Some properties of unbounded operators with closed range |
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| Authors: | S H Kulkarni M T Nair G Ramesh |
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| Institution: | (1) Department of Mathematics Indian Institute of Technology Madras, Chennai, 600 036, India |
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| 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 σ(T*T) of T*T, In addition, if H
1 = H
2 and T is self-adjoint, then
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| (ii) |
inf {‖T x‖: x ∈ D(T) ∩ N(T)⊥‖x‖ = 1} = inf {|λ|: 0 ≠ λ ∈ σ(T)}
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| (iii) |
Every isolated spectral value of T is an eigenvalue of T
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| (iv) |
Range of T is closed if and only if 0 is not an accumulation point of the spectrum σ(T) of T
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| (v) |
σ(T) bounded implies T is bounded.
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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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| Keywords: | Densely defined operator closed operator Moore-Penrose inverse reduced minimum modulus |
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