hi everyone, I have some questions, I hope someone can help..

Let {a_n, n from 0 to infinity} denotes a sequence of complex numbers that is summable (i.e. sum of |a_n| for all n, is finite)

and f is the continuous function defined on the unit circle by

f(z)= SUM (from minus infinity to plus infinity) {a_n.z^n} where z=e^it where 0<= t <= 2pi

1) Consider hilbert space H = l^2 (Z). how can we show that the convolution operator A defined by,

(A g)_n = SUM (k from minus infinity to plus infinity) {a_(n-k).g_k}

= SUM (j from minus infinity to plus infinity) {a_j.g_(n-j)}

satisfies ||A|| <= (sum of |a_n| for n from zero to infinity)

2) How can we show A is normal?

3) what is the spectrum of A in concrete terms

4) Assuming {a_n} is not trivial sequence, how can we deduce that A has no eigenvalues (i.e. no point spectrum).. when the operator A is invertible in terms of funciton f, and what is ||A|| exactly?