For the discrete Laguerre operators we compute explicitly the corresponding heat kernels by expressing them with the help of Jacobi polynomials. This enables us to show that the heat semigroup is ultracontractive and to compute the corresponding norms. On the one hand, this helps us to answer basic questions (recurrence, stochastic completeness) regarding the associated Markovian semigroup. On the other hand, we prove the analogs of the Cwiekel–Lieb–Rosenblum and the Bargmann estimates for perturbations of the Laguerre operators, as well as the optimal Hardy inequality.

中文翻译：

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离散Laguerre算子的热核

对于离散的Laguerre算子，我们借助于Jacobi多项式将它们表示出来，从而显式地计算出相应的热核。这使我们能够证明热半群是超收缩的，并能够计算出相应的范数。一方面，这有助于我们回答有关相关马尔可夫半群的基本问题（递归，随机完整性）。另一方面，我们证明了Cwiekel–Lieb–Rosenblum的类似物以及Bargmann估计对Laguerre算子的扰动以及最优的Hardy不等式的估计。