S H Kulkarni
Articles written in Proceedings – Mathematical Sciences
Volume 95 Issue 1 September 1986 pp 37-40
Analytic and harmonic maps into a topological space
The relationship between the harmonicity and analyticity of a continuous map from the open unit disc to the underlying space of a real algebra is investigated.
Volume 118 Issue 4 November 2008 pp 613-625
Some Properties of Unbounded Operators with Closed Range
S H Kulkarni M T Nair G Ramesh
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.
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November 2019
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