  We use methods of harmonic analysis and group rep- resentation theory to
estimate memory decay of the inverse oper- ators in Banach spaces. The memory
of the operators is defined using the notion of the Beurling spectrum. We
obtain a general continuous non-commutative version of the celebrated Wiener's
Tauberian lemma with estimates of the "Fourier coefficients" of inverse
operators. In particular, we generalize various estimates of the elements of
the inverse matrices. The results are illus- trated with a variety of examples
including integral and integro- differential operators.
