Suppose $H$ is a Hilbert space with orthonormal basis $\{e\_i\}\_{i\in \mathbb N}$. To every operator $T$, we associate a infinite matrix $[T\_{ij}]$, where $T\_{ij}=\left<Te\_j,e\_i\right>$. We know that for any trace class operator $T$, the trace norm is $||T||\_1=\operatorname{Tr}(|T|) $.


Q). Suppose $T$ is a trace class operator and $S$ is such that its matrix entries are either equal to the matrix entries of $T$ or they vanish (possibly at infinite number of points). Can I say that $||S||\_1\leq ||T||\_1$? If not, is there any finite upper bound to such $S$ obtained from $T$?



Trace norm of operators obtained by restricting the matrix of a trace class operator
Here's an algorithm for testing an ad-hoc conjecture $C$ about Hilbert space operators. :-)


0. Set up the runtime environment correctly by loading the information "Most conjectures are false" into short term memory.
1. Test $C$ against the zero and the identity operator.
2. Test $C$ against finite-dimensional diagonal matrices.
3. Test $C$ against multiplication operators on $\ell^2$ and $L^2$.
4. Test $C$ against the following $2 \times 2$-matrices:
$$
 \begin{pmatrix}
 0 & 1 \\ 
 0 & 0
 \end{pmatrix},
 \quad 
 \begin{pmatrix}
 1 & 1 \\ 
 0 & 1
 \end{pmatrix},
 \quad 
 \begin{pmatrix}
 0 & 1 \\ 
 1 & 0
 \end{pmatrix},
 \quad 
 \begin{pmatrix}
 0 & 1 \\ 
 -1 & 0
 \end{pmatrix},
 \quad 
 \begin{pmatrix}
 1 & 1 \\ 
 1 & 1
 \end{pmatrix}.
$$
5. Test $C$ against simple modifications (appropriate to the setting of $C$) of the matrices from Step 4.
6. Write a computer programm to test $C$ against randomly generated $2 \times 2$-matrices; make sure to restrict the matrices that your random generator creates to the set of matrices that occur in $C$.
7. Repeat Step 6 with $3 \times 3$-matrices.
8. If you have not found a counterexample yet, there might be a reason to believe that $C$ holds.


Of course this should not be taken completely seriously - but often it works.
In my experience, for many ad-hoc conjectures the algorithm stops at Step 5 or earlier. The question from the OP adds another data point to this pattern:


The matrix
$$
 T = 
 \begin{pmatrix}
 1 & 1 \\ 
 1 & 1
 \end{pmatrix}
$$
has trace norm $2$, but the matrix
$$
 S = 
 \begin{pmatrix}
 1 & 1 \\ 
 1 & 0
 \end{pmatrix}
$$
has trace norm
$$
 \frac{1 + \sqrt{5}}{2} + \frac{\lvert 1 - \sqrt{5} \rvert}{2} = \sqrt{5} > 2.
$$


