Spectral invariance of quasi-Banach algebras of matrices and pseudodifferential operators (Q6612585)

Initial reference for the authors is the work of \textit{R. Beals} [Duke Math. J. 44, 45--57 (1977; Zbl 0353.35088)] proving the following for \(P\) in different classes of Hormander's pseudo-differential operators: if \(P\) is invertible as bounded linear map on the space of square-integrable Lebesgue functions, then the inverse is again a pseudo-differential operator, belonging to the same class of \(P\). This phenomenon was studied in more general settings, under the name of spectral invariance, or Wiener lemma. Recently pseudo-differential operators \(P\) have been reconsidered in the frame of the Time-Frequency Analysis with symbol in suitable non-smooth classes and boundedness on \(p\)-modulation spaces, see [\textit{K. Gröchenig}, Foundations of time-frequency analysis. Boston, MA: Birkhäuser (2001; Zbl 0966.42020)]. In the present paper attention is fixed on the quasi-Banach \(p\)-modulation spaces, with index \(p\) in the interval \((0,1)\). \N\NThe main result can be expressed as follows: if \(P\) is invertible as bounded linear operator on a \(p\)-modulation space for some p, then the inverse is a pseudo-differential operator in the same class of \(P\). Note that the case of the Lebesgue space \(p=2\) was already considered in [\textit{E. Cordero} and \textit{G. Giacchi}, J. Pseudo-Differ. Oper. Appl. 14, No. 1, Paper No. 9, 26 p. (2023; Zbl 1504.35664)]. The case of arbitrary \(p\) is more involved since Hilbert space techniques are lost and for quasi-Banach spaces even duality arguments are not available.