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Subnormal and Hyponormal Generators of C*-algebras

To appear in J. Functional Analysis
with Paul McGuire

In this paper we consider the problem of when C*(A) has a subnormal or hyponormal generator for a given Hilbert space operator A. We give some results about operators in general and then focus mainly on operators A that are irreducible and essentially normal.

The first theorem has the flavor of a subnormal operator type of result. Namely, if the spectrum of A is "small", then C*(A) has a subnormal generator only in the trivial case where A is normal. Recall that a compact set is polynomially convex if its complement in the complex plane is connected.

Theorem 1:  If A is a bounded linear operator on a Hilbert space and the spectrum of A is polynomially convex and has no interior, then C*(A) has a subnormal generator if and only if A is normal.

After considering the above theorem, one may naturally ask what if the spectrum of A has area zero? Does a similar theorem hold? The answer is NO!

Example 1: There exists an irreducible essentially normal operator A whose spectrum has area zero, and yet C*(A) does have a subnormal generator.

Another natural question would be does C*(A) have a subnormal generator when A is hyponormal? Or equivalently, is having a subnormal generator the same as having a hyponormal generator?

Example 2: There is an irreducible hyponormal operator A with rank one self-commutator such that C*(A) does not have a subnormal generator.

Theorem 2: Let A be an irreducible essentially normal operator whose Fredholm index function is identically zero off of the essential spectrum of A. Then C*(A) has a hyponormal generator if and only if the essential spectrum of A has no isolated points.

Theorem 3: If A is an irreducible essentially normal operator whose essential spectrum is a "swiss-cheese type set", then C*(A) has a subnormal generator if and only if C*(A) has a hyponormal generator if and only if the values of the Fredholm index of A are either bounded above or bounded below.

Theorem 4: Let A be an irreducible essentially normal operator. Suppose that one of the following holds:
(a) The values of the Fredholm index of A are bounded above and if M is the maximum value and we define K(M) to be the union of the essential spectrum of A together with those points where the Fredholm index of A is strictly less than M, then the interior of K(M) is connected and dense in K(M).
or
(b) The values of the Fredholm index of A are bounded below and if m is the minimum value and we define K(m) to be the union of the essential spectrum of A together with those points where the Fredholm index of A is strictly larger than m, then the interior of K(m) is connected and dense in K(m).
If (a) or (b) holds, then C*(A) has a subnormal generator.

Another Theorem is given where additional conditions are prescribed on the essential spectrum of A so that C*(A) has a subnormal generator if and only if condition (a) or (b) above holds. Examples would include any irreducible essentially normal operator whose essential spectrum is a "checkboard" or a "grid of lines". The theorem even applies to certain infinite checkerboards!