• S H Kulkarni

      Articles written in Proceedings – Mathematical Sciences

    • Analytic and harmonic maps into a topological space

      S Arundhathi S H Kulkarni

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      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.

    • Some Properties of Unbounded Operators with Closed Range

      S H Kulkarni M T Nair G Ramesh

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      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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