Hello, I hope I am asking this in the right area of the forums. My teacher wrote the following formula down at our last meeting, and I was wondering if it was true ( [tex] \mathcal{H} [/tex] is the infinite dimension separable Hilbert space):(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \mathcal{K} (\mathcal{H}) \approx \mathcal{K} (\mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H})\approx M_{4} (\mathcal{K} (\mathcal{H})) [/tex]

I'm pretty sure this is true, but I am worried I am crazy, because I don't understand how every compact operator could secretly be 16 compact operators.

I think one formula that could aid in the proof of the above isomorphism is:

[tex] \mathcal{H} \approx \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} [/tex]

which I believe to be true considering that [tex] \mathcal{H} \cong \mathpzc{l}^2 ( \mathbb{N} ) [/tex]. So let

[tex] \mathcal{W} = \mathpzc{l}^2 (\mathbb{N}) \oplus \mathpzc{l}^2 (\mathbb{N}) \oplus \mathpzc{l}^2 (\mathbb{N}) \oplus \mathpzc{l}^2 (\mathbb{N}) \cong \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} [/tex] .

From this, we take the maximal orthonormal bases [tex] (e_n)_i \text{ of } \mathcal{H}_i = \mathcal{H} \text{ for } i = 1, 2, 3, 4 \text{ and set } (b_n) \subseteq \mathcal{W} [/tex] by

[tex] (b_1) = (e_1)_1 , (b_2) = (e_1)_2, \ldots , b_4 = (e_1)_4 , b_5 = (e_2)_1 , \ldots [/tex]

which is a countable maximal orthonormal basis for [tex] \mathcal{W} [/tex]. This shows

[tex] \mathpzc{l}^2 (\mathbb{N}) \cong \mathcal{W} \cong \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \qedsymbol [/tex]

but I can't find any proofs like this in any of my textbooks, which makes me feel like I am making a mistake. Can anyone please help me?

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# Compact Operators on a Hilbert Space

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