Crystal complex symmetric operators

Abstract

In this paper, we study properties of weak crystal operators. In particular, we show that if T is an element of L(H) is a complex symmetric operator, then T is weak crystal, crystal, or crystal-like if and only if T* is weak crystal, crystal, or crystal-like, respectively. Moreover, we prove that if T is a quasiaffinity, then T is weak crystal if and only if the Aluthge transform (T) over tilde (or the Duggal transform (T) over tilde (D)) of T is weak crystal. Finally, we give several applications of these results.