For $ A\in\mathcal{B}(X), \; B\in\mathcal{B}(Y), \; C\in\mathcal{B}(Y, X), $ let $ M = \left(\begin{array} {cc}{A} & {C}\ {0} & {B} \end{array} \right) $ denote a $ 2\times2 $ upper-triangular block operator matrix on $ X\oplus Y $. Employing the spatial decomposition technique and stability of the closed range under small perturbations, this paper investigated the $ \varepsilon $-injectivity and $ \varepsilon $-density of upper-triangular block operator matrices under upper-triangular perturbations. Based on this, we studied meticulous characterization of the pseudo-residual spectrum and pseudo-continuous spectrum of upper-triangular block operator matrices under $ 2 \times 2 $ upper-triangular perturbations.
Citation: Runshuan Shen, Guolin Hou. A meticulous characterization of the pseudo-residual spectrum and pseudo-continuous spectrum of upper-triangular block operator matrices[J]. AIMS Mathematics, 2026, 11(7): 20461-20472. doi: 10.3934/math.2026832
For $ A\in\mathcal{B}(X), \; B\in\mathcal{B}(Y), \; C\in\mathcal{B}(Y, X), $ let $ M = \left(\begin{array} {cc}{A} & {C}\ {0} & {B} \end{array} \right) $ denote a $ 2\times2 $ upper-triangular block operator matrix on $ X\oplus Y $. Employing the spatial decomposition technique and stability of the closed range under small perturbations, this paper investigated the $ \varepsilon $-injectivity and $ \varepsilon $-density of upper-triangular block operator matrices under upper-triangular perturbations. Based on this, we studied meticulous characterization of the pseudo-residual spectrum and pseudo-continuous spectrum of upper-triangular block operator matrices under $ 2 \times 2 $ upper-triangular perturbations.
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Runshuan Shen, Guolin Hou. A meticulous characterization of the pseudo-residual spectrum and pseudo-continuous spectrum of upper-triangular block operator matrices[J]. AIMS Mathematics, 2026, 11(7): 20461-20472. doi: 10.3934/math.2026832
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