A Spectral Theorem for Hermitian Operators of Meromorphic Type on Banach Spaces

Abstract

It is well known that if T is a compact self-adjoint operator on a Hilbert space whose distinct non-zero eigenvalues {λn} are arranged so that |λn|≥|λn+1| for n = 1, 2…. and if En in the spectral projection corresponding to λn, then with convergence in the uniform operator topology. With the generalisation of self-adjoint operators on Hilbert spaces to Hermitian operators on Banach spaces by Vidav and Lumer, Bonsall gave a partial analogue of this result for Banach spaces when he proved the following theorem.