Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$. An operator $T$ is said to be normaloid if its numerical radius $w(T)$ equals its operator norm $\Vert T\Vert$. In this paper, we establish several characterizations of normaloid operators in Hilbert spaces. In particular, we investigate these operators through the framework of Birkhoff–James orthogonality and norm-parallelism. Mainly, we show that $T$ is normaloid if, and only if, there exists ${\xi }_0\in \mathbb{C}$ with $|{\xi }_0|=\Vert T\Vert$ such that

$$I{\perp }_{BJ}(T-{\xi }_{0}I),$$

where ${\perp }_{BJ}$ denotes Birkhoff–James orthogonality. We also present further equivalent formulations and explore various structural consequences of these characterizations.

In this paper, we present several refinements of the operator Cauchy-Schwarz inequality for positive operators. Our main result strengthens the classical form of this inequality and serves as a foundation for deriving a series of new inequalities that both generalize and improve upon existing results in the literature. Furthermore, we investigate substantial improvements to the Cauchy–Schwarz inequality by employing orthogonal projections, leading to sharper bounds in various settings. Additionally, we obtain a new perspective on the Cauchy–Schwarz inequality by showing that both the inner product and the product of norms can be characterized as extremal values of projection-dependent expressions. Several related inequalities are also established, many of which recover or extend recent contributions by other authors.

The main purpose of this paper is to introduce and study the so-called $p$-arithmetic-mean norm for pairs of bounded linear operators on a complex Hilbert space $H$. Specifically, for a pair $(A,B)$ of bounded linear operators on $H$, with $\nu \in [0,1]$ and $p>0$, we define the following:

$$\Vert (A,B){\Vert }_{p,\nu }:=\underset{\Vert x\Vert =1}{sup}((1-\nu )\Vert Ax{\Vert }^p+\nu \Vert Bx{\Vert }^p{)}^{1/p}.$$

We establish, among other results, that for $\nu \in (0,1]$ and $p\in (0,1]$,

$$\Vert (A,B){\Vert }_{2p,\nu }^{2p}\le R\Vert A-B{\Vert }^{2p}+min\{\Vert A{\Vert }^{2(1-\nu )p}\Vert B{\Vert }^{2\nu p},\Vert (1-\nu )|A{|}^2+\nu |B{|}^2{\Vert }^p\}.$$

Applications are given to off-diagonal operator matrices, and to the particular cases $(A,B)=(T,T^{\ast })$ and $(A,B)=(\mathrm{Re}(T),\mathrm{Im}(T))$, where $T$ is a bounded linear operator on $H$, $T^{\ast }$ denotes its adjoint, $\mathrm{Re}T=\frac{1}{2}(T+T^{\ast })$ is its real part, and $\mathrm{Im}T=\frac{1}{2i}(T-T^{\ast })$ is its imaginary part.

In this paper, we aim to investigate the class of jointly hyponormal operators related to a positive operator $A$ on a complex Hilbert space $\mathcal{X}$, which is called jointly $A$-hyponormal. This notion was first introduced by Guesba et al. in [Linear and Multilinear Algebra, 69(15), 2888–2907] for $m$-tuples of operators that admit adjoint operators with respect to $A$. Mainly, we prove that if $\mathbf{B}=(B_1,\cdots ,B_m)$ is a jointly $A$-hyponormal $m$-tuple of commuting operators, then $\mathbf{B}$ is jointly $A$-normaloid. This result allows us to establish, for a particular case when $A$ is the identity operator, a sharp bound for the distance between two jointly hyponormal $m$-tuples of operators, expressed in terms of the difference between their Taylor spectra. We also aim to introduce and investigate the class of spherically $A$- $p$-hyponormal operators with $0<p<1$. Additionally, we study the tensor product of specific classes of multivariable operators in semi-Hilbert spaces.