We observe that the classical notion of numerical radius gives rise to a smoothness in space bounded linear operators on certain Banach spaces, whenever is norm. characterize Birkhoff-James orthogonality finite-dimensional space, endowed with Some examples are also discussed illustrate geometric differences between norm and usual operator norm, from viewpoint smoothness.

In this paper, we establish several characterizations of the A-parallelism bounded linear operators with respect to seminorm induced by a positive operator A acting on complex Hilbert space. Among other things, investigate relationship between A-seminorm-parallelism and A-Birkhoff–James orthogonality A-bounded operators. particular, characterize which satisfy A-Daugavet equation. addition, rela...

This paper deals with the study of Birkhoff--James orthogonality a linear operator to subspace operators defined between arbitrary Banach spaces. In case domain space is reflexive and finite dimensional we obtain complete characterization. For spaces, same under some additional conditions. an Hilbert H, also L(H), both respect norm as well numerical radius norm.

We present a short proof for the fact that if smooth real Banach spaces of dimension three or higher have isomorphic Birkhoff–James orthogonality structures, then they are (linearly) isometric to each other. This generalizes results Koldobsky and Wójcik. Moreover, in an arbitrary dimension, we construct examples non-isometric pairs non-smooth admit norm preserving homogeneous bicontinuous prese...

In this paper, the notion of rank−k numerical range of rectangular complex matrix polynomials are introduced. Some algebraic and geometrical properties are investigated. Moreover, for > 0, the notion of Birkhoff-James approximate orthogonality sets for −higher rank numerical ranges of rectangular matrix polynomials is also introduced and studied. The proposed definitions yield a natural general...

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 positive operator $A\in\B(\h)$, and number $\lambda\in [0,1]$, seminorm ${\|\cdot\|}_{(A,\lambda)}$ is defined set $\B_{A^{1/2}}(\h)$ in $\B(\h)$ having an $A^{1/2}$-adjoint. The combination sesquilinear form ${\langle \cdot, \...