$$\boldsymbol{(P,Q)}$$–$${\varepsilon}$$-Pseudo Condition Spectrum for $$\mathbf{2\times 2}$$ Matrices. Linear Operator and Application
Józef Banaś, A. B. Ali, Kamel Mahfoudhi, Bilel Saadaoui · Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences) · 2023
We define a new type of spectrum, called the $$(P,Q)$$ – $$\varepsilon$$ -pseudo condition spectra $$\Sigma_{(P,Q)-\varepsilon}^{(2)}(T)=\sigma_{(P,Q)}^{(2)}(T)\bigcup\left\{\lambda\in\mathbb{C}:||(\lambda-T)_{(P,Q)}^{(2)}||\ ||\lambda-T||>\displaystyle{\frac{1}{\varepsilon}}\right\}.$$ This $$(P,Q)$$ – $$\varepsilon$$ -pseudo condition spectrum shares some properties of the usual spectrum such as nonemptiness. Our aim in this paper is to show some properties of $$(P,Q)$$ – $$\varepsilon$$ -pseudo condition spectra of a linear operator $$T$$ in Banach spaces and reveal the relation between their $$(P,Q)$$ – $$\varepsilon$$ -pseudo condition spectra. Additionally, we investigate the $$(P,Q)$$ – $$\varepsilon$$ -pseudo condition spectrum of a block matrix in a Banach space.