{H})$ by $w_p(T_1,\ldots,T_n):= \sup_{\| x \| =1} \left(\sum_{i=1}^{n}| \langle T_i x, x \rangle |^p \right)^{\frac1p}$ for $p\geq1$. We generalize some inequalities including Euclidean operator radius of two operators...

We generalize several inequalities involving powers of the numerical radius for product of two operators acting on a Hilbert space. For any $A, B, X\in \mathbb{B}(\mathscr{H})$ such that $A,B$ are positive, we establish some numerical radius inequalities for $A^\alpha XB^\alpha$ and $A^\alpha X B^{1-\alpha}\,\,(0 \leq \alpha \leq 1)$ and Heinz mean...

A Hermite-Hadamard-Mercer type inequality is presented and then generalized to Hilbert space operators. It is shown that f M+m-Sigma(n)(i=1) x(i)A(i)) <= f(M)+f(m)-Sigma(n)(i=1)f(x(i))A(i), where f is a convex function on an interval [m,M] containing 0, x(i) is an element of [m, M], i=1,...,n, and A(i) are positive operators acting on a finite dime...

We show that the family of all operator monotone functions f on (-1,1) such that f(0)=0 and f'(0)=1 is a normal family and investigate some properties of odd operator monotone functions. We also characterize the odd operator monotone functions and even operator convex functions on (-1,1). As a consequence, we show that if f is an odd operator monot...

We introduce the non-commutative f-divergence functional for an operator convex function f, where and are continuous fields of Hilbert space operators and study its properties. We establish some relations between the perspective of an operator convex function f and the non-commutative f-divergence functional. In particular, an operator extension o...

We extend the celebrated Löwner–Heinz inequality by showing that if A,BA,B are Hilbert space operators such that A>B⩾0, thenAr-Br⩾||A||r-||A||-1||(A-B)-1||r>0for each 0<r⩽1. As an application we prove thatlogA-logB⩾log||A||-log||A||-1||(A-B)-1||>0.

We extend the celebrated L\"owner--Heinz inequality by showing that if $A, B$ are Hilbert space operators such that $A > B \geq 0$, then A^r - B^r \geq ||A||^r-(||A||- \frac{1}{||(A-B)^{-1}||})^r > 0 for each $0 < r \leq 1$. As an application we prove that \log A - \log B \geq \log||A||- \log(||A||-\frac{1}{||(A-B)^{-1}||})>0.

Some inequalities of Jensen type for Q-class functions are proved. More precisely, a refinement of the inequality f((1/P) Sigma(n)(i=1) p(i)x(i)) <= P Sigma(n)(i=1)(f(x(i))/p(i)) is given in which p(l), ..., p(n) are positive numbers, P = Sigma(n)(i=1) p(i) and f is a Q-class function. The notion of the jointly Q-class function is introduced and so...

We show that the symmetrized product AB+BA of two positive operators A and B is positive if and only if f(A+B) £ f(A)+f(B){f(A+B)\leq f(A)+f(B)} for all non-negative operator monotone functions f on [0,∞) and deduce an operator inequality. We also give a necessary and sufficient condition for that the composition f °g{f \circ g} of an operator conv...

In the framework of a pre-inner product C*-module over a unital C*-algebra, we show several reverse Cauchy-Schwarz type inequalities of additive and multiplicative types, by using some ideas in N. Elezović et al. [Math. Inequal. Appl., 8 (2005), no.2, 223-231]. We apply our results to give Klamkin-Mclenaghan, Shisha-Mond and Cassels type inequaliti...

We study the Cauchy–Schwarz and some related inequalities in a semi-inner product module over a C⁎C⁎-algebra AA. The key idea is to consider a semi-inner product AA-module as a semi-inner product AA-module with respect to another semi-inner product. In this way, we improve some inequalities such as the Ostrowski inequality and an inequality related...

We survey several significant results on the Bohr inequality and presented its generalizations in some new approaches. These are some Bohr type inequalities of Hilbert space operators related to the matrix order and the Jensen inequality. An eigenvalue extension of Bohr's inequality is discussed as well.

We establish several operator versions of the classical Aczél inequality. One of operator versions deals with the weighted operator geometric mean and another is related to the positive sesquilinear forms. Some applications including the unital positive linear maps on C*C*-algebras and the unitarily invariant norms on matrices are presented.

Suppose that is an algebra, σ, τ : → are two linear mappings such that both σ ( ) and τ ( ) are subalgebras of and 𝒳 is a ( τ ( ), σ ( ))-bimodule. A linear mapping D : → 𝒳 is called a ( σ, τ )-derivation if D ( ab ) = D ( a ) · σ ( b ) + τ ( a ) · D ( b ) ( a, b ∈ ). A ( σ, τ )-derivation D is called a ( σ, τ )-inner derivation if there exists an...

We establish a general operator parallelogram law concerning a characterization of inner product spaces, get an operator extension of Bohr's inequality and present several norm inequalities. More precisely, let ${\mathfrak A}$ be a $C^*$-algebra, $T$ be a locally compact Hausdorff space equipped with a Radon measure $\mu$ and let $(A_t)_{t\in T}$ b...

In this paper we present a new criterion on characterization of real inner product spaces. We conclude that a real normed space $(X, \|...\|)$ is an inner product space if $$\sum_{\epsilon_i \in \{-1,1\}} \|x_1 + \sum_{i=2}^k\epsilon_ix_i\|^2=\sum_{\epsilon_i \in \{-1,1\}} (\|x_1\| + \sum_{i=2}^k\epsilon_i\|x_i\|)^2,$$ for some positive integer $k\...

We show that the unit ball of a full Hilbert C∗-module is sequentially compact in a certain weak topology if and only if the underlying C∗-algebra is finite dimensional. This provides an answer to the question posed in J. Chmieliński et al [Perturbation of the Wigner equation in inner product C∗-modules, J. Math. Phys. 49 (2008), no. 3, 033519]. 1....

Let σ be a surjective ultraweakly continuous ∗-linear mapping and d be a σ-derivation on a von Neumann algebra \mathfrak M\mathfrak M. We show that there are a surjective ultraweakly continuous ∗-homomorphism S:\mathfrak M®\mathfrak M\Sigma:\mathfrak M\to\mathfrak M and a Σ-derivation D:\mathfrak M®\mathfrak MD:\mathfrak M\to\mathfrak M such...

Suppose that sigma : M -> M is an ultraweakly continuous surjective *-linear mapping and d : M -> M is an ultraweakly continuous *-sigma-derivation such that d(l) is a central element of M. We provide a Kadison-Sakai-type theorem by proving that h can be decomposed into R circle plus L and d can be factored as the form delta circle plus 2Zr, where...

We define the notion of φ-perturbation of a densely defined adjointable mapping and prove that any such mapping f between Hilbert A-modules over a fixed C* -algebra A with densely defined corresponding mapping g is A-linear and adjointable in the classical sense with adjoint g. If both f and g are everywhere defined then they are bounded. Our work...

In this paper we introduce a notion of a non-Archimedean fuzzy norm and study the stability of the Cauchy equation in the context of non-Archimedean fuzzy spaces in the spirit of Hyers–Ulam–Rassias–Găvruţa. As a corollary, the stability of the Jensen equation is established. We indeed present an interdisciplinary relation between the theory of fuzz...

Suppose that M,N are von Neumann algebras acting on a Hilbert space and M is hyperfinite. Let ρ : M → N be an ultraweakly continuous ∗-homomorphism and let δ : M → N be a ∗-ρ-derivation such that δ(I )c ommutes withρ(I). We prove that there is an element U in N withU �≤� δ� such that δ(A )= Uρ (A) − ρ(A)U for all A ∈ M.

Let T, A be operators with domains D(T) subset of D(A) in a normed space X. The operator A is called T-bounded if parallel to Ax parallel to <= a parallel to x parallel to + b parallel to Tx parallel to for some a, b >= 0 and all x is an element of D(T). If A has the Hyers-Ulam stability then under some suitable assumptions we show that both T and...

The classical Mazur–Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur–Ulam theorem in the non-Archimedean strictly convex normed spaces.