Let $D\in \mathbb{R}^{n\times n}$ denote a lower triangular matrix. With $\|\cdot\|$ denoting the spectral matrix norm, is there an estimate like
$$ \|D\| \leq C\|D+D^T\|, $$
where $C>0$ is independent of the dimension $n\in\mathbb{N}$ and $D$?
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Sign up to join this communityLet $D\in \mathbb{R}^{n\times n}$ denote a lower triangular matrix. With $\|\cdot\|$ denoting the spectral matrix norm, is there an estimate like
$$ \|D\| \leq C\|D+D^T\|, $$
where $C>0$ is independent of the dimension $n\in\mathbb{N}$ and $D$?
No. The best constant is $C_n\sim \ln n$. See, for example, this paper. In particular, for the lower bound, see example 3.3 of the paper.
The $\log n$ result mentioned by Christian Remling is a special case of the results in
Kwapień, S.; Pełczyński, A. The main triangle projection in matrix spaces and its applications. Studia Math. 34 1970 43–68.
The assumption that $D$ is triangular is not necessary (the Schur decomposition may bring a matrix to its triangular form). If the numerical range of D is contained in a sector, then Lemma 3.1 in http://www.tandfonline.com/doi/abs/10.1080/03081087.2014.933219#preview says that there is a C in terms of secant function.