How do I construct a sequence of bounded linear transformations that converge strongly to the zero operator, but do not converge to the zero operator in the operator norm? Something strange must happen for certain elements of the Hilbert space, but what sort of thing should I be looking for? An example would be most helpful, thanks in advance!
Strong Convergence Does Not Imply Operator Norm Convergence – Functional Analysis
functional-analysis
Best Answer
Hint: Consider iterated powers of the unilateral left shift operator $S\colon \ell^2(\mathbb N)\to \ell^2(\mathbb N)$ given by $$(Sa)(n)=a(n+1).$$ Can you compute $\lVert S^ka\rVert$ in terms of the entries of $a$? On the other hand, what is the operator norm of $S^k$?