Abstract
Asymmetric dual truncated Toeplitz operators acting between the orthogonal complements of two (eventually different) model spaces are introduced and studied. They are shown to be equivalent after extension to paired operators on L^2({mathbb {T}}) oplus L^2({mathbb {T}}) and, if their symbols are invertible in L^infty ({mathbb {T}}), to asymmetric truncated Toeplitz operators with the inverse symbol. Relations with Carleson’s corona theorem are also established. These results are used to study the Fredholmness, the invertibility and the spectra of various classes of dual truncated Toeplitz operators.
Highlights
Toeplitz operators have been for a long time one of the most studied classes of nonselfadjoint operators [3]
Dual Toeplitz operators are analogously defined on the orthogonal complement of H2( ), iden‐ tified as usual with a subspace of L2( ), as multiplication operators followed by projection onto L2( ) ⊖ H2( )
The symbol of a dual truncated Toeplitz operator is unique and the only compact operator of that kind is the zero operator, in sharp contrast with what happens with truncated Toeplitz operators on model spaces
Summary
Toeplitz operators have been for a long time one of the most studied classes of nonselfadjoint operators [3]. The results are applied to describe the spectra of dual truncated Toeplitz operators in sev‐ eral classes including, as particular cases, the dual truncated shift and its adjoint We do this by using a novel approach to dual truncated Toeplitz operators and their asymmetric analogues, defined between the orthogonal comple‐ ments of two possibly different model spaces. This involves proving their equiva‐ lence after extension to paired operators in L2( ) ⊕ L2( ) , defined, and establishing connections with the corona theorem. We use the previous results to study the Fred‐ holmness, invertibility and spectra of several classes of dual truncated Toeplitz operators
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
