# Compact Operators on a Hilbert Space

1. Jul 12, 2010

### lunde

Hello, I hope I am asking this in the right area of the forums. I wanted to ask if the following formula is true (assuming H is infinite dimensional and separable):

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

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

Last edited: Jul 12, 2010
2. Jul 12, 2010

### Hurkyl

Staff Emeritus
Please only post a question once.