Almost diagonalization of $\tau$-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces

In this paper we focus on the almost-diagonalization properties of $\tau$-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. A particular example is provided by the Sj\"ostrand class, for which Gr\"ochenig exhibited the almost diagonalization of Weyl operators. We shall show that such result can be extended to any $\tau$-pseudodifferential operator, for $\tau \in [0,1]$, also with symbol in weighted Wiener amalgam spaces. As a consequence, we infer boundedness, algebra and Wiener properties for $\tau$-pseudodifferential operators on Wiener amalgam and modulation spaces.