Reproducing kernels and symmetric operators A. Aleman, R. Martin, and W. Ross * Lille 2013 Aleman,Martin,Ross (*) SymmetricOperators Lille2013 1/35 Main question When are two densely defined unbounded symmetric operators on a Hilbert space unitarily equivalent? Aleman,Martin,Ross (*) SymmetricOperators Lille2013 2/35 Note about the examples Usually when one talks about unbounded symmetric operators, one talks about Tf = −(pf(cid:48))(cid:48)+qf (Sturm-Liouville) Tf = −f(cid:48)(cid:48)+Vf (Schr¨odinger) We want unbounded symmetric Toeplitz operators T on H2. ϕ Aleman,Martin,Ross (*) SymmetricOperators Lille2013 3/35 Note about the examples Usually when one talks about unbounded symmetric operators, one talks about Tf = −(pf(cid:48))(cid:48)+qf (Sturm-Liouville) Tf = −f(cid:48)(cid:48)+Vf (Schr¨odinger) We want unbounded symmetric Toeplitz operators T on H2. ϕ Aleman,Martin,Ross (*) SymmetricOperators Lille2013 3/35 Note about the examples Usually when one talks about unbounded symmetric operators, one talks about Tf = −(pf(cid:48))(cid:48)+qf (Sturm-Liouville) Tf = −f(cid:48)(cid:48)+Vf (Schr¨odinger) We want unbounded symmetric Toeplitz operators T on H2. ϕ Aleman,Martin,Ross (*) SymmetricOperators Lille2013 3/35 Note about the examples Usually when one talks about unbounded symmetric operators, one talks about Tf = −(pf(cid:48))(cid:48)+qf (Sturm-Liouville) Tf = −f(cid:48)(cid:48)+Vf (Schr¨odinger) We want unbounded symmetric Toeplitz operators T on H2. ϕ Aleman,Martin,Ross (*) SymmetricOperators Lille2013 3/35 Note about the examples Usually when one talks about unbounded symmetric operators, one talks about Tf = −(pf(cid:48))(cid:48)+qf (Sturm-Liouville) Tf = −f(cid:48)(cid:48)+Vf (Schr¨odinger) We want unbounded symmetric Toeplitz operators T on H2. ϕ Aleman,Martin,Ross (*) SymmetricOperators Lille2013 3/35 Bounded Toeplitz operators H2 Hardy space of the open unit disk D H∞ bounded analytic functions on D For ϕ ∈ H∞, T : H2 → H2, T f = ϕf. ϕ ϕ ∼ When is T = T ? ϕ1 ϕ2 When is T ∼ T ? ϕ1 ϕ2 Aleman,Martin,Ross (*) SymmetricOperators Lille2013 4/35 Bounded Toeplitz operators H2 Hardy space of the open unit disk D H∞ bounded analytic functions on D For ϕ ∈ H∞, T : H2 → H2, T f = ϕf. ϕ ϕ ∼ When is T = T ? ϕ1 ϕ2 When is T ∼ T ? ϕ1 ϕ2 Aleman,Martin,Ross (*) SymmetricOperators Lille2013 4/35 Bounded Toeplitz operators H2 Hardy space of the open unit disk D H∞ bounded analytic functions on D For ϕ ∈ H∞, T : H2 → H2, T f = ϕf. ϕ ϕ ∼ When is T = T ? ϕ1 ϕ2 When is T ∼ T ? ϕ1 ϕ2 Aleman,Martin,Ross (*) SymmetricOperators Lille2013 4/35