Normality,Subnormality And Hyponormality Of Toeplitz Operators And Products Of Toeplitz Operators

In this paper,we make a summary of research about normality subnormality and hyponormality of Toeplitz operators . The negative answer to Halmos Problem 5 make it desirable to describe symbols of subnormal Toeplitz operators and hint that it is worth to begin with the hyponormality . Therefore,attention is paid to characterize the hyponormality and find that k-hyponormality provides a measure of the gap between hyponormality and subnormality . Then we try to make an extension to products of Toeplitz operators ,and show a necessary and sufficient condition about the self-adjointness of products of two Toeplitz operators by a skillful method , which can also be used to prove the characteristic of normal Toeplitz operators .