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.

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 = \begin{bmatrix} A & B\\ 0 & C\end{bmatrix} for some bounded operators B : H^{\perp} \rightarrow H, \quad \mbox{and} \quad C : H^{\perp} \rightarrow H^{\perp}. ==Normality, quasinormality, and subnormality== 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 = \begin{bmatrix} U & I - UU^* \\ 0 & - U^* \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 N^*N = QV^*VQ = Q^2 = \begin{bmatrix} P^2 & 0 \\ 0 & P^2 \end{bmatrix}. On the other hand, N N^* = \begin{bmatrix} UP^2U^* + D_{U^*} P^2 D_{U^*} & -D_{U^*}P^2 U \\ -U^* P^2 D_{U^*} & U^* P^2 U \end{bmatrix}. Because UP = PU and P is self adjoint, we have U*P = PU* and D U* P = D U* 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== B (\ldots, a_{-1}, {\hat a_0}, a_1, \ldots) = (\ldots, {\hat a_{-1}}, a_0, a_1, \ldots), where ˆ denotes the zero-th position. B can be expressed in terms of the operator matrix B = \begin{bmatrix} A & I - AA^* \\ 0 & A^* \end{bmatrix}. Another normal extension is given by the unitary dilation B' of A defined above: B' = \begin{bmatrix} A & I - AA^* \\ 0 & - A^* \end{bmatrix} whose action is described by B' (\ldots, a_{-2}, a_{-1}, {\hat a_0}, a_1, a_2, \ldots) = (\ldots, - a_{-2}, {\hat a_{-1}}, a_0, a_1, a_2, \ldots). ===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 U: K_1 \rightarrow K_2. Also, the following intertwining relationship holds: U B_1 = B_2 U. \, This can be shown constructively. Consider the set S consisting of vectors of the following form: \sum_{i=0}^n (B_1^*)^i h_i = h_0+ B_1 ^* h_1 + (B_1^*)^2 h_2 + \cdots + (B_1^*)^n h_n \quad \text{where} \quad h_i \in H. 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: U \sum_{i=0}^n (B_1^*)^i h_i = \sum_{i=0}^n (B_2^*)^i h_i Because \left\langle \sum_{i=0}^n (B_1^*)^i h_i, \sum_{j=0}^n (B_1^*)^j h_j\right\rangle = \sum_{i j} \langle h_i, (B_1)^i (B_1^*)^j h_j\rangle = \sum_{i j} \langle (B_2)^j h_i, (B_2)^i h_j\rangle = \left\langle \sum_{i=0}^n (B_2^*)^i h_i, \sum_{j=0}^n (B_2^*)^j h_j\right\rangle , , 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) \text{if } g = \sum_{i=0}^n (B_1^*)^i h_i , \text{then } U B_1 g = B_2 U g = \sum_{i=0}^n (B_2^*)^i A h_i. 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. ==References==

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