Subnormal operator

In operator theory, subnormal operators are bounded operators on a Hilbert space defined by weakening the requirements for normal operators.

Definition

Let H be a Hilbert space. A bounded operator A on H is said to be subnormal if A has a normal extension. In other words, A is subnormal if there exists a Hilbert space K such that H can be embedded in K and there exists a normal operator N of the form

$N={\begin{bmatrix}A&B\\0&C\end{bmatrix}}$

for some bounded operators

$B:H^{\perp }\rightarrow H,\quad C:H^{\perp }\rightarrow H^{\perp }.$