Powers of Catalan generating functions for bounded operators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916110067040256 |
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| author | Miana, Pedro J. Romero, Natalia |
| author_facet | Miana, Pedro J. Romero, Natalia |
| contents | Let $c=(C_n)_{n\ge 0}$ be the Catalan sequence and $T$ a linear and bounded operator on a Banach space $X$ such $4T$ is a power-bounded operator. The Catalan generating function is defined by the following Taylor series, $$ C(T):=\sum_{n=0}^\infty C_nT^n. $$ Note that the operator $C(T)$ is a solution of the quadratic equation $TY^2-Y+I=0.$ In this paper we define powers of the Catalan generating function $C(T)$ in terms of the Catalan triangle numbers. We obtain new formulae which involve Catalan triangle numbers; the spectrum of $c^{\ast j}$ and the expression of $c^{-\ast j}$ for $j\ge 1$ in terms of Catalan polynomials ($\ast$ is the usual convolution product in sequences). In the last section, we give some particular examples to illustrate our results and some ideas to continue this research in the future. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_16493 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Powers of Catalan generating functions for bounded operators Miana, Pedro J. Romero, Natalia Functional Analysis Operator Algebras Catalan triangle numbers, generating function, powers of bounded operators, quadratic equation Let $c=(C_n)_{n\ge 0}$ be the Catalan sequence and $T$ a linear and bounded operator on a Banach space $X$ such $4T$ is a power-bounded operator. The Catalan generating function is defined by the following Taylor series, $$ C(T):=\sum_{n=0}^\infty C_nT^n. $$ Note that the operator $C(T)$ is a solution of the quadratic equation $TY^2-Y+I=0.$ In this paper we define powers of the Catalan generating function $C(T)$ in terms of the Catalan triangle numbers. We obtain new formulae which involve Catalan triangle numbers; the spectrum of $c^{\ast j}$ and the expression of $c^{-\ast j}$ for $j\ge 1$ in terms of Catalan polynomials ($\ast$ is the usual convolution product in sequences). In the last section, we give some particular examples to illustrate our results and some ideas to continue this research in the future. |
| title | Powers of Catalan generating functions for bounded operators |
| topic | Functional Analysis Operator Algebras Catalan triangle numbers, generating function, powers of bounded operators, quadratic equation |
| url | https://arxiv.org/abs/2401.16493 |