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 $T$?

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