Compression (functional analysis)

In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, the compression of a <a href="/facts/Linear_operator/Q1PBKNVV">linear operator</a> T on a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> to a <a href="/facts/Linear_subspace/2wtKTgHL">subspace</a> K is the operator

P
 
 K
 
 
 T
 
 |
 
 K
 
 
 :
 K
 →
 K
 
 
 {\displaystyle P_{K}T\vert _{K}:K\rightarrow K}
 
,
where 
 
 
 
 
 P
 
 K
 
 
 :
 H
 →
 K
 
 
 {\displaystyle P_{K}:H\rightarrow K}
 
 is the <a href="/facts/Orthogonal_projection/sElXGkxD">orthogonal projection</a> 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 <a href="/facts/Invariant_subspace/51Vm1W2X">invariant subspace</a> for T, then the compression of T to K is the <a href="/facts/Restriction_(mathematics)/JV59DhKy">restricted</a> operator K→K sending k to Tk.
More generally, for a linear operator T on a Hilbert space 
 
 
 
 H
 
 
 {\displaystyle H}
 
 and an <a href="/facts/Isometry/K5wX8ah6">isometry</a> V on a subspace 
 
 
 
 W
 
 
 {\displaystyle W}
 
 of 
 
 
 
 H
 
 
 {\displaystyle H}
 
, define the compression of T to 
 
 
 
 W
 
 
 {\displaystyle W}
 
 by

T
 
 W
 
 
 =
 
 V
 
 ∗
 
 
 T
 V
 :
 W
 →
 W
 
 
 {\displaystyle T_{W}=V^{*}TV:W\rightarrow W}
 
,
where 
 
 
 
 
 V
 
 ∗
 
 
 
 
 {\displaystyle V^{*}}
 
 is the <a href="/facts/Hermitian_adjoint/gN6RHphd">adjoint</a> of V. If T is a <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operator</a>, then the compression 
 
 
 
 
 T
 
 W
 
 
 
 
 {\displaystyle T_{W}}
 
 is also self-adjoint.
When V is replaced by the <a href="/facts/Inclusion_map/VUFNdaDH">inclusion map</a> 
 
 
 
 I
 :
 W
 →
 H
 
 
 {\displaystyle I:W\to H}
 
, 
 
 
 
 
 V
 
 ∗
 
 
 =
 
 I
 
 ∗
 
 
 =
 
 P
 
 K
 
 
 :
 H
 →
 W
 
 
 {\displaystyle V^{*}=I^{*}=P_{K}:H\to W}
 
, and we acquire the special definition above.

Compression (functional analysis) open-in-new

Compression (functional analysis)