Subnormal operator: Difference between revisions

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{{Unreferenced|date=November 2006}}

In [[mathematics]], especially [[operator theory]], '''subnormal operators''' are [[bounded operator]]s on a [[Hilbert space]] defined by weakening the requirements for [[normal operator]]s. <ref name="Conway1991">{{citation|author=John B. Conway|title=The Theory of Subnormal Operators|url=https://books.google.com/books?id=Ho7yBwAAQBAJ|accessdate=15 June 2017|year=1991|publisher=American Mathematical Soc.|isbn=978-0-8218-1536-6|page=27|chapter=11}}</ref> Some examples of subnormal operators are [[isometry|isometries]] and [[Toeplitz operator]]s 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

:<math>N = \begin{bmatrix} A & B\\ 0 & C\end{bmatrix}</math>

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=== 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 operator]]s. A unitary operator is an isometry with [[dense set|dense]] [[range (mathematics)of a function|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

:<math>H \oplus H</math>

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===Quasinormal operators===

An operator ''A'' is said to be '''[[quasinormal operator|quasinormal]]''' if ''A'' commutes with ''A*A''.<ref>{{citation|author=John B. Conway|title=The Theory of Subnormal Operators|url=https://books.google.com/books?id=Ho7yBwAAQBAJ|accessdate=15 June 2017|year=1991|publisher=American Mathematical Soc.|isbn=978-0-8218-1536-6|page=29|chapter=11}}</ref> 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.<ref name="ConwayOlin1977">{{citation|author1=John B. Conway|author2=Robert F. Olin|title=A Functional Calculus for Subnormal Operators II|url=https://books.google.com/books?id=yQXUCQAAQBAJ|accessdate=15 June 2017|year=1977|publisher=American Mathematical Soc.|isbn=978-0-8218-2184-8|page=51}}</ref>

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'',

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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

:<math>B (\cdotsldots, a_{-1}, {\hat a_0}, a_1, \cdotsldots) = (\cdotsldots, {\hat a_{-1}}, a_0, a_1, \cdotsldots),</math>

where ˆ denotes the zero-th position. ''B'' can be expressed in terms of the operator matrix

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:<math>

B' (\cdotsldots, a_{-2}, a_{-1}, {\hat a_0}, a_1, a_2, \cdotsldots) = (\cdotsldots, - a_{-2}, {\hat a_{-1}}, a_0, a_1, a_2, \cdotsldots).

</math>

===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*''.)<ref>{{citation|author=John B. Conway|title=The Theory of Subnormal Operators|url=https://books.google.com/books?id=Ho7yBwAAQBAJ&pg=PA38|accessdate=15 June 2017|year=1991|publisher=American Mathematical Soc.|isbn=978-0-8218-1536-6|pages=38–}}</ref>

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

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:<math>U: K_1 \rightarrow K_2.</math>

Also, the following interwiningintertwining relationship holds:

:<math>U B_1 = B_2 U. \,</math>

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:<math>

\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 \mboxtext{where} \quad h_i \in H.

</math>

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:<math>

\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 ,

</math>

, 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)

:<math>\mboxtext{if} \quad} g = \sum_{i=0}^n (B_1^*)^i h_i ,</math>

:<math>\mboxtext{then} \quad} U B_1 g = B_2 U g = \sum_{i=0}^n (B_2^*)^i A h_i.</math>

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]].

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{{DEFAULTSORT:Subnormal Operator}}

[[Category:Operator theory]]

[[Category:Linear operators]]