Complex symmetry of Toeplitz operators with continuous symbols

作者：Noor S Waleed^*

来源：ARCHIV DER MATHEMATIK, 2017, 109(5): 455-460.

DOI：10.1007/s00013-017-1101-9

• Toeplitz operator
• Complex symmetric operator
• Normal operator
• Hardy-Hilbert space
• Nowhere winding curve

In this note, we consider the question of when a Toeplitz operator on the Hardy-Hilbert space H-2 of the open unit disk D is complex symmetric, focusing on symbols phi : T -> C that are continuous on the unit circle T =alpha D. A closed curve phi is called nowhere winding if the winding number of phi is 0 about every point not in the range of phi. It is then shown that if T-phi is complex symmetric, then phi must be nowhere winding. Hence if phi is a simple closed curve, then T-phi cannot be a complex symmetric operator. The spectrum and invertibility of complex symmetric Toeplitz operators with continuous symbols are then described. Finally, given any continuous curve gamma : [a, b] -> C, it is shown that there exists a complex symmetric Toeplitz operator with continuous symbol whose spectrum is precisely the range of gamma.

• 出版日期2017-11