Definition:Space of Compact Linear Transformations

Definition

Let $H, K$ be Hilbert spaces.

Let $\Bbb F \in \left\{{\R, \C}\right\}$ be the ground field of $K$.

The space of compact linear transformations from $H$ to $K$, $B_0 \left({H, K}\right)$, is the set of all compact linear transformations:

$B_0 \left({H, K}\right):= \left\{{T: H \to K: T \text{ compact}}\right\}$

endowed with pointwise addition and ($\F$)-scalar multiplication.

It is a Banach space, as proven on Space of Compact Linear Transformations is Banach Space.

The notation resembles that for the space of bounded linear transformations $B \left({H, K}\right)$.

This is appropriate as a Compact Linear Transformation is Bounded; i.e., $B_0 \left({H, K}\right) \subseteq B \left({H, K}\right)$.

Space of Compact Linear Operators

When $H$ is equal to $K$, one speaks about the space of compact (linear) operators instead.

One writes $B_0 \left({H}\right)$ for $B_0 \left({H, H}\right)$.