# Projection is trace-decreasing?

I'm studying Mark Wilde's "Quantum Information Theory" and the author sometimes use the inequality $$\mathrm{Tr}(\prod_\mathcal{H'}Y) \leq \mathrm{Tr}(Y)$$ where $$Y\in \mathcal{H}'$$ is a density matrix and $$\prod_\mathcal{H'}=V^{\dagger}V$$ with isometry $$V: \mathcal{H} \rightarrow \mathcal{H}'$$. As far as I know, this inequality does not hold for general matrix $$Y$$. So I tried to prove the inequality using the positive semi-definite condition of $$Y$$, but I cannot grasp any clue. Does the inequality really holds? And if it does, how can I prove it? I appreciate any help.

Hint: Write $$\Pi_{\mathcal H'}=\sum \lambda_i|i\rangle\langle i|$$ in an eigenbasis $$|i\rangle$$, with $$\lambda_i=0,1$$. Then use that the trace is cyclic, and $$\langle i|Y|i\rangle\ge0$$.
(Note that the inequality cannot hold for a general matrix: If it holds for $$Y=Y_0$$, it cannot hold for $$Y=-Y_0$$.)