On the essential spectrum of the sum of self-adjoint operators and the closedness of the sum of operator ranges

Abstract

Let $\mathcal{H}$ be a complex Hilbert space, and $A_1,\ldots,A_N$ be bounded self-adjoint operators in $\mathcal{H}$ such that $A_i A_j$ is compact for any $i\neq j$. It is well-known that $\sigma_e(\sum_{i=1}^N A_i)\setminus\{0\}=(\cup_{i=1}^N\sigma_e(A_i))\setminus\{0\}$, where $\sigma_e(B)$ stands for the essential spectrum of a bounded self-adjoint operator $B$. In this paper we get necessary and sufficient conditions for $0\in\sigma_e(\sum_{i=1}^N A_i)$. This conditions are formulated in terms of the projection valued spectral measures of $A_i$, $i=1,\ldots,N$. Using this result, we obtain necessary and sufficient conditions for the sum of ranges of $A_i$, $i=1,\ldots,N$ to be closed.