## Banach Journal of Mathematical Analysis

- Banach J. Math. Anal.
- Volume 8, Number 1 (2014), 55-63.

### 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.

#### Article information

**Source**

Banach J. Math. Anal., Volume 8, Number 1 (2014), 55-63.

**Dates**

First available in Project Euclid: 14 October 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.bjma/1381782087

**Digital Object Identifier**

doi:10.15352/bjma/1381782087

**Mathematical Reviews number (MathSciNet)**

MR3161682

**Zentralblatt MATH identifier**

1310.47003

**Subjects**

Primary: 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.)

Secondary: 46C07: Hilbert subspaces (= operator ranges); complementation (Aronszajn, de Branges, etc.) [See also 46B70, 46M35]

**Keywords**

self-adjoint operator compact operator essential spectrum sum of operator ranges closedness

#### Citation

Feshchenko, Ivan S. On the essential spectrum of the sum of self-adjoint operators and the closedness of the sum of operator ranges. Banach J. Math. Anal. 8 (2014), no. 1, 55--63. doi:10.15352/bjma/1381782087. https://projecteuclid.org/euclid.bjma/1381782087