- See also
- References
In functional analysis, the compression of a linear operator T on a Hilbert space to a subspace K is the operator
,
where 
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 
and an isometry V on a subspace 
of , define the compression of T to by
,
where 
is the adjoint of V. If T is a self-adjoint operator, then the compression 
is also self-adjoint.
When V is replaced by the inclusion map 
, 
, and we acquire the special definition above.
See also
- Dilation
References
- P. Halmos, A Hilbert Space Problem Book, Second Edition, Springer-Verlag, 1982.
1 : Functional analysis