Subnormal operator

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
for some bounded operators
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
by
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",
Define
The operator "N" = "VQ" is clearly an extension of "A". We show it is a normal extension via direct calculation. Unitarity of "V" means
On the other hand,
Because "UP = PU" and "P" is self adjoint, we have "U*P = PU*" and "DU*P = DU*P". Comparing entries then shows "N" is normal. This proves quasinormality implies subnormality.
For a counter example that shows the converse is not true, consider again the unilateral shift "A". The operator "B" = "A" + "s" for some scalar "s" remains subnormal. But if "B" is quasinormal, a straightforward calculation shows that "A*A = AA*", which is a contradiction.
Minimal normal extension.
Non-uniqueness of normal extensions.
Given a subnormal operator "A", its normal extension "B" is not unique. For example, let "A" be the unilateral shift, on "l"2(N). One normal extension is the bilateral shift "B" on "l"2(Z) defined by
where ˆ denotes the zero-th position. "B" can be expressed in terms of the operator matrix
Another normal extension is given by the unitary dilation "B' " of "A" defined above:
whose action is described by
Minimality.
Thus one is interested in the normal extension that is, in some sense, smallest. More precisely, a normal operator "B" acting on a Hilbert space "K" is said to be a minimal extension of a subnormal "A" if " K' " ⊂ "K" is a reducing subspace of "B" and "H" ⊂ " K' ", then "K' " = "K". (A subspace is a reducing subspace of "B" if it is invariant under both "B" and "B*".)
One can show that if two operators "B"1 and "B"2 are minimal extensions on "K"1 and "K"2, respectively, then there exists a unitary operator
Also, the following intertwining relationship holds:
This can be shown constructively. Consider the set "S" consisting of vectors of the following form:
Let "K' " ⊂ "K"1 be the subspace that is the closure of the linear span of "S". By definition, "K' " is invariant under "B"1* and contains "H". The normality of "B"1 and the assumption that "H" is invariant under "B"1 imply "K' " is invariant under "B"1. Therefore, "K' " = "K"1. The Hilbert space "K"2 can be identified in exactly the same way. Now we define the operator "U" as follows:
Because
, the operator "U" is unitary. Direct computation also shows (the assumption that both "B"1 and "B"2 are extensions of "A" are needed here)
When "B"1 and "B"2 are not assumed to be minimal, the same calculation shows that above claim holds verbatim with "U" being a partial isometry.