• M T Nair

Articles written in Proceedings – Mathematical Sciences

• Some Properties of Unbounded Operators with Closed Range

Let $H_1, H_2$ be Hilbert spaces and 𝑇 be a closed linear operator defined on a dense subspace $D(T)$ in $H_1$ and taking values in $H_2$. In this article we prove the following results:

(i) Range of 𝑇 is closed if and only if 0 is not an accumulation point of the spectrum $\sigma(T^\ast T)$ of $T^\ast T$,

In addition, if $H_1=H_2$ and 𝑇 is self-adjoint, then

(ii) $\inf \{\| Tx\|:x\in D(T)\cap N(T)^\perp \| x\|=1\}=\inf\{| \lambda|:0\neq\lambda\in\sigma(T)\}$,

(iii) Every isolated spectral value of 𝑇 is an eigenvalue of 𝑇,

(iv) Range of 𝑇 is closed if and only if 0 is not an accumulation point of the spectrum $\sigma(T)$ of 𝑇,

(v) $\sigma(T)$ bounded implies 𝑇 is bounded.

We prove all the above results without using the spectral theorem. Also, we give examples to illustrate all the above results.

• # Proceedings – Mathematical Sciences

Volume 131, 2021
All articles
Continuous Article Publishing mode

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019