Let H be a separable Hilbert space, and (e,) C Ha total orthonormal sequence in H....

  

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Let H be a separable Hilbert space, and (e,) C Ha total orthonormal sequence in H. Let T: H H be the right shift operator, defined as Tx = (x, en-1)en for every x = >(x, en)en- n=2 n=1 Note that, in particular, we have Te, = en+l, for every n e N. (a) Show that T is linear. (b) Find the range R(T) of T. (c) Find the null space N(T) of T. (d) Show that T is bounded and find ||7||. (e) Find the adjoint T" of T. Let H be a separable Hilbert space, and (e,) C Ha total orthonormal sequence in H. Let T: H H be the right shift operator, defined as Tx = (x, en-1)en for every x = >(x, en)en- n=2 n=1 Note that, in particular, we have Te, = en+l, for every n e N. (a) Show that T is linear. (b) Find the range R(T) of T. (c) Find the null space N(T) of T. (d) Show that T is bounded and find ||7||. (e) Find the adjoint T" of T.

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