We introduce a numerical radius operator space $(X, \mathcal{W}_n)$. The conditions to be a numerical radius operator space are weaker than the Ruan's axiom for an operator space $(X, \mathcal{O}_n)$. Let $w(\cdot)$ be the numerical radius norm on $\mathbb{B}(\mathcal{H})$. It is shown that if $X$ admits a norm $\mathcal{W}_n(\cdot)$ on the matrix space $\mathbb{M}_n(X)$ which satisfies the conditions, then there is a complete isometry, in the sense of the norms $\mathcal{W}_n(\cdot)$ and $w_n(\cdot)$, from $(X, \mathcal{W}_n)$ into $(\mathbb{B}(\mathcal{H}), w_n)$. We study the relationship between the operator space $(X, \mathcal{O}_n)$ and the numerical radius operator space $(X, \mathcal{W}_n)$. The category of operator spaces can be regarded as a subcategory of numerical radius operator spaces.
Let $H$ be a complex separable Hilbert space of dimension $\geq 2$, ${\mathcal B}_s(H)$ the space of all self-adjoint operators on $H$. We give a complete classification of non-linear surjective maps on $\mathcal B_s(H)$ preserving respectively numerical radius and numerical range of Lie product.
We study a factorization of bounded linear maps from an operator space $A$ to its dual space $A^*$. It is shown that $T : A \longrightarrow A^*$ factors through a pair of a column Hilbert spaces $\mathcal{H}_c$ and its dual space if and only if $T$ is a bounded linear form on $A \otimes A$ by the canonical identification equipped with a numerical radius type Haagerup norm. As a consequence, we characterize a bounded linear map from a Banach space to its dual space, which factors through a pair of Hilbert spaces.
This article is devoted to studying some new numerical radius inequalities for Hilbert space operators. Our analysis enables us to improve an earlier bound of numerical radius due to Kittaneh.
Some elementary inequalities providing upper bounds for the difference of the norm and the numerical radius of a bounded linear operator on Hilbert spaces under appropriate conditions are given.
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 their off-diagonal parts. One of our main results states that if $(X, (O_n))$ is an operator space, then \begin{align*} \frac12\max\big(W_{\max}(x_1+x_2)&, W_{\max}(x_1-x_2) \big)\\ &\le W_{\max}\Big(\begin{bmatrix} 0 & x_1 \\ x_2 & 0 \end{bmatrix}\Big)…
In this paper, we define a new concept of numerical range $W_{o}(\cdot)$ and prove its basic results. We also define the numerical radius $ω_{o}(\cdot)$ and prove that $$ω_{o}(T)\leq||| T|||\leq 2ω_{o}(T).$$
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--Wielandt radius. Some other related results are also discussed.
We prove numerical radius inequalities involving commutators of $G_{1}$ operators and certain analytic functions. Among other inequalities, it is shown that if $A$ and $X$ are bounded linear operators on a complex Hilbert space, then \begin{equation*} w(f(A)X+X\bar{f}(A))\leq {\frac{2}{d_{A}^{2}}}w(X-AXA^{\ast }), \end{equation*} where $A$ is a $G_{1}$ operator with $σ(A)\subset \mathbb{D}$ and $f$ is analytic on the unit disk $\mathbb{D}$ such that $\textrm{Re}(f)>0$ and $f(0)=1$.
We generalize several inequalities involving powers of the numerical radius for off-diagonal part of $2\times2$ operator matrices of the form $T=\left[\begin{array}{cc} 0&B, C&0 \end{array}\right]$, where $B, C$ are two operators. In particular, if $T=\left[\begin{array}{cc} 0&B, C&0 \end{array}\right]$, then we get \begin{align*} {1\over 2^{{3\over2}(r-1)}}\max\{ \| μ\|, \| η\| \} \leq w^{r}(T)\leq \frac{1}{2^{r+1}} \max\{ \| μ\|, \| η\| \}, \end{align*} where $r\geq 2$ and $ μ=|(C-B^{*})+i(C+B^{*})|^{r}+|(B^{*}-C)+i(C+B^{*})|^{r}$, $ η=|(B-C^{*})+i(B+C^{*})|^{r}+|(C^{*}-B)+i(B+C^{*})|^{r}$.
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 $A^αXB^α$ and $A^αX B^{1-α}\,\,(0 \leq α\leq 1)$ and Heinz means under mild conditions.
Let $a$ be a positive element in a unital $C^*$-algebra $\mathfrak{A}$. We define a semi-norm on $\mathfrak{A}$, which generalizes the $a$-operator semi-norm and the $a$-numerical radius. We investigate basic properties of this semi-norm and prove inequalities involving it. Further, we derive new upper and lower bounds for the $a$-numerical radii of elements in $\mathfrak{A}$. Some other related results are also discussed.
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 to those involving $w_p$. Further we obtain some lower and upper bounds for $w_p$. Our main result states that if $f$ and $g$ are nonnegative continuous functions on $\left[ 0,\infty \right) $ satisfying $f\left( t\right) g\left(t\right) =t$ for all …
The problems considered in this paper come as a natural continuation of our program to develop a free analogue of Sz.-Nagy-Foias theory, for row contractions. The paper is structured as follows: Introduction Part I. Unitary invariants for n-tuples of operators 1. Joint numerical radius 2. Euclidean operator radius 3. Joint numerical range and spectrum 4. $ρ$-operator radius Part II. Joint operator radii, inequalities, and applications 5. von Neumann inequalities 6. Constrained von Neumann inequalities 7. Multivariable Haagerup-de la Harpe inequalities 8. Multivariable Fejer inequalities Refere…
In this paper, we study the hyperbolic geometry of noncommutative balls generated by the joint operator radius $ω_ρ$, $ρ\in (0,\infty]$, for $n$-tuples of bounded linear operators on a Hilbert space. In particular, $ω_1$ is the operator norm, $ω_2$ is the joint numerical radius, and $ω_\infty$ is the joint spectral radius. We provide mapping theorems, von Neumann inequalities, and Schwarz type lemmas for free holomorphic functions on noncommutative balls, with respect to the hyperbolic metric $δ_ρ$, the Carath\' eodory metric $d_K$, and the joint operator radius $ω_ρ$.
