New refinements of numerical radius inequalities for Hilbert space operators

This study explores the power vector inequalities for a pair of operators ( B , C ) \left(B,C) in a Hilbert space. By utilizing a Mitrinović-Pečarić-Fink-type inequality for inner products and norms, we derive various power vector inequalities. Specifically, we consider the cases where ( B , C ) \left(B,C) is equal to ( A , A * ) \left(A,{A}^{* }) or ( Re ( A ) , Im ( A ) ) (\hspace{0.1em}\text{Re}\hspace{0.1em}\left(A),\hspace{0.1em}\text{Im}\hspace{0.1em}\left(A)) for an operator A A in B ( H ) B\left(H) , where H H is a Hilbert space. This leads to the derivation of vector, norm, and numerical radius inequalities for a single operator. Furthermore, we obtain power inequalities for the s s - r r -norm and s s - r r -numerical radius of the operator pair ( B , C ) ∈ B ( H ) \left(B,C)\in B\left(H) , which generalizes the Euclidean norm and Euclidean numerical radius. Finally, we apply these results to derive the corresponding inequalities for a single operator A ∈ B ( H ) A\in B\left(H) .

Some inequalities for normal operators in Hilbert spaces are given. For this purpose, some results for vectors in inner product spaces due to Buzano, Dunkl-Williams, Hile, Goldstein-Ryff-Clarke, Dragomir-Sandor and the author are employed.

In this paper we obtain some new power and hölder type trace inequalities for positive operators in Hilbert spaces. As tools, we use some recent reverses and refinements of Young inequality obtained by several authors.

Let T be a normal bounded operator on a Hilbert space and let ω(T) denote the numerical radius of T. In this paper, we give new inequalities numerical radius of normal operatos on a Hilbert space, one of these inequalties we prove that where T = T 1 + iT 2 the Cartesian decomposition of T and . Moreover, some other related results are also obtained.

We develop upper and lower bounds for the numerical radius of $$2\times 2$$ off-diagonal operator matrices, which generalize and improve on some existing ones. We also show that if A is a bounded linear operator on a complex Hilbert space, then for all $$r\ge 1$$ , $$\begin{aligned} w^{2r}(A) \le \frac{1}{4} \big \Vert |A|^{2r}+|A^*|^{2r} \big \Vert + \frac{1}{2} \min \left\{ \big \Vert \Re \big (|A|^r\, |A^*|^r \big ) \big \Vert , w^r(A^2) \right\} \end{aligned}$$ where w(A), $$\Vert A\Vert $$ and $$\Re (A)$$ , respectively, stand for the numerical radius, the operator norm and the real part of A. This (for $$r=1$$ ) improves on some existing well-known numerical radius inequalities.

Let T and S be bounded linear operators on a complex Hilbert space H. In this paper, we define a new quantity K(T) which is less than the numerical radius w(T) of T. We employ this quantity to provide some new refinements of the numerical radii of products TS, commutators TS ? ST, and anticommutators TS + ST, which give an improvement to the important results by A. Abu-Omar and F. Kittaneh (Studia Mathematica, 227 (2), (2015)). Furthermore, we extend these results to the case of semi-Hilbertian space operators in order to improve some results of A. Zamani (Linear Algebra and its Applications, 578, (2019)).

A numerical radius inequality due to Shebrawi and Albadawi says that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then for all r≥1. We give sharper numerical radius inequality which states that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then where . Moreover, we give many numerical radius inequalities which are sharper than related inequalities proved recently, and several applications are given.

For compact Hilbert space operators A and B, the singular values of $A^ * B$ are shown to be dominated by those of $\frac{1}{2}(AA^* + BB^* )$.

The objective of this chapter is to introduce a new class of operator s-Godunova-Levin-Dragomir convex functions. We also derive some new Hermite-Hadamard-like inequalities for operator s-Godunova-Levin-Dragomir convex functions of positive operators in Hilbert spaces.

Hermite–Hadamard’s type inequalities for convex functions of selfadjoint operators in Hilbert spaces

Some vector and operator generalized trapezoidal inequalities for continuous functions of selfadjoint operators in Hilbert spaces are given. Applications for power and logarithmic functions of operators are provided as well.

In this article, we present generalized improvements of certain Cauchy-Schwarz type inequal-ities. As applications of our results, we provide refinements of some numerical radius inequalities for Hilbert space operators. Finally, we obtain certain numerical radius inequalities of Hilbert space operators involving geometrically convex functions.

On utilizing the spectral representation of self-adjoint operators in Hilbert spaces, some inequalities for the composite operator , where and for various classes of continuous functions are given. Applications for the power function and the logarithmic function are also provided.

In this article, we employ certain properties of the transform $C_{M,m}(A)=(MI-A^*)(A-mI)$ to obtain new inequalities for the bounded linear operator $A$ on a complex Hilbert space $\mathcal{H}$. In particular, we obtain new relations among $|A|,|A^*|,|\mathfrak{R}A|$ and $|\mathfrak{I}A|$. Further numerical radius inequalities that extend some known inequalities will be presented too.

In this paper, we establish several inner product inequalities that characterize the positivity of block operator matrices. By examining specific blocks of positive operator matrices involving the Moore–Penrose inverse, we derive new inner product inequalities that refine recently established bounds. As an application, we further refine certain numerical radius inequalities from the literature. Our findings generalize and extend several well-known results in this field, contributing to ongoing advancements in operator inequalities.