Catalan generating functions for bounded operators
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909086361059328 |
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| author | Miana, Pedro J. Romero, Natalia |
| author_facet | Miana, Pedro J. Romero, Natalia |
| contents | In this paper we study the solution of the quadratic equation $TY^2-Y+I=0$ where $T$ is a linear and bounded operator on a Banach space $X$. We describe the spectrum set and the resolvent operator of $Y$ in terms of operator $T$. In the case that $ 4T$ is a power-bounded operator, we show that a solution (named Catalan generating function) is given by the Taylor series $$ C(T):=\sum_{n=0}^\infty C_nT^n, $$ where the sequence $(C_n)_{n\ge 0}$ is the well-known Catalan numbers. We express $C(T)$ by means of an integral representation which involves the resolvent operator $(λ-T)^{-1}$. Some particular examples to illustrate our results are given, in particular an iterative method defined for square matrices $T$ which involves Catalan numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16415 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Catalan generating functions for bounded operators Miana, Pedro J. Romero, Natalia Functional Analysis Catalan numbers, generating function, power bounded operators, quadratic equation, iterative methods In this paper we study the solution of the quadratic equation $TY^2-Y+I=0$ where $T$ is a linear and bounded operator on a Banach space $X$. We describe the spectrum set and the resolvent operator of $Y$ in terms of operator $T$. In the case that $ 4T$ is a power-bounded operator, we show that a solution (named Catalan generating function) is given by the Taylor series $$ C(T):=\sum_{n=0}^\infty C_nT^n, $$ where the sequence $(C_n)_{n\ge 0}$ is the well-known Catalan numbers. We express $C(T)$ by means of an integral representation which involves the resolvent operator $(λ-T)^{-1}$. Some particular examples to illustrate our results are given, in particular an iterative method defined for square matrices $T$ which involves Catalan numbers. |
| title | Catalan generating functions for bounded operators |
| topic | Functional Analysis Catalan numbers, generating function, power bounded operators, quadratic equation, iterative methods |
| url | https://arxiv.org/abs/2401.16415 |