Let $A$ be a positive bounded operator on a Hilbert space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$. The semi-inner product ${\langle x, y\rangle}_A := \langle Ax, y\rangle$, $x, y\in\mathcal{H},$ induces a seminorm ${\|\cdot\|}_A$ on $\mathcal{H}$. Let ${\|T\|}_A,\ w_A(T),$ and $c_A(T)$ denote the $A$-operator seminorm, the $A$-numerical radius, and the $A$-Crawford number of an operator $T$ in the semi-Hilbertian space $\big(\mathcal{H}, {\|\cdot\|}_A\big)$, respectively. In this paper, we present some seminorm inequalities and equalities for semi-Hilbertian space operators. More precisely, we give some necessary and sufficient conditions for two orthogonal semi-Hilbertian operators satisfy Pythagoras' equality. In addition, we derive new upper and lower bounds for the numerical radius of operators in semi-Hilbertian spaces. In particular, we show that \begin{align*} \frac{1}{16} {\|TT^{\sharp_{A}} + T^{\sharp_{A}}T\|}^{2}_{A} + \frac{1}{16}c_{A}\Big(\big(T^2 + (T^{\sharp_{A}})^2\big)^2\Big) \leq w^4_{A}(T) \leq \frac{1}{8} {\|TT^{\sharp_{A}} + T^{\sharp_{A}}T\|}^{2}_{A} + \frac{1}{2}w^2_{A}(T^2), \end{align*} where $T^{\sharp_A}$ is a distinguished $A$-adjoint operator of $T$. Some applications of our inequalities are also provided.
We give new inequalities for $A$-operator seminorm and $A$-numerical radius of semi-Hilbertian space operators and show that the inequalities obtained here generalize and improve on the existing ones. Considering a complex Hilbert space $\mathcal{H}$ and a non-zero positive bounded linear operator $A$ on $\mathcal{H},$ we show with among other seminorm inequalities, if $S,T,X\in \mathcal{B}_A(\mathcal{H})$, i.e., if $A$-adjoint of $S,T,X$ exist then $$2\|S^{\sharp_A}XT\|_A \leq \|SS^{\sharp_A}X+XTT^{\sharp_A}\|_A.$$ Further, we prove that if $T\in \mathcal{B}_A(\mathcal{H})$ then \begin{eqnarr…
Let $\mathbb{B}(\mathcal{H})$ denote the $C^{\ast}$-algebra of all bounded linear operators on a Hilbert space $\big(\mathcal{H}, \langle\cdot, \cdot\rangle\big)$. Given a positive operator $A\in\B(\h)$, and a number $λ\in [0,1]$, a seminorm ${\|\cdot\|}_{(A,λ)}$ is defined on the set $\B_{A^{1/2}}(\h)$ of all operators in $\B(\h)$ having an $A^{1/2}$-adjoint. The seminorm ${\|\cdot\|}_{(A,λ)}$ is a combination of the sesquilinear form ${\langle \cdot, \cdot\rangle}_A$ and its induced seminorm ${\|\cdot\|}_A$. A characterization of Birkhoff--James orthogonality for operators with respect to th…
In this paper we generalize the concept of Davis-Wielandt shell of operators on a Hilbert space when a semi-inner product induced by a positive operator $A$ is considered. Moreover, we investigate the parallelism of $A$-bounded operators with respect to the seminorm and the numerical radius induced by $A$. Mainly, we characterize $A$-normaloid operators in terms of their $A$-Davis-Wielandt radii. In addition, a connection between $A$-seminorm-parallelism to the identity operator and an equality condition for the $A$-Davis-Wielandt radius is proved. This generalizes the well-known results in \c…
In this paper, we aim to introduce the notion of the spectral radius of bounded linear operators acting on a complex Hilbert space $\mathcal{H}$, which are bounded with respect to the seminorm induced by a positive operator $A$ on $\mathcal{H}$. Mainly, we show that $r_A(T)\leq ω_A(T)$ for every $A$-bounded operator $T$, where $r_A(T)$ and $ω_A(T)$ denote respectively the $A$-spectral radius and the $A$-numerical radius of $T$. This allows to establish that $r_A(T)=ω_A(T)=\|T\|_A$ for every $A$-normaloid operator $T$, where $\|T\|_A$ is denoted to be the $A$-operator seminorm of $T$. Moreover,…
Let $A$ be a positive bounded operator on a Hilbert space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$. The semi-inner product ${\langle x, y\rangle}_A := \langle Ax, y\rangle$, $x, y\in\mathcal{H},$ induces a seminorm ${\|\cdot\|}_A$ on $\mathcal{H}$. Let $w_A(T)$ denote the $A$-numerical radius of an operator $T$ in the semi-Hilbertian space $\big(\mathcal{H}, {\|\cdot\|}_A\big)$. In this paper, for any semi-Hilbertian operators $T$ and $S$, we show that $w_A(TR) = w_A(SR)$ for all ($A$-rank one) semi-Hilbertian operator $R$ if and only if $A^{1/2}T = λA^{1/2}S$ for some complex uni…
New inequalities for the $A$-numerical radius of the products and sums of operators acting on a semi-Hilbert space, i.e. a space generated by a positive semidefinite operator $A$, are established. In particular, it is proved for operators $T$ and $S,$ having $A$-adjoint, that $$ ω_A(TS) \leq \frac{1}{2}ω_A(ST)+\frac{1}{4}\Big(\|T\|_A\|S\|_A+\|TS\|_A\Big),$$ where $ω_A(T)$ and $\|T\|_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$.
Let $A$ be a positive (semidefinite) operator on a complex Hilbert space $\mathcal{H}$ and let $\mathbb{A}=\left(\begin{array}{cc} A & O O & A \end{array}\right).$ We obtain upper and lower bounds for the $A$-Davis-Wielandt radius of semi-Hilbertian space operators, which generalize and improve on the existing ones. We also obtain upper bounds for the $\mathbb{A}$-Davis-Wielandt radius of $2 \times 2$ operator matrices. Finally, we determine the exact value for the $\mathbb{A}$-Davis-Wielandt radius of two operator matrices $\left(\begin{array}{cc} I & X\\ 0 & 0 \end{array}\right)$ and $\left(…
In this work, the mixed Schwarz inequality for semi-Hilbertian space operators is proved. Namely, for every positive Hilbert space operator $A$. If $f$ and $g$ are nonnegative continuous functions on $\left[0,\infty\right)$ satisfying $f(t)g(t) =t$ $(t\ge0)$, then \begin{align*} \left| {\left\langle {T x,y} \right\rangle_A } \right| \le \left\| {f\left( {\left| T \right|_A x} \right)} \right\|_A \left\| {g\left( {\left| {T^{\sharp_A } } \right|_A y} \right)} \right\|_A \end{align*} for every Hilbert space operator $T$ such that the range of $T^* A$ is a subset in the range of $A$, such that $A…
In this paper we introduce and prove some properties of $(α;β)$-normal operators according to semi-Hilbertian space structures. Furthermore we s,ate various inequalities between the A-operator norm and A-numerical radius of $(α,β)$-normal operators in semi Hilbertian spaces.
Let $A$ be a positive bounded operator on a Hilbert space $\big(\mathcal{H}, \langle \cdot, \cdot\rangle \big)$. The semi-inner product ${\langle x, y\rangle}_A := \langle Ax, y\rangle$, $x, y\in\mathcal{H}$ induces a semi-norm ${\|\cdot\|}_A$ on $\mathcal{H}$. Let ${\|T\|}_A$ and $w_A(T)$ denote the $A$-operator semi-norm and the $A$-numerical radius of an operator $T$ in semi-Hilbertian space $\big(\mathcal{H}, {\|\cdot\|}_A\big)$, respectively. In this paper, we prove the following characterization of $w_A(T)$ \begin{align*} w_A(T) = \displaystyle{\sup_{α^2 + β^2 = 1}} {\left\|α\frac{T + T^…
In this paper, the concept of Birkhoff--James orthogonality of operators on a Hilbert space is generalized when a semi-inner product is considered. More precisely, for linear operators $T$ and $S$ on a complex Hilbert space $\mathcal{H}$, a new relation $T\perp^B_A S$ is defined if $T$ and $S$ are bounded with respect to the seminorm induced by a positive operator $A$ satisfying ${\|T + γS\|}_A\geq {\|T\|}_A$ for all $γ\in \mathbb{C}$. We extend a theorem due to R. Bhatia and P. Šemrl, by proving that $T\perp^B_A S$ if and only if there exists a sequence of $A$-unit vectors $\{x_n\}$ in $\math…
In this survey, we shall present characterizations of some distinguished classes of Hilbertian bounded linear operators (namely, normal operators, selfadjoint operators, and unitary operators) in terms of operator inequalities related to the arithmetic-geometric mean inequality. For the class of all normal operators, we shall present new general characterizations.
Let $\mathbb{A}=\left( \begin{array}{cc} A & 0 \\ 0 & A \\ \end{array} \right)$ be the $2\times2$ diagonal operator matrix determined by a positive bounded operator $A$. For semi-Hilbertian operators $X$ and $Y$, we first show that \begin{align*} w^2_{\mathbb{A}}\left(\begin{bmatrix} 0 & X \\ Y & 0 \end{bmatrix}\right) &\leq \frac{1}{4}\max\Big\{{\big\|XX^{\sharp_A} + Y^{\sharp_A}Y\big\|}_{A}, {\big\|X^{\sharp_A}X + YY^{\sharp_A}\big\|}_{A}\Big\} + \frac{1}{2}\max\big\{w_{A}(XY), w_{A}(YX)\big\}, \end{align*} where $w_{\mathbb{A}}(\cdot)$, ${\|\cdot\|}_{A}$ and $w_{A}(\cdot)$ are the $\mathbb{…
Let $A$ be a positive bounded linear operator acting on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$. Let $ω_A(T)$ and ${\|T\|}_A$ denote the $A$-numerical radius and the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbertian space $\big(\mathcal{H}, {\langle \cdot\mid \cdot\rangle}_A\big)$ respectively, where ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$ for all $x, y\in\mathcal{H}$. In this paper, we show that \begin{equation*}\label{m1} \tfrac{1}{4}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A\le ω_A^2\left(T\right) \le \tfrac{1}{2}\|T^{\sh…
We present sharp lower bounds for the A-numerical radius of semi-Hilbertian space operators. We also present an upper bound. Further we compute new upper bounds for the $B$-numerical radius of $2 \times 2$ operator matrices where $B = \textit{diag}(A,A)$, $A$ being a positive operator. As an application of the A-numerical radius inequalities, we obtain a bound for the zeros of a polynomial which is quite a bit improvement of some famous existing bounds for the zeros of polynomials.
Let ($\mathcal{H}, \langle . , .\rangle )$ be a complex Hilbert space and $A$ be a positive bounded linear operator on it. Let $w_A(T)$ be the $A$-numerical radius and $\|T\|_A$ be the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbertian space $(\mathcal{H}, \langle .,.\rangle_A),$ where $\langle x, y\rangle_A:=\langle Ax, y\rangle$ for all $x,y\in \mathcal{H}$. In this article, we establish several upper and lower bounds for $B$-numerical radius of $2\times 2$ operator matrices, where $B=\begin{bmatrix} A & 0 0 & A \end{bmatrix}$. Further, we prove some refinements of earlie…
Let $\mathbb{A}= \begin{pmatrix} A & 0 \\ 0 & A \\ \end{pmatrix} $ be a $2\times2$ diagonal operator matrix whose each diagonal entry is a bounded positive (semidefinite) linear operator $A$ acting on a complex Hilbert space $\mathcal{H}$. In this paper, we derive several $\mathbb{A}$-numerical radius inequalities for $2\times 2$ operator matrices whose entries are bounded with respect to the seminorm induced by the positive operator $A$ on $\mathcal{H}$. Some applications of our inequalities are also given.
We obtain Lebesgue-type inequalities for the greedy algorithm for arbitrary complete seminormalized biorthogonal systems in Banach spaces. The bounds are given only in terms of the upper democracy functions of the basis and its dual. We also show that these estimates are equivalent to embeddings between the given Banach space and certain discrete weighted Lorentz spaces. Finally, the asymptotic optimality of these inequalities is illustrated in various examples of non necessarily quasi-greedy bases.
Let $r_A(T)$ denote the $A$-spectral radius of an operator $T$ which is bounded with respect to the seminorm induced by a positive operator $A$ on a complex Hilbert space $\mathcal{H}$. In this paper, we aim to establish some $A$-spectral radius inequalities for products, sums and commutators of $A$-bounded operators. Moreover, under suitable conditions on $T$ and $A$ we show that \begin{equation*} r_A\left( \sum_{k=0}^{+\infty}c_{k}T^{k}\right) \leq \sum_{k=0}^{+\infty}|c_{k}|\left[r_A(T)\right]^{k}, \end{equation*} where $c_k$ are complex numbers for all $k\in \mathbb{N}$.
Let $A$ be a positive (semidefinite) bounded linear operator acting on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$. The semi-inner product ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$, $x, y\in\mathcal{H}$ induces a seminorm ${\|\cdot\|}_A$ on $\mathcal{H}$. Let $T$ be an $A$-bounded operator on $\mathcal{H}$, the $A$-numerical radius of $T$ is given by \begin{align*} ω_A(T) = \sup\Big\{\big|{\langle Tx\mid x\rangle}_A\big|: \,\,x\in \mathcal{H}, \,{\|x\|}_A = 1\Big\}. \end{align*} In this paper, we establish several inequalities for $ω_\mathbb{A…
Several new improvements of the $A$-numerical radius inequalities for operators acting on a semi-Hilbert space, i.e., a space generated by a positive operator $A$, are proved. In particular, among other inequalities, we show that \begin{align*} \frac{1}{4}\|T^{\sharp_A} T+TT^{\sharp_A}\|_A \leq\frac{1}{4}\Big(2ω_A^2(T)+γ(T)\Big) \leq ω_A^2(T), \end{align*} where $$γ(T)=\sqrt{\left(\|\Re_A(T)\|_A^2-\|\Im_A(T)\|_A^2\right)^2+4\|\Re_A(T)\Im_A(T)\|_A^2}.$$ Here $ω_A(X)$ and $\|X\|_A$ denote respectively the $A$-numerical radius and the $A$-seminorm of an operator $X$. Also, $\Re_A(T):=\frac{T+T^{\…
Let $A$ be a positive bounded linear operator on a complex Hilbert space $\mathcal{H}$ and $\mathcal{B}_{A}(\mathcal{H})$ be the subspace of all operators which admit $A$-adjoints operators. In this paper, we establish some inequalities involving the commutator and the anticommutator of operators in semi-Hilbert spaces, i.e. spaces generated by positive semidefinite sesquilinear forms. Mainly, among other inequalities, we prove that for $T, S\in\mathcal{B}_{A}(\mathcal{H})$ we have \begin{align*} ω_A(TS \pm ST) \leq 2\sqrt{2}\min\Big\{f_A(T,S), f_A(S,T) \Big\}, \end{align*} where $$f_A(X,Y)=\|…
Let $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$ be a complex Hilbert space and $A$ be a positive (semidefinite) bounded linear operator on $\mathcal{H}$. The semi-inner product induced by $A$ is given by ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$, $x, y\in\mathcal{H}$ and defines a seminorm ${\|\cdot\|}_A$ on $\mathcal{H}$. This makes $\mathcal{H}$ into a semi-Hilbert space. The $A$-joint numerical radius of two $A$-bounded operators $T$ and $S$ is given by \begin{align*} ω_{A,\text{e}}(T,S) = \sup_{\|x\|_A= 1}\sqrt{\big|{\langle Tx\mid x\rangle}_A\big|^2+\big|{\langl…
For a given bounded positive (semidefinite) linear operator $A$ on a complex Hilbert space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle \big)$, we consider the semi-Hilbertian space $\big(\mathcal{H}, \langle \cdot\mid \cdot\rangle_A \big)$ where ${\langle x\mid y\rangle}_A := \langle Ax\mid y\rangle$ for every $x, y\in\mathcal{H}$. The $A$-numerical radius of an $A$-bounded operator $T$ on $\mathcal{H}$ is given by \begin{align*} ω_A(T) = \sup\Big\{\big|{\langle Tx\mid x\rangle}_A\big|\,; \,\,x\in \mathcal{H}, \,{\langle x\mid x\rangle}_A= 1\Big\}. \end{align*} Our aim in this paper is t…
