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Let $mathcal{H}_0=V, mathcal{H}_1=B+V$ and $mathcal{H}_2=mathcal{L}+V$ be the operators on the Heisenberg group $mathbb{H}^n$, where $V$ is the operator of multiplication growing like $|g|^kappa, 0<kappa<1$, $B$ is a bounded linear operator and $mathcal{L}$ is the sublaplacian on $mathbb{H}^n$. In this paper we prove Szego limit theorem for the operators $mathcal{H}_0, mathcal{H}_1$ and $mathcal{H}_2$ on $L^2(mathbb{H}^n).$
We prove an analogue of Chernoffs theorem for the sublaplacian on the Heisenberg group and use it prove a version of Inghams theorem for the Fourier transform on the same group.
We characterize even measures $mu=wdx+mu_s$ on the real line with finite entropy integral $int_{R} frac{log w(t)}{1+t^2}dt>-infty$ in terms of $2times 2$ Hamiltonian generated by $mu$ in the sense of inverse spectral theory. As a corollary, we obtain
Although convolution on Euclidean space and the Heisenberg group satisfy the same $L^p$ bounds with the same optimal constants, the former has maximizers while the latter does not. However, as work of Christ has shown, it is still possible to charact
We establish the necessary and sufficient conditions for those symbols $b$ on the Heisenberg group $mathbb H^{n}$ for which the commutator with the Riesz transform is of Schatten class. Our main result generalises classical results of Peller, Janson-
Versions of well known function theoretic operator theory results of Szego and Widom are established for the Neil algebra. The Neil algebra is the subalgebra of the algebra of bounded analytic functions on the unit disc consisting of those functions whose derivative vanishes at the origin.