On approximate \(A\)-seminorm and \(A\)-anumerical radius orthogonality of operators (Q6607772)

If \((\mathcal{H},\langle\cdot,\cdot\rangle)\) is a Hilbert space and \(T,S\in\mathbb{B}(\mathcal{H})\) two bounded operators, then they are Birkhoff-James orthogonal, denoted as \(T\perp^B S\Leftrightarrow \Vert T+\lambda S\Vert \ge \Vert T\Vert \), \(\forall \lambda\in\mathbb{C}\). For a non-negative \(A\in\mathbb{B}(\mathcal{H})\), a semidefinite form \(\langle x,y\rangle_A:=\langle Ax,y\rangle\) with corresponding notions of adjoint, norm, boundedness and orthogonality are defined.\par Exploring these concepts, the authors rely on, but also generalize, older results by, e.g. \textit{B. Magajna} [J. Lond. Math. Soc., II. Ser. 47, No. 3, 516--532 (1993; Zbl 0742.47010)].\par Another goal of the paper is to establish further concepts of approximate orthogonality and approximate spectral radius orthogonality in this Birkhoff-James \(A\)-setting. Here approximate \((\varepsilon,A)\) orthogonality, means that \(\Vert T+\lambda S\Vert ^2_A\ge \Vert T\Vert_A^2 -2\varepsilon \Vert T\Vert_A\Vert \lambda S\Vert_A\), \(\forall \lambda\in\mathbb{C}\). Approximate spectral radius orthogonality is less straightforward. Let \(\omega_A(T)=\sup_{\Vert x\Vert_A=1} \langle Tx,x\rangle_A\) be the \(A\)-spectral radius of an operator \(T\), then \(T\) is approximate spectral radius orthogonal to operator \(S\) if \(\omega_A^2(T+\lambda S)\ge \omega_A^2(T)-2\varepsilon \omega_A(T)\omega_A(\lambda S)\), \(\forall \lambda\in\mathbb{C}\). For the latter concept, the authors rely on more recent work by \textit{J. Sen} and \textit{K. Paul} [Math. Slovaca 73, No. 1, 147--158 (2023; Zbl 1517.46015)].