Characterization of smooth symbol classes by Gabor matrix decay

For $m\in\mathbb{R}$ we introduce the symbol classes $S^m$, $m\in\mathbb{R}$, consisting of smooth functions $\sigma$ on $\mathbb{R}^{2d}$ such that $|\partial^\alpha \sigma(z)|\leq C_\alpha (1+|z|^2)^{m/2}$, $z\in\mathbb{R}^{2d}$, and we show that can be characterized by an intersection of different types of modulation spaces. In the case $m=0$ we recapture the H\"{o}rmander class $S^0_{0,0}$ that can be obtained by intersection of suitable Besov spaces as well. Such spaces contain the Shubin classes $\Gamma^m_\rho$, $0<\rho\leq1$, and can be viewed as their limit case $\rho=0$. We exhibit almost diagonalization properties for the Gabor matrix of $\tau$-pseudodifferential operators with symbols in such classes, extending the characterization proved by Gr\"{o}chenig and Rzeszotnik. Finally, we compute the Gabor matrix of a Born-Jordan operator, which allows to prove new boundedness results for such operators.