IGNOU MMT 6 2026 - Give one example of each of the following. Also justify your choice of

Question

Give one example of each of the following. Also justify your choice of example.
i) A self-adjoint operator on $\ell^2$ .
ii) A normal operator on a Hilbert space which is not unitary.

09 Jan 2026

Answer :

Word Count : 230

i) Consider the operator (T: \ell^2 \to \ell^2) defined by (T(x_1, x_2, x_3, \dots) = (x_1, 2x_2, 3x_3, \dots)). This operator is diagonal with real entries (1, 2, 3, \dots) on the diagonal. For any (x = (x_n)) and (y = (y_n)) in (\ell^2), we have [ \langle Tx, y \rangle = \sum_{n=1}^\infty n x_n \overline{y_n} = \sum_{n=1}^\infty x_n \overline{n y_n} = \langle x, Ty \rangle, ]

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