We construct an unbounded hyponormal composition operator

The class of unbounded composition operators in

In the present paper we provide an example of a hyponormal composition operator acting in

The first example of a nonsymmetric hyponormal operator such that the domain of its square is trivial was provided within the class of weighted shifts on directed trees (cf. [

We denote by

If

We recall now some information on composition operators in

The following lemma delivers few basic properties of composition operators in

The following conditions are equivalent

Moreover, if

By definition,

That (ii) implies (i) is obvious. The converse can be proved as follows. Let

For the proof of the “moreover” statement we recall a general characterization of hyponormality of a densely defined composition operator induced by a nonsingular transformation of a measure space

By inspecting the part “(i) implies (ii)” of the proof above we can get the following description of when the domain of a composition operator in

The main result of this paper is the existence of a hyponormal composition operator whose square has trivial domain.

There exists a countably infinite set

Set

Now, we define an atomic measure

Now, we derive some complementary information on

It is easy to see that the operator

(1) It is known that there are closed symmetric operators whose squares possess trivial domains (cf. [

(2) By using [

(3) A considerable amount of attention has been given to the study of

The authors would like to thank both referees for pointing out an error in earlier version of the paper and for valuable comments that helped to improve the presentation. The research was supported by the NCN (National Science Center) Grant DEC-2011/01/D/ST1/05805. This paper is dedicated to Professor J. Stochel on the occasion of his sixtieth birthday.