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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/118/04/0613-0625

    • Keywords

       

      Densely defined operator; closed operator; Moore–Penrose inverse; reduced minimum modulus.

    • Abstract

       

      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.

    • Author Affiliations

       

      S H Kulkarni1 M T Nair1 G Ramesh1

      1. Department of Mathematics Indian Institute of Technology Madras, Chennai 600 036, India
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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