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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.17124 |
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| _version_ | 1866913486516256768 |
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| author | Battistoni, Francesco Molteni, Giuseppe |
| author_facet | Battistoni, Francesco Molteni, Giuseppe |
| contents | For every $α\in (0,+\infty)$ and $p,q \in (1,+\infty)$ let $T_α$ be the operator $L^p[0,1]\to L^q[0,1]$ defined via the equality $(T_αf)(x) := \int_0^{x^α} f(y) d y$. We study the norms of $T_α$ for every $p$, $q$. In the case $p=q$ we further study its spectrum, point spectrum, eigenfunctions, and the norms of its iterates. Moreover, for the case $p=q=2$ we determine the point spectrum and eigenfunctions for $T^*_αT_α$, where $T^*_α$ is the adjoint operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_17124 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A one parameter family of Volterra-type operators Battistoni, Francesco Molteni, Giuseppe Functional Analysis Classical Analysis and ODEs For every $α\in (0,+\infty)$ and $p,q \in (1,+\infty)$ let $T_α$ be the operator $L^p[0,1]\to L^q[0,1]$ defined via the equality $(T_αf)(x) := \int_0^{x^α} f(y) d y$. We study the norms of $T_α$ for every $p$, $q$. In the case $p=q$ we further study its spectrum, point spectrum, eigenfunctions, and the norms of its iterates. Moreover, for the case $p=q=2$ we determine the point spectrum and eigenfunctions for $T^*_αT_α$, where $T^*_α$ is the adjoint operator. |
| title | A one parameter family of Volterra-type operators |
| topic | Functional Analysis Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2408.17124 |