with Paul McGuire

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!