In mathematics, especially operator theory, subnormal operators are bounded operators on a Hilbert space defined by weakening the requirements for normal operators. Some examples of subnormal operators are isometries and Toeplitz operators with analytic symbols.
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 = [ A B 0 C ] {\displaystyle N={\begin{bmatrix}A&B\\0&C\end{bmatrix}}}
for some bounded operators
B : H ⊥ → H , and C : H ⊥ → H ⊥ . {\displaystyle B:H^{\perp }\rightarrow H,\quad {\mbox{and}}\quad C:H^{\perp }\rightarrow H^{\perp }.}
Normality, quasinormality, and subnormality
Normal operators Every normal operator is subnormal by definition, but the converse is not true in general. A simple class of examples can be obtained by weakening the properties of unitary operators. A unitary operator is an isometry with dense range. Consider now an isometry A whose range is not necessarily dense. A concrete example of such is the unilateral shift, which is not normal. But A is subnormal and this can be shown explicitly. Define an operator U on
H ⊕ H {\displaystyle H\oplus H}
by
U = [ A I − A A ∗ 0 − A ∗ ] . {\displaystyle U={\begin{bmatrix}A&I-AA^{*}\\0&-A^{*}\end{bmatrix}}.}
Direct calculation shows that U is unitary, therefore a normal extension of A. The operator U is called the unitary dilation of the isometry A.
Quasinormal operators An operator A is said to be quasinormal if A commutes with A*A. A normal operator is thus quasinormal; the converse is not true. A counter example is given, as above, by the unilateral shift. Therefore, the family of normal operators is a proper subset of both quasinormal and subnormal operators. A natural question is how are the quasinormal and subnormal operators related. We will show that a quasinormal operator is necessarily subnormal but not vice versa. Thus the normal operators is a proper subfamily of quasinormal operators, which in turn are contained by the subnormal operators. To argue the claim that a quasinormal operator is subnormal, recall the following property of quasinormal operators: Fact: A bounded operator A is quasinormal if and only if in its polar decomposition A = UP, the partial isometry U and positive operator P commute. Given a quasinormal A, the idea is to construct dilations for U and P in a sufficiently nice way so everything commutes. Suppose for the moment that U is an isometry. Let V be the unitary dilation of U,
V = [ U I − U U ∗ 0 − U ∗ ] = [ U D U ∗ 0 − U ∗ ] . {\displaystyle V={\begin{bmatrix}U&I-UU^{*}\\0&-U^{*}\end{bmatrix}}={\begin{bmatrix}U&D_{U^{*}}\\0&-U^{*}\end{bmatrix}}.}
Define
Q = [ P 0 0 P ] . {\displaystyle Q={\begin{bmatrix}P&0\\0&P\end{bmatrix}}.}
The operator N = VQ is clearly an extension of A. We show it is a normal extension via direct calculation. Unitarity of V means
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