Recently, I needed to estimate the operator norm of the *tridiagonal operator,* but I am sure answers much more refined than my simple observations must be known.


Let $T$ be the linear operator that maps a square matrix to its tridiagonal part. Thus, the action of $T$ on a matrix $X$ can be defined by the Hadamard product $M \circ X$, where $m\_{ij}=1$ if $|i-j| \le 1$, and $m\_{ij}=0$ otherwise. 



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> What is the operator 2-norm of $T$? (or what is a good approximation thereof?)
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The observation $\|M \circ X\| \le \|M\| \|X\|$ shows that $\|T\| \le 3$. A more refined estimate follows from Theorem 5.5.3 of Horn and Johnson's *Topics in Matrix Analysis*, which says that $\|M\circ X\| \le r\_1(M)c\_1(X)$, where $r\_1$ is maximum row-length (Euclidean norm) and $c\_1$ is max column length. This result then implies that $\|T\| \le \sqrt{3}$.


I am sure that significantly more refined estimates of $\|T\|$ are available, and will be thankful if you can provide me a reference, or maybe a short proof itself.



Norm of tridiagonal operator
A good answer is given by R. Bhatia: *Pinching, trimming, truncating, and averaging of matrices*. Amer. Math. Monthly **107** (2000), no. 7, 602–608.


If you consider the operator $T\_r$ that retains the diagonals defined by $|i-j|\le r$ (yours is $T\_1$), its norm is accurately bounded by 
$$L\_r=\frac{1}{2\pi}\int\_{-\pi}^\pi |D\_r(\theta)|d\theta,$$
where $D\_r$ is the Dirichlet kernel.


