Asymmetric truncated Toeplitz operators of rank one Bartosz anucha - PowerPoint PPT Presentation

Asymmetric truncated Toeplitz operators of rank one Bartosz Łanucha Maria Curie-Skłodowska University, Lublin, Poland IWOTA 2017, August 14–18 Technische Universität Chemnitz Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

Classical Toeplitz operators H 2 - the Hardy space for the unit disc D , the orthogonal projection from L 2 ( ∂ D ) onto H 2 , P - the classical Toeplitz operator on H 2 : T ϕ - f ∈ H ∞ ⊂ H 2 , T ϕ f = P ( ϕ f ) , densely defined for ϕ ∈ L 2 ( ∂ D ) , bounded if and only if ϕ ∈ L ∞ ( ∂ D ) . In particular, the shift operator on H 2 , S = T z - S ∗ = T z - the backward shift , S ∗ f ( z ) = f ( z ) − f ( 0 ) Sf ( z ) = zf ( z ) , . z Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

An inner function/ADC We say that α is an inner function if: α ∈ H ∞ , | α | = 1 a.e. on ∂ D . We say that α has an angular derivative in the sense of Carathéodory (ADC) at w ∈ ∂ D if there exist complex numbers α ( w ) and α ′ ( w ) such that α ′ ( z ) → α ′ ( w ) α ( z ) → α ( w ) ∈ ∂ D and whenever z → w nontangentially (with | z − w | / ( 1 − | z | ) bounded). Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

Shift invariant subspaces of H 2 All shift-invariant subspaces of H 2 were described by A. Beurling in 1949. He used the notion of an inner function. Beurling, 1949 A non-zero closed subspace M ⊂ H 2 is S -invariant, S ( M ) ⊂ M , if and only if M = α H 2 for some inner function α . Since S ( M ) ⊂ M if and only if M = α H 2 , then S ∗ ( M ) ⊂ M if and only if M = ( α H 2 ) ⊥ . Corollary All the S ∗ -invariant subspaces of H 2 are of the form K α = ( α H 2 ) ⊥ = H 2 ⊖ α H 2 , α − inner . K α is called the model space corresponding to the inner function α . Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

The model space K α The model space corresponding to the inner function α : K α = H 2 ⊖ α H 2 . K α is a closed S ∗ -invariant subspace of H 2 . K α is a reproducing kernel Hilbert space with the reproducing kernel given by: w ( z ) = 1 − α ( w ) α ( z ) k α w ∈ D , , 1 − wz that is, f ( w ) = � f , k α w � for all f ∈ K α , w ∈ D . Note that if α ( w ) = 0, then k α w ( z ) = k w ( z ) = ( 1 − wz ) − 1 . K α ∩ H ∞ is a dense subset of K α . Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

The model space K α The conjugate kernel w ( z ) = α ( z ) − α ( w ) � k α z − w belongs to K α for all w ∈ D . w and � If α has an ADC at w ∈ ∂ D , then k α k α w belong to K α . Moreover, � k α w = α ( w ) wk α w . Examples: 1 α ( z ) = z n , n ≥ 1: K α = P n − 1 = { polynomials of degree ≤ n − 1 } , 2 α ( z ) = a finite Blaschke product with distinct zeros a 1 , . . . , a n : K α = span { k a 1 , . . . , k a n } . The space K α is finite-dimensional, dim K α = n < ∞ , if and only if α is a finite Blaschke product with n zeros (not necessarily distinct). Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

Asymmetric truncated Toeplitz operators Let ϕ ∈ L 2 ( ∂ D ) . The classical Toeplitz operator T ϕ : f ∈ H ∞ ⊂ H 2 , T ϕ f = P ( ϕ f ) , P - the orthogonal projection from L 2 ( ∂ D ) onto H 2 . Let α and β be two inner functions. The asymmetric truncated Toeplitz operator (ATTO) A α,β is defined by ϕ A α,β f ∈ K α ∩ H ∞ , ϕ f = P β ( ϕ f ) , P β - the orthogonal projection from L 2 ( ∂ D ) onto K β . In particular, A α ϕ = A α,α is called a truncated Toeplitz operator ϕ (TTO). The operator S α = A α z is called the compressed shift . Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

Asymmetric truncated Toeplitz operators T ( α, β ) Systematic study of truncated Toeplitz operators was started by D. Sarason in 2007. Asymmetric truncated Toeplitz operators were recently introduced by C. Câmara, J. Partington (for the half-plane) and C. Câmara, J. Jurasik, K. Kliś-Garlicka, M. Ptak (for the unit disk). Put T ( α, β ) = { A α,β ϕ : ϕ ∈ L 2 ( ∂ D ) and A α,β is bounded } . ϕ Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

More on T ( α, β ) Although similar in definition, TTO’s and ATTO’s differ from the classical Toeplitz operators. T ϕ = 0 if and only if ϕ = 0, Câmara-Partington/Câmara-Jurasik-Kliś–Garlicka-Ptak ( β ≤ α ), Ł.-Jurasik, 2016 = 0 if and only if ϕ ∈ α H 2 + β H 2 . A α,β ϕ T ϕ is bounded if and only if ϕ is in L ∞ ( ∂ D ) , Baranov-Chalendar-Fricain-Mashreghi-Timotin, 2010 There exist bounded truncated Toeplitz operators without bounded symbols. the only compact Toeplitz operator is the zero operator, Sarason, 2007 There are nonzero compact truncated Toeplitz operators (in particular, rank-one truncated Toeplitz operators). Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

Rank-one TTO’s Rank-one TTO’s were described by D. Sarason. Recall that w ( z ) = 1 − α ( w ) α ( z ) w ( z ) = α ( z ) − α ( w ) � k α k α , 1 − wz z − w and f ⊗ g ( h ) = � h , g � f . Sarason, 2007 w ⊗ � w and � (a) For w in D , the operators k α k α k α w ⊗ k α w belong to T ( α, α ) . (b) If α has an ADC at the point w of ∂ D , then the operator k α w ⊗ k α w belongs to T ( α, α ) . (c) The only rank-one operators in T ( α, α ) are the nonzero scalar multiples of the operators in (a) and (b). Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

Rank-one ATTO’s Câmara-Partington ( β ≤ α ), Ł.-Jurasik, 2016 (a) For w in D , the operators k β w ⊗ � w and � k β k α w ⊗ k α w belong to T ( α, β ) . (b) If both α and β have an ADC at the point w of ∂ D , then the operator k β w ⊗ k α w belongs to T ( α, β ) . Are the nonzero scalar multiples of the operators in (a) and (b) the only rank-one operators in T ( α, β ) ? Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

Rank-one ATTO’s Recall that in (b): � k α w = α ( w ) wk α w and so w ⊗ � w ) = β ( w ) w ( � k β w ⊗ k α w = α ( w ) w ( k β k α k β w ⊗ k α w ) . Are the nonzero scalar multiples of the w ⊗ � w and � operators k β k α k β w ⊗ k α w , w ∈ D , the only rank-one operators in T ( α, β ) ? Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

The trivial case: dim K α = dim K β = 1 Let α and β be two inner functions such that dim K α = dim K β = 1 . Then f = c 1 k α f ∈ K α ⇒ 0 , g = c 2 � k β g ∈ K β ⇒ 0 , and g ⊗ f = c ( k β 0 ⊗ � k α 0 ) . So here the answer is yes. This is not always the case. Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

A counterexample Let a ∈ D \ { 0 } and let α ( z ) = z z − a z + a β ( z ) = z . 1 + az , 1 − az Then K α = span { 1 , k a , k − a } , P 0 = { λ : λ ∈ C } K β = w = � (note that k β k β w = 1). Since dim K β = 1, every linear operator from K α into K β is of rank one. Put ϕ = 1 + k a ∈ K ∞ α . Then A α,β = 1 ⊗ ( 1 + k a ) . ϕ Indeed, for every f ∈ K α , z ∈ D , A α,β = � P β ( ϕ f ) , k β z � = � ϕ f , k β z � ϕ f ( z ) = � ( 1 + k a ) f , 1 � = � f , 1 + k a � = ( 1 ⊗ ( 1 + k a ))( f ) . Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

A counterexample If = 1 ⊗ ( 1 + k a ) = c ( � A α,β k β w ⊗ k α w ) , ϕ for some w ∈ D , then (since � k β w = 1) 1 + k a = ck α w . Equivalently,  ⇒ c = 2 � ck α � 1 + k a , 1 � = w , 1 �  a ⇒ � ck α w = � 1 + k a , k a � = w , k a � . 2 −| a | 2  a � ck α � 1 + k a , k − a � = w , k − a � ⇒ w = 2 + | a | 2 This means that 1 + k a is not a scalar multiple of a reproducing kernel and = 1 ⊗ ( 1 + k a ) � = c ( � A α,β k β w ⊗ k α w ) . ϕ Similarly, 1 + k a is not a scalar multiple of a conjugate kernel and w ⊗ � A α,β = 1 ⊗ ( 1 + k a ) � = c ( k β k α w ) . ϕ Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one

dim K α = 1 or dim K β = 1 Note that if dim K α = 1 or dim K β = 1, then every bounded linear operator from K α into K β (a) is of rank one, (b) is an asymmetric truncated Toeplitz operator. Proof of (b): Câmara-Jurasik-Kliś–Garlicka-Ptak ( β ≤ α ), Gu-Ł.-Michalska, 2017 Let A be a bounded linear operator from K α into K β . Then A ∈ T ( α, β ) if and only if there exist ψ ∈ K β and χ ∈ K α such that 0 + k β A − S β AS ∗ α = ψ ⊗ k α 0 ⊗ χ A − S β AS ∗ If dim K α = 1 or dim K β = 1, then α = g ⊗ f . f = ck α If dim K α = 1, then 0 ( ψ = cg , χ = 0). g = ck β If dim K β = 1, then 0 ( ψ = 0, χ = cf ). Bartosz Łanucha Asymmetric truncated Toeplitz operators of rank one