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\(\boldsymbol{(P,Q)}\)–\({\varepsilon}\)-Pseudo Condition Spectrum for \(\mathbf{2\times 2}\) Matrices. Linear Operator and Application

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Abstract

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.

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Authors and Affiliations

  1. Rzeszáw University of Technology, Rzeszów, Poland

    J. Banaś, A. B. Ali, K. Mahfoudhi & B. Saadaoui

  2. University of Sfax, Faculty of Sciences of Sfax, Tunis, Tunisia

    J. Banaś, A. B. Ali, K. Mahfoudhi & B. Saadaoui

  3. University of Sousse, Tunis, Tunisia

    J. Banaś, A. B. Ali, K. Mahfoudhi & B. Saadaoui

Authors
  1. J. Banaś
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  2. A. B. Ali
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  3. K. Mahfoudhi
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  4. B. Saadaoui
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Corresponding authors

Correspondence to J. Banaś, A. B. Ali, K. Mahfoudhi or B. Saadaoui.

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Banaś, J., Ali, A.B., Mahfoudhi, K. et al. \(\boldsymbol{(P,Q)}\)–\({\varepsilon}\)-Pseudo Condition Spectrum for \(\mathbf{2\times 2}\) Matrices. Linear Operator and Application. J. Contemp. Mathemat. Anal. 58, 217–225 (2023). https://doi.org/10.3103/S1068362323040027

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  • DOI: https://doi.org/10.3103/S1068362323040027

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