Matrices Representing Bounded Operators and Absolute Values Let $A=(a_{ij})_{i,j=1}^{\infty}$ be an infinite matrix of complex numbers. For every positive integer $n$, we shall denote with $A_n$ the $n \times n$ matrix $A_n=(a_{i,j})_{i,j=1}^{n}$, and if $x \in \mathbb{C}^{n}$, we shall write $||x|| = \sqrt{\sum_{i=1}^{n} |x_i|^2}$. In the following, each $x \in \mathbb{C}^{n}$ will be considered as usual a column vector, and the transpose of a matrix $B$ will be denoted by $B^{\top}$. Finally, we shall denote with $|A|$ the infinite matrix $|A|=(|a_{i,j}|)_{i,j=1}^{\infty}$. 
Assume $A$ is hermitian, that is $a_{i,j}=\bar{a}_{j,i}$ for all $i,j=1,2,\dots$.
We shall say that $A$ satisfies the (BO) condition if there exists $M > 0$ such that for each positive integer $n$, by denoting with $x$ a vector in $\mathbb{C}^{n}$, we have 
\begin{equation}
\sup_{||x|| \leq 1} \left| x^{\top} A_n \bar{x} \right| \leq M.
\end{equation}
My questions are the following ones: can we find a hermitian infinite matrix $A$ such that $A$ satisfies the (BO) condition, but $|A|$ does not? Can we choose $A$ to be real symmetric?
Thank you very much in advance for your help.
P.S. I will explain the relevance of this question in the following two remarks.
Remark 1. First of all, let us note that for each fixed positive integer $n$, $A_n$ is an $n \times n$ hermitian matrix. So by the Spectral Theorem, there exist a unitary $n \times n$ matrix $U$ such that $\bar{U}^{\top} A_n U = \Lambda$ is a diagonal matrix, having as diagonal elements the eigenvalues $\lambda_1,\dots, \lambda_n$ of $A_n$. Let $L = \max \{ |\lambda_1|,\dots,|\lambda_n| \}$. If we put $ \xi = U^{\top} x$, we get
\begin{equation}
\sup_{||x|| \leq 1} \left| x^{\top} A_n \bar{x} \right| = \sup_{||\xi|| \leq 1} \left| \xi^{\top} \Lambda \bar{\xi} \right| = L.
\end{equation}
Let us also note that if $y \in \mathbb{C}^{n}$ and we put $\upsilon = U^{\top} y$, we have
\begin{equation}
\sup_{||x|| \leq 1, ||y|| \leq 1} \left| x^{\top} A_n \bar{y} \right| = \sup_{||\xi|| \leq 1, ||\upsilon|| \leq 1} \left| \xi^{\top} \Lambda \bar{\upsilon} \right| = L.
\end{equation}
From these observations, we derive two conditions equivalent to the (BO) condition. First, $A$ satisfies the (BO) condition if and only if there exists $M > 0$ such that for each positive integer $n$, by denoting with $x, y$ vectors in $\mathbb{C}^{n}$,  we have 
\begin{equation}
\sup_{||x|| \leq 1, ||y|| \leq 1} \left| x^{\top} A_n \bar{y} \right| \leq M.
\end{equation}
Second, $A$ satisfies the (BO) condition if and only if there exists $M > 0$ such that for each positive integer $n$ and each eigenvalue $\lambda$ of $A_n$, we have $|\lambda | \leq M$.
Remark 2. Let $C=(c_{i,j})_{i,j=1}^{\infty}$ be an infinite matrix of complex numbers (hermitian or not). Moreover let $\mathcal{H}$ be a separable Hilbert space and fix an orthonormal basis $(e_j)_{j=1}^{\infty}$ of $\mathcal{H}$. Then a classical result (see Akhiezer and Glazman, Theory of Linear Operators in Hilbert Space, Volume I, Section 26) asserts that $C$ represents a bounded operator defined on all $\mathcal{H}$ with respect to the basis $(e_j)_{j=1}^{\infty}$ if and only if there exists $M > 0$ such that for each positive integer $n$, by denoting with $x, y$ vectors in $\mathbb{C}^{n}$, we have 
\begin{equation}
\sup_{||x|| \leq 1, ||y|| \leq 1} \left| x^{\top} C_n \bar{y} \right| \leq M.
\end{equation}
Note that if $C$ is hermitian, in view of the previous remark, $C$ represents a bounded operator defined on all $\mathcal{H}$ with respect to the basis $(e_j)_{j=1}^{\infty}$ if and only if $C$ satisfies the (BO) condition (from this fact we derived the name "BO", which stands for Bounded Operator).
We have the following simple result.
Theorem
Let $C=(c_{i,j})_{i,j=1}^{\infty}$ be an infinite matrix of complex numbers. If $|C|$ represents a bounded operator defined on all $\mathcal{H}$ with respect to the basis $(e_j)_{j=1}^{\infty}$, then also $C$ does.
Proof. Let $M > 0$ be such that for each positive integer $n$, by denoting with $x, y$ vectors in $\mathbb{C}^{n}$, we have 
\begin{equation}
\sup_{||x|| \leq 1, ||y|| \leq 1} \left| x^{\top} |C|_n \bar{y} \right| \leq M.
\end{equation}
If $x=(x_1,\dots,x_n)$ and $y=(y_1,\dots,y_n)$, then we have 
\begin{equation}
\left| x^{\top} |C|_n \bar{y} \right| = \left| \sum_{i,j=1}^{n} |c_{i,j}| x_{i} \bar{y}_j \right| \leq 
\sum_{i,j=1}^{n} |c_{i,j}| |x_{i}| |\bar{y}_j|.
\end{equation}
On the other hand, if $x_1,\dots, x_n$ and $y_1,\dots,y_n$ are real and nonnegative (we write succinctly $x \geq 0$, and $y \geq 0$), we have
\begin{equation}
\left| \sum_{i,j=1}^{n} |c_{i,j}| x_{i} \bar{y}_j \right| = 
\sum_{i,j=1}^{n} |c_{i,j}| |x_{i}| |\bar{y}_j|.
\end{equation}
Since we  clearly have
\begin{equation}
\sup_{||x|| \leq 1, ||y|| \leq 1} \sum_{i,j=1}^{n} |c_{i,j}| |x_{i}| |\bar{y}_j| = \sup_{\substack{||x|| \leq 1, ||y|| \leq 1 \\ x \geq 0, y \geq 0}} \sum_{i,j=1}^{n} |c_{i,j}| |x_{i}| |\bar{y}_j|,
\end{equation}
we get
\begin{equation}
\sup_{||x|| \leq 1, ||y|| \leq 1} \sum_{i,j=1}^{n} |c_{i,j}| |x_{i}| |\bar{y}_j| = \sup_{||x|| \leq 1, ||y|| \leq 1} \left| x^{\top} |C |_n \bar{y} \right| \leq M.
\end{equation}
From this we conclude that
\begin{equation}
\sup_{||x|| \leq 1, ||y|| \leq 1} \left| x^{\top} C_n \bar{y} \right| \leq \sup_{||x|| \leq 1, ||y|| \leq 1} \sum_{i,j=1}^{n} |c_{i,j}| |x_{i}| |\bar{y}_j| \leq M.
\end{equation}
QED
I think that the converse of this result does not hold, even if we make the additional assumption that $C$ is hermitian or real symmetric. Anyway any effort I made to find a counterexample has failed up to now.
 A: As requested, I've moved my comments to an answer.
The question is equivalent to the following:

Suppose that a doubly-infinite matrix $A=(A_{ij})_{i,j\in {\bf Z}}$ represents a bounded hermitian operator on $\ell^2$. Does the matrix $(|A_{ij}|)$ also represent a bounded operator on $\ell^2$ ?

(The original question just considers matrices indexed by ${\bf N}\times {\bf N}$ but the version I've stated is equivalent.)
To see that the answer is negative, we consider doubly-infinite Toeplitz matrices: that is, there is a sequence $(a_n)\subset {\bf C}$ such that $A_{ij}=a_{i-j}$ for all $i,j\in {\bf Z}$. (To get a Hermitian operator we need $a_{-n}=\overline{a_n}$ for all $n\in\bf Z$.)
Classical operator theory tells us that $A$ is bounded if and only if $(a_n)$ is the Fourier series of some $f\in L^\infty({\bf T})$, which is necessarily real-valued if $A$ is Hermitian. It is known that we can find such $f$ with the following property: the Fourier series
$$ g(t) \sim \sum_{n\in{\bf Z}} \vert a_n \vert e^{int} $$
does not represent an essentially bounded function. Details should be in Katznelson's book, for instance, and no doubt also in Zygmund somewhere.
(An original version of this answer tried to give an explict example, but contained several stupid errors. See Trieu Le's comment to the main question for an explicit example.)
A: By following Yemon Choi's hint, I give here a complete answer to my question.
We denote by $\mathbb{Z}$ the set of all integer numbers.
Let $f:[-\pi, \pi) \rightarrow \mathbb{R}$ be defined by $f(x)=x$. The Fourier coefficients of $f$ are:
\begin{equation}
c_{0}=\frac{1}{2\pi} \int_{-\pi}^{\pi} x dx = 0,
\end{equation}
\begin{equation}
c_{n}= \frac{1}{2 \pi} \int_{-\pi}^{\pi} x e^{-inx} dx = - \frac{xe^{-inx}}{2 \pi i n} \bigg|_{-\pi}^{\pi} + \int_{-\pi}^{\pi} \frac{e^{-inx}}{2 \pi i n} dx = \frac{i (-1)^{|n|}}{n}, \qquad (n= \pm 1, \pm 2, \dots).
\end{equation}
So $(c_n)_{n=-\infty}^{\infty}$ are the Fourier coefficients of a real bounded function. We shall prove that $(|c_n|)_{n=-\infty}^{\infty}$ are not the Fourier coefficients of an essentially bounded function.  The partial sums of the Fourier series corresponding to the coefficients $(|c_n|)_{n=-\infty}^{\infty}$ are
\begin{equation}
s_N(x) = \sum_{n=-N}^{N} |c_n| e^{inx} = \sum_{n=1}^{N} \frac{e^{inx} }{n} + \sum_{n=1}^{N} \frac{e^{-inx} }{n} \quad (x \in [-\pi,\pi) \quad \textrm{and} \quad N=1,2,\dots).
\end{equation}
Now note that for every complex number $z$, with $|z| < 1$, we have the power expansion
\begin{equation}
\sum_{n=1}^{\infty} \frac{z^n}{n} = - \sum_{n=1}^{\infty} (-1)^{n+1} \frac{(-z)^n}{n} = -\log(1-z).
\end{equation}
Moreover by [R], Theorem (3.44) the series
\begin{equation}
\sum_{n=1}^{\infty} \frac{z^n}{n}
\end{equation}
converges for each complex number $z$ such that $|z|=1$, with $z \neq 1$. Fix $z$ such that $|z|=1$, with $z \neq 1$. Then for any real $t \in (-1,1)$, an application of Abel's Theorem to the series
\begin{equation}
\sum_{n=1}^{\infty} \frac{z^n}{n} t^n =-\log(1-tz),
\end{equation}
shows that
\begin{equation}
\sum_{n=1}^{\infty} \frac{z^n}{n} =-\log(1-z).
\end{equation}
By replacing $z$ by $\bar{z}$ we also get that for any complex number $z$ such that $|z| \leq 1$, with $z \neq 1$, we have
\begin{equation}
\sum_{n=1}^{\infty} \frac{\bar{z}^n}{n} =-\log(1-\bar{z}).
\end{equation}
In particular we get that for any $x \in [-\pi,\pi) \backslash \{0\}$ we have 
\begin{equation}
\lim_{N \rightarrow \infty} s_N(x) =-2 \mathfrak{R}(\log(1-e^{ix})).
\end{equation}
Now let $g$ be a function in the equivalent class $[g] \in L^{2}([-\pi,\pi))$ which has $(|c_n|)_{n=-\infty}^{\infty}$ has Fourier coefficients (note that for sure $(|c_n|)_{n=-\infty}^{\infty} \in \ell^{2}(\mathbb{Z})$ as we can immediately check and as it must be since the $(c_n)_{n=-\infty}^{\infty}$ are the Fourier coefficients of $[f] \in L^{2}([-\pi,\pi))$). Since $(s_{N})_{N=1}^{\infty}$ converges to $g$ in $L^{2}([-\pi,\pi))$, there is a subsequence $(s_{N_k})_{k=1}^{\infty}$ which converges pointwise a.e. to $g$. We conclude that
\begin{equation}
g(x)=-2 \mathfrak{R}(\log(1-e^{ix})) \quad \textrm{a.e. on}\quad  [-\pi,\pi),
\end{equation}
so that $g$ is not essentially bounded.
Now take $A=(a_{i,j})_{i,j=0}^{\infty}$, with $a_{i,j}=c_{i-j}$ for all $i,j=0,1,2,\dots$. The result we have proved combined with Toeplitz's Theorem (see e.g. [BG] , Theorem (1.1)) shows that $A$ is a hermitian matrix with the required properties. 
To get an example with $A$ real symmetric, let us consider the following even real-valued function $f:[-\pi, \pi) \rightarrow \mathbb{R}$ defined by
\begin{equation}
f(x) = \begin{cases}
1 \quad \textrm{if} & x \in \left[ - \frac{\pi}{2}, \frac{\pi}{2} \right), \\
-1 \quad \textrm{otherwise}.
\end{cases}
\end{equation}
The Fourier coefficients of $f$ are:
\begin{equation}
c_n = \begin{cases}
0 \quad \textrm{if} & n=2k \quad (k \in \mathbb{Z}), \\
4 \frac{(-1)^{|k|}}{n} \quad \textrm{if} & n=2k+1 \quad (k \in \mathbb{Z}).
\end{cases}
\end{equation}
We shall now prove that $(|c_n|)_{n=-\infty}^{\infty}$ are not the Fourier coefficients of an essentially bounded function. From we have proved above, we get that for any complex number $z$ such that $|z| \leq 1$, with $z \neq 1$, we have
\begin{equation}
\sum_{k=0}^{\infty} \frac{z^{2k+1}}{2k+1} = \frac{1}{2} \left[ \log(1+z) - \log(1-z) \right].
\end{equation}
By replacing $z$ by $\bar{z}$, we also have that for any complex number $z$ such that $|z| \leq 1$, with $z \neq 1$
\begin{equation}
\sum_{k=0}^{\infty} \frac{\bar{z}^{2k+1}}{2k+1} = \frac{1}{2} \left[ \log(1+\bar{z}) - \log(1-\bar{z}) \right].
\end{equation}
So if we consider the partial sums of the Fourier series corresponding the coefficients $(|c_n|)_{n=-\infty}^{\infty}$:
\begin{equation}
s_N(x) = \sum_{n=-N}^{N} |c_n| e^{inx} \quad (x \in [-\pi,\pi) \quad \textrm{and} \quad N=1,2,\dots),
\end{equation}
we get that for any $x \in [-\pi,\pi) \backslash \{0\}$ we have 
\begin{multline}
\lim_{N \rightarrow \infty} s_N(x) = 2 \left[ \log(1+e^{ix}) - \log(1 - e^{ix}) \right] + 2 \left[ \log(1+e^{-ix}) - \log(1 - e^{-ix}) \right] = \\ =  2 \left[ \mathfrak{R}(\log(1+e^{ix})) -  \mathfrak{R}(\log(1 - e^{ix})) \right].
\end{multline}
Since the function
\begin{equation}
g(x) = 2 \left[ \mathfrak{R}(\log(1+e^{ix}))  - \mathfrak{R}(\log(1 - e^{ix})) \right] \quad (x \in [-\pi,\pi))
\end{equation}
is not essentially bounded, by proceeding as in the previous example we conlcude that $(|c_n|)_{n=-\infty}^{\infty}$ are not the Fourier coefficients of an essentially bounded function. Finally, take $A=(a_{i,j})_{i,j=0}^{\infty}$, with $a_{i,j}=c_{i-j}$ for all $i,j=0,1,2,\dots$. The result we have proved combined with Toeplitz's Theorem (see e.g. [BG] , Theorem (1.1)) shows that $A$ is a real symmetric matrix with the required properties.
References
[R] Rudin, Principles of Mathematical Analysis, Third Edition
[BG] Böttcher, Albrecht; Grudsky, Sergei M. (2012), Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis
