Let $A$ be a Toeplitz matrix , associated to a bounded operator from $\ell^2$ to itself. We consider its associated banded matrices, that is, those matrices that have a band of diagonals equal to the corresponding diagonals of $A$, and the rest of the diagonals are zero.
The question is: Is there any relation between the norm of $A$ and the norm of those banded matrices? Is $\|A\|$ (operator norm) always bigger than the norm of any of its banded matrices? Furthermore, can $\|A\|$ be calculated as the supremum of the norms of those banded matrices?
Take $A = \left[\begin{array}{ccc}1 & 1 & -1\\ 1 & 1 & 1 \\ -1 & 1 & 1\end{array}\right]$ and $B = \left[\begin{array}{ccc}1 & 1 & 0\\ 1 & 1 & 1 \\ 0 & 1 & 1\end{array}\right]$. We have $\|A\|_2 = 2$ and $\|B\|_2 = 1 + \sqrt{2} > 2$. Thus, norm of a matrix can be smaller than norm of its banded matrix even for symmetric Toeplitz matrices. I suppose, that the same holds in infinite dimensional case.