Brief Thought: Compact Operators on Hilbert Spaces

Most proofs that a compact Hilbert-space operator T:\mathbb{H}\rightarrow\mathbb{H} is the limit (in the operator topology) of finite-rank operators T_{n}:\mathbb{H}\rightarrow\mathbb{H} seem to restrict to the case that \mathbb{H} is separable. But this is unnecessary hypothesis.

Indeed, let B:=\left\{x\in\mathbb{H}:\left\|x\right\|\leq 1\right\}. Since the image T(B) has compact closure–some authors use the term relatively compactT(B) is necessarily totally bounded: for any \epsilon > 0 , there exist finitely many elements y_{1},\cdots,y_{n}, where n=n(\epsilon), such that

\displaystyle \forall x\in B,\exists j\in\left\{1,\cdots,n\right\}\text{ such that }\left\|Tx-y_{j}\right\|<\epsilon

We define P_{\epsilon}:\mathbb{H}\rightarrow\mathbb{H} to be the orthogonal projection onto the finite-dimensional subspace spanned by y_{1},\cdots,y_{n}, and we can define a finite-rank operator by T_{\epsilon} :=P_{\epsilon}T.

I claim that \left\|T-T_{\epsilon}\right\|\rightarrow 0 as \epsilon\rightarrow 0. Recall that for all \left\|x\right\|\leq 1, T_{\epsilon}x is the unique element in \mathbb{H}_{\epsilon}:=\text{span}\left\{y_{1},\cdots,y_{n(\epsilon)}\right\} such that \left\|Tx-T_{\epsilon}x\right\|=\inf_{y\in\mathbb{H}_{\epsilon}}\left\|Tx-y\right\|, so that

\displaystyle\left\|(T-T_{\epsilon})x\right\|\leq\min_{1\leq j\leq n}\left\|Tx-y_{j}\right\|<\epsilon

Taking the supremum over all \left\|x\right\|\leq 1, we obtain the estimate \left\|T-T_{\epsilon}\right\|\leq\epsilon, which tends to 0 as \epsilon\rightarrow 0.

I wonder whether a similar characterization was true for compact operators between possibly nonseparable Banach spaces, but there answer is no. Apparently, this was an open question for many years until the Swedish mathematician Per Enflo showed that there are compact operators on separable, reflexive Banach spaces which are not the limit of finite-rank operators. The interested reader can consult Enflo, Per. “A counterexample to the approximation problem in Banach spaces.” Acta Mathematica 130.1 (1973): 309-317.

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