In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operator
P
K
T
|
K
:
K
→
K
,
where
P
K
:
H
→
K
is the orthogonal projection onto K. This is a natural way to obtain an operator on K from an operator on the whole Hilbert space. If K is an invariant subspace for T, then the compression of T to K is the restricted operator K→K sending k to Tk.
More generally, for a linear operator T on a Hilbert space
H
and an isometry V on a subspace
W
of
H
, define the compression of T to
W
by
T
W
=
V
∗
T
V
:
W
→
W
,
where
V
∗
is the adjoint of V. If T is a self-adjoint operator, then the compression
T
W
is also self-adjoint. When V is replaced by the inclusion map
I
:
W
→
H
,
V
∗
=
I
∗
=
P
K
:
H
→
W
, and we acquire the special definition above.
