Trace norm of operators obtained by restricting the matrix of a trace class operator 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 $T$?
 A: Here's an algorithm for testing an ad-hoc conjecture $C$ about Hilbert space operators. :-)


*Set up the runtime environment correctly by loading the information "Most conjectures are false" into short term memory.


*Test $C$ against the zero and the identity operator.


*Test $C$ against finite-dimensional diagonal matrices.


*Test $C$ against multiplication operators on $\ell^2$ and $L^2$.


*Test $C$ against the following $2 \times 2$-matrices:
$$
  \begin{pmatrix}
    0 & 1 \\ 
    0 & 0
  \end{pmatrix},
  \quad 
  \begin{pmatrix}
    1 & 1 \\ 
    0 & 1
  \end{pmatrix},
  \quad 
  \begin{pmatrix}
    0 & 1 \\ 
    1 & 0
  \end{pmatrix},
  \quad 
  \begin{pmatrix}
     0 & 1 \\ 
    -1 & 0
  \end{pmatrix},
  \quad 
  \begin{pmatrix}
    1 & 1 \\ 
    1 & 1
  \end{pmatrix}.
$$


*Test $C$ against simple modifications (appropriate to the setting of $C$) of the matrices from Step 4.


*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 $C$.


*Repeat Step 6 with $3 \times 3$-matrices.


*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
$$
  \frac{1 + \sqrt{5}}{2} + \frac{\lvert 1 - \sqrt{5} \rvert}{2} = \sqrt{5} > 2.
$$
