Toeplitz operators associated with the hypergeometric Gabor transform and applications (Q6577066)

Fix a root system, let \(k\) be the multiplicity function and let \(A_k(x)\) be the corresponding weight function. Also let \(F_{\lambda}(x)\) be the Opdam hypergeometric function associated with the root system. Define the hypergeometric transform as \N\[ \N\mathcal{H}_k(f)=\int_{\mathbb{R}^d}f(x)F_{\lambda}(x) A_k(x) \,dx, \N\] \Nand the hypergeometric convolution as\N\[ \Nf\star g(\xi)=\int_{\mathbb{R}^d} \mathcal{H}_k(f)^{-1}(x)\mathcal{H}^{-1}_{g}F_{\xi}(x) A_k(x) \,dx. \N\] \NThe hypergeometric Gabor transform is a result of the hypergeometric convolution with a specific \(W\)-invariant function.\N\NThe first aim of the paper is to expose and study the boundedness and compactness of Toeplitz operators associated with the hypergeometric Gabor transform. Next, spectral theorems associated with the concentration operator associated with the hypergeometric Gabor transform are proved.