Seminorm and numerical radius inequalities of operators in semi-Hilbertian spaces

Abstract

Let $A$ be a positive bounded operator on a Hilbert space $(\mathcal{H},⟨\cdot ,\cdot ⟩)$. The semi-inner product ${⟨x,y⟩}_A:=⟨Ax,y⟩$, $x,y\in \mathcal{H},$ induces a seminorm ${\Vert \cdot \Vert }_A$ on $\mathcal{H}$. Let ${\Vert T\Vert }_A,w_A(T),$ 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 $(\mathcal{H},{\Vert \cdot \Vert }_A)$, 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 $T^{{♯}_A}$ is a distinguished $A$-adjoint operator of $T$. Some applications of our inequalities are also provided.

References in corpus (2)

• Positivity of $2\times 2$ block matrices of operators
• On inequalities for A-numerical radius of operators

Cited by in corpus (15)

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• Improved inequalities for the numerical radius via Cartesian decomposition
• Further results on $\mathbb{A}$-numerical radius inequalities
• Some numerical radius inequalities for semi-Hilbert space operators
• Refinement of seminorm and numerical radius inequalities of semi-Hilbertian space operators
• Refinements of generalized Euclidean operator radius inequalities of 2-tuple operators
• An extension of Birkhoff--James orthogonality relations in semi-Hilbertian space operators
• On A-numerical radius inequalities for $2\times 2$ operator matrices
• New semi-norm of semi-Hilbertian space operators and its application
• The generalized Schwarz inequality for semi-Hilbertian space operators and Some $A$-numerical radius inequalities
• $A$-numerical radius : New inequalities and characterization of equalities
• Improvement of $A$-numerical radius inequalities of semi-Hilbertian space operators
• On $\mathbb{A}$-numerical radius inequalities for $2\times 2$ operator matrices-II
• Some upper bounds for the $\mathbb{A}$-numerical radius of $2\times 2$ block matrices