# Hypercyclic and Supercyclic Cohyponormal Operators

Acta Sci. Math. (Szeged) 68 (2002), no. 1-2, 303--328.
with Vivien & Len Miller
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This paper gives a characterization of the hyponormal operators with hypercyclic or supercyclic adjoints.  An operator T is hypercyclic if there is a vector with dense orbit; so if T acts on H, then T is hypercyclic if there is a vector x in H such that {x, Tx, T^2x, ...} is dense in H.  An operator is supercyclic if there is a vector x such that {cT^nx : c is a scalar, n = 0, 1, 2, ...} is dense - another way to say this is that T has a one-dimensional subspace with dense orbit.

If S is an operator, then a part of S, is (S|M), for some invariant subspace M of S.  A part of the spectrum of S is a compact set of the form s(S|M), where M is an invariant subspace for S.  So the parts of S are all operators obtained by restricting S to an invariant subspace; and the parts of the spectrum of S are the spectra of all parts of S.

## Hypercyclic Operators

Theorem:  If S is a subdecomposable operator and every part of the spectrum of S intersects {z :|z|<1} and {z:|z|>1}, then S* is hypercyclic.

Corollary:  If S is a hyponormal operator, then S* is hypercyclic if and only if every part of the spectrum of S intersects {z :|z|<1} and {z:|z|>1}.

Remark:  In the previous two results, it suffices to check hyperinvariant parts.

Corollary:  Let S and T be hyponormal operators.
(a)  If S* and T* are hypercyclic, then (S* direct sum T*) is also hypercyclic.
(b)  If S* is finitely hypercyclic, then S* is hypercyclic.
(c)  If S* is hypercyclic and f(z) is an inner function analytic near s(S), then f(S)* is also hypercyclic.

## Supercyclic Operators

Theorem:  If S is any supercyclic operator, then there exists a circle C = {z:|z| = r}, where r is non-negative, such that every part of the spectrum of S* intersects C.  In particular, every component of the spectrum of S must intersect C.

Theorem:  If S is a subdecomposable operator and there exists a circle C = {z:|z|=r}, r  non-negative, such that either:

(a) every part of the spectrum of S intersects C and the interior of C; or

(b) every part of the spectrum of S intersects C and the exterior of C,

Then S* is supercyclic.

Corollary:  If S is a pure hyponormal operator, then S* is supercyclic if and only if there exists a circle C = {z:|z|=r}, r non-negative, such that either:
(a) every part of the spectrum of S intersects C and the interior of C; or
(b) every part of the spectrum of S intersects C and the exterior of C.

Remark:  Notice that the radius of the circle may be zero!

Terminology:  If S is hyponormal and S* is supercyclic, then we say that S (or S*) is inner or outer depending on whether S satisfies (a) or (b) respectively.  The circle associated with S is called a supercyclicity circle for S (or S*) and its radius is a supercyclicity radius for S (or for S*).

Corollary:  Let S and T be pure hyponormal operators.

(a)  If S* and T* are supercyclic, then (S* direct sum T*) is supercyclic if and only if both S* and T* have a common supercyclicity circle and they both have the same type (inner or outer).

(b)  If S* is finitely supercyclic, then S* is supercyclic.

(c)  If S* is supercyclic with supercyclicity radius 0 and f(z) is a function analytic near s(S) satisfying f(0) = 0, then f(S)* is also supercyclic.

We say that an operator is purely supercyclic, if it is pure, supercyclic and no multiple of it is hypercyclic.

Corollary:  If S is hyponormal and S* is purely supercyclic, then its supercyclicity circle is unique.

Corollary:  If S is hyponormal, then S* is both inner and outer if and only if a multiple of S* is hypercyclic.