# Thread: [SOLVED] convolution operator, functional analysis, linear algebra

1. ## [SOLVED] convolution operator, functional analysis, linear algebra

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?

2. The Fourier transform converts convolution to pointwise multiplication. By Parseval's theorem, it also transforms the sequence space $\displaystyle \ell^2(\mathbb{Z})$ to the space $\displaystyle L^2(\mathbb{T})$ of square-integrable functions on the unit circle. Under this isometric isomorphism, the convolution operator A gets mapped to the operator $\displaystyle M_f$ of multiplication by f. The adjoint of $\displaystyle M_f$ is the multiplication operator $\displaystyle M_{\bar{f}}$ corresponding to the complex conjugate of f. Since multiplication operators commute with each other, it is clear that $\displaystyle M_f$ commutes with its adjoint, hence so does A (so A is normal).

The spectrum of a multiplication operator is the range of the function, so $\displaystyle \text{sp}(A) = \text{sp}(M_f) = \{f(z):z\in\mathbb{T}\}$.

The norm of a multiplication operator is the sup norm of the function, so $\displaystyle \|A\| = \|M_f\| = \sup\{|f(z)|:z\in\mathbb{T}\}$.

3. thanks for the reply Opalg..
I need to clear a few points in my mind, may be someone can help..
for the fourth question , thanks to Opalg, we know what is the norm of A, but for the remaining part what should we do?

there was a hint saying that using third question, we assume there is a function f in L^2(unit circle) such that Af=\lambda f, for some complex number \lambda, and then use some measure theory plus a little inspiration to show that this leads to a contradiction (because lebesgue measure on the unit circle has no atoms)

can someone explain the details in this hint??
thanks