We prove that if T : X → Y is a bounded linear operator between Hubert spaces X and Y, then the range of T is closed in Y if and only if 0 is not an accumulation point of the spectrum of T*T. We supply three proofs of this result as applications of different concepts in basic operator theory.

S H Kulkarni

Department of Mathematics

M.T. Nair

Indian Institute of Technology Palakkad&nbsp;(IIT Palakkad or IIT PKD) is a public autonomous&nbsp;engineering&nbsp;and&nbsp;research&nbsp;institute located in&nbsp;Palakkad,&nbsp;Kerala. It was&nbsp;one of the five new&nbsp;IITs&nbsp;proposed in the&nbsp;2014 Union budget of India. The campus was inaugurated on 3 August 2015. The institute has 94 faculty, 978 students, and 63 non-teaching staffs.

IT Palakkad currently offers four-year Bachelor of Technology (B.Tech) and two years MS programmes in 8 different engineering disciplines including&nbsp;Computer Science, Electrical, Mechanical, Civil among others.

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Journal	Indian Journal of Pure and Applied Mathematics
ISSN	00195588