Krein space representation of submodules in H 2 ( D 2 ) Michio Seto 1 - PowerPoint PPT Presentation

Krein space representation of submodules in H 2 ( D 2 ) Michio Seto 1 National Defense Academy mseto@nda.ac.jp This work was inspired by Rongwei Yang (SUNY, Albany). 1 Supported by JSPS KAKENHI Grant Number 15K04926. | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 1 / 14

Introduction D : the open unit disk in C , T : the boundary of D . My interest I have been interested in ( φ 1 , φ 2 , φ 3 ) satisfying 1 φ 1 , φ 2 , φ 3 are holomorphic on D 2 , 2 | φ 1 ( λ ) | 2 + | φ 2 ( λ ) | 2 − | φ 3 ( λ ) | 2 ≤ 1 ( λ ∈ D 2 ), 3 | φ 1 ( λ ) | 2 + | φ 2 ( λ ) | 2 − | φ 3 ( λ ) | 2 → 1 a.e. as λ tends radially to T 2 . | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 2 / 14

Introduction (in D ) Inner function φ is called an inner function if 1 φ is holomorphic on D , 2 | φ ( λ ) | 2 ≤ 1 ( λ ∈ D ), 3 | φ ( λ ) | 2 → 1 a.e. as λ tends radially to T . The following functions are inner: ( z − λ ) / (1 − λ z ) ( λ ∈ D ), exp(( z + e i θ ) / ( z − e i θ )) ( θ ∈ [0 , 2 π )). | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 3 / 14

Introduction (my interest again) My interest 1 φ 1 , φ 2 , φ 3 are holomorphic on D 2 , 2 | φ 1 ( λ ) | 2 + | φ 2 ( λ ) | 2 − | φ 3 ( λ ) | 2 ≤ 1 ( λ ∈ D 2 ), 3 | φ 1 ( λ ) | 2 + | φ 2 ( λ ) | 2 − | φ 3 ( λ ) | 2 → 1 a.e. as λ tends radially to T 2 . | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 4 / 14

Examples Trivial example ( φ 1 , φ 2 , φ 3 ) = ( z 1 , z 2 , z 1 z 2 ) For λ 1 , λ 2 ∈ D , ∵ 1 − ( | λ 1 | 2 + | λ 2 | 2 − | λ 1 λ 2 | 2 ) = (1 − | λ 1 | 2 )(1 − | λ 2 | 2 ) ≥ 0 . and | λ 1 | 2 + | λ 2 | 2 − | λ 1 λ 2 | 2 → 1 + 1 − 1 = 1 . | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 5 / 14

Examples Non-trivial example For inner functions q 0 ( z 1 ) and q 1 ( z 1 ) on D satisfying 1 q 0 / q 1 is also inner, √ 1 − | ( q 0 / q 1 )(0) | 2 ̸ = 0. 2 α := ( φ 1 , φ 2 , φ 3 ) := ( q 0 , −√ 1 − α q 0 + √ 1 + α q 1 √ 1 + α q 0 − √ 1 − α q 1 √ √ z 2 , z 2 ) 2 α 2 α | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 6 / 14

How to find examples H 2 ( D 2 ): the Hardy space over D 2 , H 2 ( D 2 ) is a Hilbert module over C [ z 1 , z 2 ]. ↓ M ⊂ H 2 (a submodule) ↓ ∆ M (the defect operator of M ) ↓ if rank ∆ M =3 ↓ ∆ M = φ 1 ⊗ φ 1 + φ 2 ⊗ φ 2 − φ 3 ⊗ φ 3 (the spectral resolution of ∆ M ) | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 7 / 14

Further examples In general, rank ∆ M = 2 N + 1 ( N = 0 , 1 , 2 , . . . , ∞ ) ( R . Yang ). Hence we have the following cases: | φ 1 ( λ ) | 2 ≤ 1 | φ 1 ( λ ) | 2 + | φ 2 ( λ ) | 2 − | φ 3 ( λ ) | 2 ≤ 1 | φ 1 ( λ ) | 2 + | φ 2 ( λ ) | 2 + | φ 3 ( λ ) | 2 − | φ 4 ( λ ) | 2 − | φ 5 ( λ ) | 2 ≤ 1 ( ← hard) . . . | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 8 / 14

Setting In our construction, ( φ 1 , φ 2 , φ 3 ) has the following additional properties: Additional properties φ j ∈ H 2 ( D 2 ), φ i and φ j are orthogonal in H 2 ( D 2 ). Remark φ j might be unbounded (by Rudin). Setting we deal with ( φ 1 , φ 2 , φ 3 ) obtained by our construction, we will assume that each φ j is bounded. | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 9 / 14

Theorem 1 (representation of modules) Let M be a submodule in H 2 ( D 2 ) with ( φ 1 , φ 2 , φ 3 ). Then ∃K (= H + ⊕ H − ): a Krein space s.t. dim K = 3 ∃ D : K ⊗ H 2 ( D 2 ) → M a module map s.t. DD ♯ = P M = T φ 1 T ∗ φ 1 + T φ 2 T ∗ φ 2 − T φ 3 T ∗ φ 3 . Further, I H 2 = T ∗ φ 1 T φ 1 + T ∗ φ 2 T φ 2 − T ∗ φ 3 T φ 3 . Remark (Beurling) In H 2 ( D ), M is a submodule iff M = φ H 2 ( D ) where φ is inner. Further, P M = T φ T ∗ φ and I H 2 = T ∗ φ T φ . | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 10 / 14

Theorem 2 (homomorphisms) U = ( u ij ): a J -inner matrix valued function (that is, UJU ∗ = J a.e. on T 2 and UJU ∗ ≤ J on D 2 ) with H ∞ -entries. Then ∃ U : a module map on K ⊗ H 2 s.t. 1 U ♯ U = I K⊗ H 2 , that is, U is ♯ -isometric, 2 ∥ ( U F )( λ ) ∥ 2 K ≤ ∥ F ( λ ) ∥ 2 K for any λ in D 2 . The converse is also true if U is continuous. | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 11 / 14

Back to examples If ( φ 1 , φ 2 , φ 3 ) := ( q 0 , −√ 1 − α q 0 + √ 1 + α q 1 √ 1 + α q 0 − √ 1 − α q 1 √ z 2 , √ z 2 ) 2 α 2 α then 1 0 0       q 0 φ 1 √ 1+ α √ 1 − α 0 −  =  . √ √ φ 2 q 1 z 2   2 α 2 α   √ 1 − α √ 1+ α   q 0 z 2 φ 3 0 − √ √ 2 α 2 α | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 12 / 14

Theorem 3 (local representation) For any ( φ 1 , φ 2 , φ 3 ), ∃H : a Hilbert space, ∃ V = ( V ij ) : C ⊕ C ⊕ H ⊕ H → C ⊕ C ⊕ H ⊕ H , an isometry s.t. for k = 1 , 2, λ = ( λ 1 , λ 2 ) ∈ D 2 , φ k ( λ ) = V k 1 + V k 2 φ 3 ( λ ) + ( λ 1 V k 3 + λ 2 V k 4 )( I H − λ 1 V 33 − λ 2 V 34 ) − 1 ( V 31 + V 32 φ 3 ( λ )) where | λ 1 | and | λ 2 | are sufficiently small. | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 13 / 14

Summary In operator theory on H 2 ( D 2 ), Krein space approach will be useful as Yang suggested 2 to me. Triplet ( φ 1 , φ 2 , φ 3 ) will be manageable. However, we should not avoid unbounded cases toward general theory. Our approach can be applied to other spaces. 2 His approach is different from that given here. | ϕ 1 ( λ ) | 2 + | ϕ 2 ( λ ) | 2 − | ϕ 3 ( λ ) | 2 ≤ 1 Michio Seto (National Defense Academy) mseto@nda.ac.jp 14 / 14