Compact Operators on a Hilbert Space

In summary, a compact operator on a Hilbert space is a linear operator that maps bounded sets to relatively compact sets. This is different from a bounded operator, which maps bounded sets to other bounded sets. Not every linear operator on a Hilbert space can be represented as a compact operator, and compact operators have properties such as continuity, completeness, self-adjointness, and a discrete spectrum. They are important in mathematical analysis, especially in functional analysis and differential equations, and have applications in fields like quantum mechanics and signal processing.
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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):

[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
 
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Please only post a question once.
 

FAQ: Compact Operators on a Hilbert Space

What is a compact operator on a Hilbert space?

A compact operator on a Hilbert space is a linear operator between two Hilbert spaces that maps bounded sets to relatively compact sets. In other words, the image of a bounded set under a compact operator is a set that has a compact closure, meaning it contains all of its limit points.

How are compact operators different from bounded operators?

While both compact and bounded operators are linear operators on a Hilbert space, the main difference is in the behavior of their images of bounded sets. A bounded operator maps bounded sets to bounded sets, while a compact operator maps bounded sets to relatively compact sets.

Can every linear operator on a Hilbert space be represented as a compact operator?

No, not every linear operator on a Hilbert space can be represented as a compact operator. For example, the identity operator is bounded but not compact.

What are some properties of compact operators?

Compact operators have several important properties, including the fact that they are continuous, completely continuous, and self-adjoint. They also have a discrete spectrum and are compact perturbations of the identity operator.

How are compact operators used in mathematical analysis?

Compact operators are an important tool in mathematical analysis, particularly in the study of functional analysis and differential equations. They are used to prove results such as the Fredholm alternative and the spectral theorem, and they have applications in areas such as quantum mechanics and signal processing.

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