Convolution operators in discrete Ces` aro spaces Werner Ricker - PowerPoint PPT Presentation

Convolution operators in discrete Ces` aro spaces Werner Ricker Paweł Doma´ nski Memorial Conference Banach Center in Bedlewo Poland 1-7 July 2018 Werner Ricker Convolution in discrete Ces` aro spaces Doma´ nski Memorial Conference 0 / 12

The Ces` aro operator C aro operator C : C N → C N given by Consider the Ces` � � x 0 , x 0 + x 1 , x 0 + x 1 + x 2 x = ( x n ) ∞ 0 ∈ C N . C ( x ) ≔ , . . . , 2 3 ∀ x ∈ C N (with | x | ≔ ( | x n | ) ∞ | C ( x ) | ≤ C ( | x | ) , 0 ) C is a vector space isomorphism of C N onto C N . The discrete Ces` aro space for 1 < p < ∞ (early 1970’s):   � �   � �    ∞  � n �   �       � � x ∈ C N : � x � ces( p ) ≔     1 � � ces ( p ) ≔  | x k |  = � C ( | x | ) � p < ∞      � �   n + 1   � �  k = 0 0 p Intense research by G. Bennett, Mem. Amer. Math. Soc.120 (1996): “Factorizing the classical inequalities” Werner Ricker Convolution in discrete Ces` aro spaces Doma´ nski Memorial Conference 1 / 12

Properties of ces ( p ) , 1 < p < ∞ ( ces ( p ) , � · � ces( p ) ) is a reflexive Banach lattice. e k ≔ ( δ nk ) ∞ n = 0 , k ∈ N are an unconditional basis : � � � e k � ces( p ) ≃ ( k + 1 ) − 1 / p ′ , 1 p + 1 p ′ = 1 k ∈ N , . Equivalent norms in ces ( p ) : � � �� ∞ � ∞ � � � � | x k | (a) x �→ � p . � k = n k + 1 n = 0 � � p � 1 / p j = 0 2 j ( 1 − p ) �� 2 j + 1 − 1 | x 0 | p + � ∞ (b) x �→ k = 2 j | x k | . Hardy’s inequality for 1 < p < ∞ : � C ( | x | ) � p ≤ p ′ � x � p , x ∈ ℓ p . ⇒ C maps ℓ p → ℓ p continuously (operator norm is p ′ ). Moreover, the inclusion ℓ p ⊆ ces ( p ) is proper , continuous and C maps ces ( p ) → ℓ p continuously (isometrically). Hence, also C maps ces ( p ) → ces ( p ) continuously. Werner Ricker Convolution in discrete Ces` aro spaces Doma´ nski Memorial Conference 2 / 12

Further properties of ces ( p ) , 1 < p < ∞ Let 1 < p < ∞ and x ∈ C N . Remarkable property (G. Bennett): x ∈ ces ( p ) if and only if C ( | x | ) ∈ ces ( p ) ℓ p , 1 < p < ∞ , do not have this property. G. Curbera (2014): The largest of all those solid Banach lat- tices X ⊆ C N with ℓ p ⊆ X s.t. C maps X → ℓ p cont. is ces ( p ) . G. Curbera (2014): Largest amongst the class of spaces ℓ r ( 1 < r < ∞ ) satisfying ℓ r ⊆ ces ( p ) is the space ℓ p . Dual Banach space ces ( p ) ∗ identified by A.A. Jagers (1974). Rather complicated: G. Bennett (1996) showed: � � x ∈ C N : � ∞ n = 0 sup k ≥ n | x k | p ′ < ∞ ces ( p ) ∗ ≃ d ( p ′ ) = for the equivalent (but not dual) norm � � � � � � � � � ( sup k ≥ n | x k | ) ∞ � p ′ ≔ � ˆ � p ′ . � x � d ( p ′ ) ≔ x n = 0 Werner Ricker Convolution in discrete Ces` aro spaces Doma´ nski Memorial Conference 3 / 12

Convolution operators in ces ( p ) Fix b = ( b n ) ∞ n = 0 ∈ C N . Each element   ∞  �  n       x ∈ C N ,   x ∗ b ≔ x j b n − j ,     j = 0 n = 0 again belongs to C N . So, we have the convolution operator T b : x �→ x ∗ b = b ∗ x , which is well defined and linear from C N → C N . Moreover, b , c ∈ C N . T b T c = T c T b = T b ∗ c , Relevant are the following identities: T e 0 = I (Identity operator), i.e. x ∗ e 0 = x , ∀ x ∈ C N . e n = e 1 ∗ e 1 ∗ . . . ∗ e 1 ( n terms ) , ∀ n ≥ 1 . We first recall when T b acts in the spaces ℓ p , 1 < p < ∞ . Werner Ricker Convolution in discrete Ces` aro spaces Doma´ nski Memorial Conference 4 / 12

Convolution p -multipliers for ℓ p , 1 ≤ p ≤ ∞ b ∈ C N is a (convolution) p -multiplier for ℓ p (write b ∈ M ( ℓ p )) if x ∗ b ∈ ℓ p , ∀ x ∈ ℓ p . Closed graph theorem ⇒ T b : ℓ p → ℓ p is continuous. Facts: [N.K. Nikolskii (1966) & ( p = 2) I. Schur (1917)] M ( ℓ 1 ) = M ( ℓ ∞ ) = ℓ 1 . M ( ℓ p ) = M ( ℓ p ′ ) If 1 ≤ p 1 < p 2 ≤ 2, then M ( ℓ p 1 ) � M ( ℓ p 2 ) . ℓ 1 � M ( ℓ p ) � � H ∞ , whenever 1 < p < ∞ (and M ( ℓ p ) � ℓ p ) . Schur: M ( ℓ 2 ) = � H ∞ , i.e., each element of M ( ℓ 2 ) is the sequence of Taylor coefficients of some function from H ∞ ( D ) . What if we replace ℓ p , 1 < p < ∞ , with ces ( p ) , 1 < p < ∞ ? Werner Ricker Convolution in discrete Ces` aro spaces Doma´ nski Memorial Conference 5 / 12

Convolution p -multipliers for ces ( p ) , 1 < p < ∞ Which elements b ∈ C N satisfy (for 1 < p < ∞ fixed) x ∗ b ∈ ces ( p ) , ∀ x ∈ ces ( p )? Equivalently, when does T b map ces ( p ) → ces ( p ) continuously? Answer is dramatically different than for ℓ p spaces. Proposition 1 [Curbera (2014)] Let 1 < p < ∞ and b ∈ C N . Then T b : C N → C N maps ces ( p ) into ces ( p ) , i.e. b ∈ M ( ces ( p )) , if and only if b ∈ ℓ 1 . In this case � ∞ � T b � op = � b � ℓ 1 = | b n | . n = 0 L p ≔ ( L ( ces ( p )) , �·� op ) is a unital, non-commutative Banach algebra (for composition of operators from ces ( p ) → ces ( p ) ). How does one identify the spectrum σ ( T b ) of T b ∈ L p ? Werner Ricker Convolution in discrete Ces` aro spaces Doma´ nski Memorial Conference 6 / 12

� T b : b ∈ M ( ces ( p )) = ℓ 1 � Proposition 1 implies M p ≔ is a unital, commutative, closed (proper) subalgebra of ( L p , �·� op ) . M p is isometrically isomorphic to the B-algebra A ≔ ( ℓ 1 , ∗ ) . Here, ∗ is convolution: the unit is e 0 = ( 1 , 0 , 0 , . . . ) . R ⊆ C ( D ) is the unital, commutative B-algebra of functions � ∞ b n z n , ϕ b ( z ) ≔ z ∈ D , n = 0 for all b ∈ ℓ 1 . We use pointwise operations of scalar multiplication, addition and product: the norm is � ϕ b � R ≔ � b � ℓ 1 . ϕ b ϕ c = ϕ b ∗ c , the map Unit of R is the constant function 1 . As b �→ ϕ b is an isometric B-algebra isomorphism of A onto R . For a B-algebra B with unit e and u ∈ B , define σ B ( u ) ≔ { λ ∈ C : u − λ e not invertible } and ρ B ( u ) ≔ C \ σ B ( u ) . Werner Ricker Convolution in discrete Ces` aro spaces Doma´ nski Memorial Conference 7 / 12

The B-algebras M p ≃ A ≃ R ( 1 < p < ∞ ) are all isometrically isomorphic. So the spectrum of T b satisfies ∀ b ∈ ℓ 1 . σ M p ( T b ) = σ A ( b ) = σ R ( ϕ b ) , ( ⋆ ) Maximal ideal space of R is D . Gelfand theory and ( ⋆ ) imply: σ M p ( T b ) = ϕ b ( D ) = � ϕ b ( z ) : | z | ≤ 1 � , b ∈ ℓ 1 ( 1 < p < ∞ ) . M p is a closed, unital subalgebra of the non-commutative unital B-algebra L p . Consequently, b ∈ ℓ 1 . σ L p ( T b ) ⊆ σ M p ( T b ) , If the more traditional notation σ ( T b ) is used for � � σ L p ( T b ) = λ ∈ C : ( T b − λ I ) not invertible in L p , the previous containment becomes σ ( T b ) ⊆ � ϕ b ( z ) : | z | ≤ 1 � , b ∈ ℓ 1 . How does one deduce this is an equality? Werner Ricker Convolution in discrete Ces` aro spaces Doma´ nski Memorial Conference 8 / 12

The shift-operator in ces ( p ) , 1 < p < ∞ The right-shift operator S p in ces ( p ) , given by S p (( x n ) ∞ n = 0 ) ≔ ( 0 , x 0 , x 1 , . . . ) , x ∈ ces ( p ) , � � � � � S p � op = 1 and the identity ( with e 1 = ( 0 , 1 , 0 , 0 , . . . )) satisfies S p ( x ) = T e 1 ( x ) = x ∗ e 1 , x ∈ ces ( p ) . This formula, in turn, implies that S n p = T e n = T e 1 ∗ ... e 1 , ( n times convolution. ) Via the isomorphism ces ( p ) ∗ ≃ d ( p ′ ) , the adjoint operator p : ces ( p ) ∗ → ces ( p ) ∗ can be identified with the left-shift operator S ∗ L p ′ in d ( p ′ ) given by u �→ L p ′ (( u n ) ∞ u ∈ d ( p ′ ) . n = 0 ) ≔ ( u 1 , u 2 , u 3 , . . . ) , Werner Ricker Convolution in discrete Ces` aro spaces Doma´ nski Memorial Conference 9 / 12

Spectrum of the operator T b in ces ( p ) Direct calculation: Every λ ∈ D is eigenvalue of S ∗ p x λ = ( 1 , λ, λ 2 , λ 3 , . . . ) ∈ d ( p ′ ) . with eigenvector ⇒ D ⊆ σ pt ( S ∗ p ) = σ ( S p ) . ⇒ For all b = � ∞ n = 0 b n e n ∈ ℓ 1 we have T b = � ∞ n = 0 b n T e n = � ∞ n = 0 b n S n (absolute conv. in �·� op ). b = � ∞ p ⇒ T ∗ n = 0 b n ( S ∗ p ) n (absolute conv. in �·� op ). p ) n x λ = λ n x λ , ∀ n ∈ N , implies T ∗ Since ( S ∗ b x λ = ϕ b ( λ ) x λ , we have ϕ b ( λ ) ∈ σ ( T ∗ b ) = σ ( T b ) , ∀ λ ∈ D . This implies the reverse inclusion to yield: Proposition 2. Let 1 < p < ∞ . Then, for each b ∈ ℓ 1 , we have � � σ ( T b ) = ϕ b ( D ) = ϕ b ( z ) : z ∈ D . Werner Ricker Convolution in discrete Ces` aro spaces Doma´ nski Memorial Conference 10 / 12