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I am having some trouble proving this: Let $T_1\in H_1$ and $T_2\in H_2$ where $H_1,H_2$ are Hilbert spaces. Let $T=T_1\oplus T_2$ on $H_1\oplus H_2$. I need to show $T$ is compact iff $T_1$ and $T_2$ are compact.

I was able to prove it in the reverse direction, but I am having trouble proving that if T is compact then $T_1$ and $T_2$ are compact.

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up vote 3 down vote accepted

Define $P_j\colon T_1\otimes T_2\to T_j$ by $P(x_1,x_2)=x_j$, $j\in \{1,2\}$. Then $T_j=P_jT$, and $P_j$ is linear and continuous.

Conversely, denote $B_j(0,1)$ the closed unit ball of $H_j$. If $H$ is endowed with the norm $\lVert (x_1,x_2)\rVert_H=\lVert x_1\rVert_{H_1}+\lVert x_2\rVert_{H_2}$, then $B_H(0,1)\subset B_1(0,1)\times B_2(0,1)$ hence $$T(B_H(0,1))\subset T_1(B_1(0,1))\times T_2(B_2(0,1)),$$ which have a compact closure as a product of such sets.

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