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Operator matrices have played a significant role in the study of properties of the numerical radii of Hilbert space operators. This paper presents several new sharp upper bounds for the numerical radii of operator matrices in terms of certain complex combinations. The obtained results reveal many interesting properties of the numerical radius.

In this paper, we introduce the f-operator radius of Hilbert space operators as a generalization of the Euclidean operator radius and the q-operator radius. The properties of the newly defined radius are discussed, emphasizing how it extends some known results in the literature.

M. Lin defined a binary operation for two positive semi-definite matrices in studying certain determinantal inequalities that arise from diffusion tensor imaging. This operation enjoys some interesting properties similar to the operator geometric mean. We study this operation further and present numerous properties emphasizing the relationship with the operator geometric mean. In the end, we present an application toward Tsallis relative operator entropy.

We study Birkhoff-James orthogonality and isosceles orthogonality of bounded linear operators between Hilbert spaces and Banach spaces. We explore Birkhoff-James orthogonality of bounded linear operators in light of a new notion introduced by us and also discuss some of the possible applications in this regard. We also study isosceles orthogonality of bounded (positive) linear operators on a Hilbert space and some of the related properties, including that of operators having disjoint support. We further explore the relations between Birkhoff-James orthogonality and isosceles orthogonality in a general Banach space.

In this paper, we present various inequalities regarding the linear combinations of orthogonal projections. These results aim to generalize and refine well-known inequalities, such as those due to Buzano and Ostrowski. Additionally, we investigate a specific case of these linear combinations and introduce new refinements of the Cauchy–Schwarz inequality. Furthermore, we establish some findings related to the covariance and variance of bounded linear operators. Moreover, as applications of some of our results, we establish several inequalities involving the product of three operators, one of which is a linear combination of an orthogonal projection and the identity operator. Finally, we introduce a new positive operator construction in terms of an orthogonal projection and the identity operator, and we derive some norms and numerical radius inequalities involving it.

In this note, we give a new proof of the following well-known norm formula which holds for any two orthogonal projections PT,PSon a Hilbert H,‖PT+PS‖=1+‖PTPS‖,unless PT=PS=0.This equality was proved by Duncan and Taylor (Proc R Soc Edinb Sect A 75(2):119–129, 1975). We derive this formula from the relationship between the spectra of the sum and product of any two idempotents, as well as various norm inequalities for positive operators defined on H.Applications of our results are given.

Let Abe a positive (semidefinite) bounded linear operator on a complex Hilbert space (H,⟨·,·⟩). The semi-inner product induced by Ais defined by ⟨x,y⟩A:=⟨Ax,y⟩for all x,y∈Hand defines a seminorm ‖·‖Aon H. This makes Hinto a semi-Hilbert space. For p∈[1,+∞), the generalized A-joint numerical radius of a d-tuple of operators T=(T1,…,Td)is given by ωA,p(T)=sup‖x‖A=1∑k=1d|〈Tkx,x〉A|p1p.Our aim in this paper is to establish several bounds involving ωA,p(·). In particular, under suitable conditions on the operators tuple T, we generalize the well-known inequalities due to Kittaneh (Studia Math 168(1):73–80, 2005).