Motivated by Federico's recent question: Norm of upper triangular matrix of all ones I decided to ask a question about the norm of a matrix that is, in a sense, dual to the all ones upper triangular matrix.

Let $C_n$ be the $n \times n$ matrix (transpose of the Cesàro matrix)

$$[C_n]_{ij} = \begin{cases} 1/j & i \le j\\ 0 & i > j \end{cases}$$

What is $\|C_n\|_2$?

Notes:

1. From results on Cesàro operators, it seems to follow that $\|C_n\| \le 2$ for all $n \ge 1$.
2. It is also worth noting that $\|C_n\|_1=1$ and $\|C_n\|_{\infty} = H_n$, where $H_n$ denotes the $n$-th Harmonic number.
3. Computing $\frac{e^TC_n^TC_ne}{e^Te}=2-H_n/n$ (e is the all ones vector) we obtain a simple lower-bound on $\|C_n\|_2$.
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# Norm of Cesaro matrix

Motivated by Federico's recent question: Norm of upper triangular matrix of all ones I decided to ask a question about the norm of a matrix that is, in a sense, dual to the all ones upper triangular matrix.

Let $C_n$ be the $n \times n$ matrix (transpose of the Cesàro matrix)

$$[C_n]_{ij} = \begin{cases} 1/j & i \le j\\ 0 & i > j \end{cases}$$

What is $\|C_n\|_2$?

Notes:

1. From results on Cesàro operators, it seems to follow that $\|C_n\| \le 2$ for all $n \ge 1$.
2. It is also worth noting that $\|C_n\|_1=1$ and $\|C_n\|_{\infty} = H_n$, where $H_n$ denotes the $n$-th Harmonic number.