Compact Operators on a Hilbert Space

[lunde]

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 ( $$\mathcal{H}$$ is the infinite dimension separable Hilbert space):

$$\mathcal{K} (\mathcal{H}) \approx \mathcal{K} (\mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H})\approx M_{4} (\mathcal{K} (\mathcal{H}))$$

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:

$$\mathcal{H} \approx \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H}$$

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

$$\mathcal{W} = \mathpzc{l}^2 (\mathbb{N}) \oplus \mathpzc{l}^2 (\mathbb{N}) \oplus \mathpzc{l}^2 (\mathbb{N}) \oplus \mathpzc{l}^2 (\mathbb{N}) \cong \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H}$$ .

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

$$(b_1) = (e_1)_1 , (b_2) = (e_1)_2, \ldots , b_4 = (e_1)_4 , b_5 = (e_2)_1 , \ldots$$

which is a countable maximal orthonormal basis for $$\mathcal{W}$$. This shows

$$\mathpzc{l}^2 (\mathbb{N}) \cong \mathcal{W} \cong \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \qedsymbol$$

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?

 

[Hurkyl]

They sound believable. It looks like just another manifestation of the fact a countably infinite set can be split into two disjoint countably infinite sets.

 

[DrRocket]

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 ( $$\mathcal{H}$$ is the infinite dimension separable Hilbert space):

$$\mathcal{K} (\mathcal{H}) \approx \mathcal{K} (\mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H})\approx M_{4} (\mathcal{K} (\mathcal{H}))$$

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:

$$\mathcal{H} \approx \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H}$$

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

$$\mathcal{W} = \mathpzc{l}^2 (\mathbb{N}) \oplus \mathpzc{l}^2 (\mathbb{N}) \oplus \mathpzc{l}^2 (\mathbb{N}) \oplus \mathpzc{l}^2 (\mathbb{N}) \cong \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H}$$ .

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

$$(b_1) = (e_1)_1 , (b_2) = (e_1)_2, \ldots , b_4 = (e_1)_4 , b_5 = (e_2)_1 , \ldots$$

which is a countable maximal orthonormal basis for $$\mathcal{W}$$. This shows

$$\mathpzc{l}^2 (\mathbb{N}) \cong \mathcal{W} \cong \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \qedsymbol$$

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?

What you have stated is true, but the point is rather mysterious.

You can, of course, break any infinite-dimensional separable Hilbert space into any finite (or countable) number of direct summands -- simply look at the closed span of some partition of a Hilbert space basis. This is essentially what you have above.

The question is, "why ?".

 

[lunde]

Yeah, the idea of partitioning the basis makes total sense. For some reason I just hadn't been able to find anyone saying that this statement was true, so I started to get worried. But the reason I was interested in a Hilbert space being equal to the direct sum of 4 had to do with the compact operator question I posted at the top of the page. Thanks for your help.

 

[rhit2013]

that's not right...

 

[lunde]

Which part isn't right?

 

[xinisey]

how to prove that the set of all compact operators on a hilbert space is a vector space.. help..