Compressions of Multiplication Operators and Their Characterizations

Abstract

Dual truncated Toeplitz operators and other restrictions of the multiplication by the independent variable $$M_z$$ M z on the classical $$L^2$$ L 2 space on the unit circle are investigated. Commutators are calculated and commutativity is characterized. A necessary and sufficient condition for any operator to be a dual truncated Toeplitz operator is established. A formula for recovering its symbol is stated.

Shift invariance and reflexivity of compressions of multiplication operators

“…θ ∩ L ∞ is also a dense subset of K ⊥ θ as it was observed in [8]. For nonconstant inner functions θ, α and for ϕ ∈ L 2 define…”

“…For the definitions of properties mentioned above and classes of operators see Section 2. In Section 3 (Theorem 3.4) we give a characterization of asymmetric dual truncated Toeplitz operators (ADTTO) extending the characterization given by the same authors in [8] for the symmetric case. This result is a crucial tool used in Section 5.…”

“…The importance of this class of operators in both symmetric and asymmetric cases was shown by many important papers (for instance [32,17,2,3,7,20]). The history of the dual case is shorter, however there are many results obtained in this direction (for example [16,21,30,31,8,11,12]). For arguments for the importance of (asymmetric) dual truncated Toeplitz operators and their connection to physics one can search in [8].…”

“…The history of the dual case is shorter, however there are many results obtained in this direction (for example [16,21,30,31,8,11,12]). For arguments for the importance of (asymmetric) dual truncated Toeplitz operators and their connection to physics one can search in [8].…”

Intertwining property for compressions of multiplication operators

“…Note that if the operator D given above is bounded, then necessarily ϕ 1 , ϕ 4 ∈ L ∞ . This is a consequence of the fact that T θ,α ϕ 1 and Ťϕ 4 are determined by the classical Toeplitz operators T ϕ 1 and T φ4 , respectively (see [5,Proposition 20]), and that a classical Toeplitz operator is bounded if and only if its symbol is from L ∞ . On the other hand, even if D is bounded, the functions ϕ 2 and ϕ 3 may not belong to L ∞ (since a classical Hankel operator may be bounded even if its symbol is not, see [22, Chapter 1] for details).…”