Localization operators on discrete Orlicz modulation spaces

In this paper, we introduce Orlicz spaces on ${\mathbb{Z}}^n\times {\mathbb{T}}^n$ and Orlicz modulation spaces on ${\mathbb{Z}}^n$, and present some basic properties such as inclusion relations, convolution relations, and duality of these spaces. We show that the Orlicz modulation space $M^{\mathrm{\Phi }}({\mathbb{Z}}^n)$ is close to the modulation space $M^2({\mathbb{Z}}^n)$ for some particular Young function $\mathrm{\Phi }$. Then, we study a class of pseudo-differential operators known as time-frequency localization operators on ${\mathbb{Z}}^n$, which depend on a symbol $\varsigma$ and two windows functions $g_1$ and $g_2$. Using appropriate classes for symbols, we study the boundedness of the localization operators on Orlicz modulation spaces on ${\mathbb{Z}}^n$. Also, we show that these operators are compact and in the Schatten–von Neumann classes.