The operator M is a weighted shift (check that link for some useful guidance). Itshiftseach basis vector to the next one multiplying it by theweightSince its value at each basis vector is specified, it must be unique (you know its value at each finite linear combination of basis vectors and hence, by continuity, at every vector).

The adjoint operator is a backward weighted shift, taking each to a multiple of and sending to 0. The advantage of looking at the adjoint is that it has many eigenvalues, whereas M itself does not.

For convenience, write , and let . If then , and

That shows that . By looking at for each k, you should be able to show the reverse inequality.