Basic properties of Toeplitz and Hankel operators in non-algebraic - PowerPoint PPT Presentation

Basic properties of Toeplitz and Hankel operators in non-algebraic setting Karol Leśnik Poznań University of Technology Paweł Domański Memorial Conference, Będlewo 2018

L 0 := L 0 ( T , m ) - the space of all measurable complex-valued, almost everywhere finite functions on T . quasi-Banach function space (q-B.f.s.) A quasi Banach space X ⊂ L 0 such that ◮ if f ∈ X , g ∈ L 0 and | g | � | f | -a.e., then g ∈ X and � g � X � � f � X , ◮ χ E ∈ X for each measurable set E ⊂ T (i.e. L ∞ ⊂ X ). If, in addition, X is a Banach space, we will use the abbreviation B.f.s. for X .

K¨ othe dual othe dual X ′ is defined as the space of functions For a q-B.f.s. X , its K¨ g ∈ L 0 satisfying �� � � g � X ′ = sup | f ( t ) g ( t ) | dm ( t ) : � f � X � 1 < ∞ . T Fatou property A B.f.s. X has the Fatou property iff X = X ′′ Order continuity f ∈ X is said to be an order continuous element if for each ( f n ) n ∈ N ⊂ X , 0 � f n � | f | with f n → 0 a.e., there holds � f n � X → 0. The subspace of order continuous elements of X is denoted by X o .

K¨ othe dual othe dual X ′ is defined as the space of functions For a q-B.f.s. X , its K¨ g ∈ L 0 satisfying �� � � g � X ′ = sup | f ( t ) g ( t ) | dm ( t ) : � f � X � 1 < ∞ . T Fatou property A B.f.s. X has the Fatou property iff X = X ′′ Order continuity f ∈ X is said to be an order continuous element if for each ( f n ) n ∈ N ⊂ X , 0 � f n � | f | with f n → 0 a.e., there holds � f n � X → 0. The subspace of order continuous elements of X is denoted by X o .

K¨ othe dual othe dual X ′ is defined as the space of functions For a q-B.f.s. X , its K¨ g ∈ L 0 satisfying �� � � g � X ′ = sup | f ( t ) g ( t ) | dm ( t ) : � f � X � 1 < ∞ . T Fatou property A B.f.s. X has the Fatou property iff X = X ′′ Order continuity f ∈ X is said to be an order continuous element if for each ( f n ) n ∈ N ⊂ X , 0 � f n � | f | with f n → 0 a.e., there holds � f n � X → 0. The subspace of order continuous elements of X is denoted by X o .

The distribution function µ f of f ∈ L 0 is given by µ f ( λ ) = m { t ∈ T : | f ( t ) | > λ } , λ � 0 . f , g ∈ L 0 are equimeasurable if µ f ≡ µ g The non-increasing rearrangement f ∗ of f ∈ L 0 is defined by f ∗ ( x ) = inf { λ : µ f ( λ ) � x } , x � 0 . Rearrangement invariant space A q-B.f.s. X is called rearrangement-invariant (r.i. q-B.f.s. for short) if for every pair of equimeasurable functions f , g ∈ L 0 f ∈ X ⇒ g ∈ X and � f � X = � g � X .

The distribution function µ f of f ∈ L 0 is given by µ f ( λ ) = m { t ∈ T : | f ( t ) | > λ } , λ � 0 . f , g ∈ L 0 are equimeasurable if µ f ≡ µ g The non-increasing rearrangement f ∗ of f ∈ L 0 is defined by f ∗ ( x ) = inf { λ : µ f ( λ ) � x } , x � 0 . Rearrangement invariant space A q-B.f.s. X is called rearrangement-invariant (r.i. q-B.f.s. for short) if for every pair of equimeasurable functions f , g ∈ L 0 f ∈ X ⇒ g ∈ X and � f � X = � g � X .

The distribution function µ f of f ∈ L 0 is given by µ f ( λ ) = m { t ∈ T : | f ( t ) | > λ } , λ � 0 . f , g ∈ L 0 are equimeasurable if µ f ≡ µ g The non-increasing rearrangement f ∗ of f ∈ L 0 is defined by f ∗ ( x ) = inf { λ : µ f ( λ ) � x } , x � 0 . Rearrangement invariant space A q-B.f.s. X is called rearrangement-invariant (r.i. q-B.f.s. for short) if for every pair of equimeasurable functions f , g ∈ L 0 f ∈ X ⇒ g ∈ X and � f � X = � g � X .

The distribution function µ f of f ∈ L 0 is given by µ f ( λ ) = m { t ∈ T : | f ( t ) | > λ } , λ � 0 . f , g ∈ L 0 are equimeasurable if µ f ≡ µ g The non-increasing rearrangement f ∗ of f ∈ L 0 is defined by f ∗ ( x ) = inf { λ : µ f ( λ ) � x } , x � 0 . Rearrangement invariant space A q-B.f.s. X is called rearrangement-invariant (r.i. q-B.f.s. for short) if for every pair of equimeasurable functions f , g ∈ L 0 f ∈ X ⇒ g ∈ X and � f � X = � g � X .

Dilation operator Let X be a r.i. q-B.f. space. For each s ∈ R + the dilation operator D s is defined as � f ( e i θ s ) , θ s ∈ [ 0 , 2 π ) , ( D s f )( e i θ ) = θ ∈ [ 0 , 2 π ) . 0 , θ s �∈ [ 0 , 2 π ) , Boyd indices The limits log � D 1 / s � X → X log � D 1 / s � X → X α X = lim , β X = lim log s log s s → 0 + s →∞ are called the lower and upper Boyd indices of X , respectively. We say that the Boyd indices are nontrivial if α X , β X ∈ ( 0 , 1 ) .

Dilation operator Let X be a r.i. q-B.f. space. For each s ∈ R + the dilation operator D s is defined as � f ( e i θ s ) , θ s ∈ [ 0 , 2 π ) , ( D s f )( e i θ ) = θ ∈ [ 0 , 2 π ) . 0 , θ s �∈ [ 0 , 2 π ) , Boyd indices The limits log � D 1 / s � X → X log � D 1 / s � X → X α X = lim , β X = lim log s log s s → 0 + s →∞ are called the lower and upper Boyd indices of X , respectively. We say that the Boyd indices are nontrivial if α X , β X ∈ ( 0 , 1 ) .

Pointwise multipliers Let X and Y be B.f.s. The space of pointwise multipliers M ( X , Y ) is defined by M ( X , Y ) = { f ∈ L 0 : fg ∈ Y for all g ∈ X } (1) with the norm � f � M ( X , Y ) = sup {� fg � Y : g ∈ X , � g � X � 1 } . (2) Each f ∈ M ( X , Y ) is the symbol of multiplication operator M f : g �→ fg , M f : X → Y .

Pointwise multipliers Let X and Y be B.f.s. The space of pointwise multipliers M ( X , Y ) is defined by M ( X , Y ) = { f ∈ L 0 : fg ∈ Y for all g ∈ X } (1) with the norm � f � M ( X , Y ) = sup {� fg � Y : g ∈ X , � g � X � 1 } . (2) Each f ∈ M ( X , Y ) is the symbol of multiplication operator M f : g �→ fg , M f : X → Y .

Examples � E , L 1 � ≡ E ′ - K¨ ◮ M othe dual of E . ◮ If 1 � q < p < ∞ , 1 / r = 1 / q − 1 / p , then M ( L p , L q ) ≡ L r . ◮ Let 1 � p < q < ∞ , then M ( L p , L q ) = { 0 } . ◮ M ( L ϕ 1 , L ϕ ) = L ϕ 2 , where ϕ 2 ( u ) = sup { ϕ ( uv ) − ϕ 1 ( v ) } . v > 0

Examples � E , L 1 � ≡ E ′ - K¨ ◮ M othe dual of E . ◮ If 1 � q < p < ∞ , 1 / r = 1 / q − 1 / p , then M ( L p , L q ) ≡ L r . ◮ Let 1 � p < q < ∞ , then M ( L p , L q ) = { 0 } . ◮ M ( L ϕ 1 , L ϕ ) = L ϕ 2 , where ϕ 2 ( u ) = sup { ϕ ( uv ) − ϕ 1 ( v ) } . v > 0

Examples � E , L 1 � ≡ E ′ - K¨ ◮ M othe dual of E . ◮ If 1 � q < p < ∞ , 1 / r = 1 / q − 1 / p , then M ( L p , L q ) ≡ L r . ◮ Let 1 � p < q < ∞ , then M ( L p , L q ) = { 0 } . ◮ M ( L ϕ 1 , L ϕ ) = L ϕ 2 , where ϕ 2 ( u ) = sup { ϕ ( uv ) − ϕ 1 ( v ) } . v > 0

Examples � E , L 1 � ≡ E ′ - K¨ ◮ M othe dual of E . ◮ If 1 � q < p < ∞ , 1 / r = 1 / q − 1 / p , then M ( L p , L q ) ≡ L r . ◮ Let 1 � p < q < ∞ , then M ( L p , L q ) = { 0 } . ◮ M ( L ϕ 1 , L ϕ ) = L ϕ 2 , where ϕ 2 ( u ) = sup { ϕ ( uv ) − ϕ 1 ( v ) } . v > 0

Pointwise product For a given two B.f. spaces X and Y we define pointwise product space X ⊙ Y as X ⊙ Y = { xy : x ∈ X and y ∈ Y } , (3) with the quasi-norm �·� X ⊙ Y given by the formula � z � X ⊙ Y = inf {� x � X � y � Y : z = xy , x ∈ X and y ∈ Y } . (4)

For n ∈ Z and t ∈ T , let χ n ( t ) := t n . The Fourier coefficients of a function f ∈ L 1 are given by � f ( n ) := � f , χ n � , n ∈ Z , where � � f , g � := f ( t ) g ( t ) dm ( t ) . T Hardy spaces Let X be a r.i. q-B.f.s. such that X ⊂ L 1 . Hardy space H [ X ] is defined as � � f ∈ X : � H [ X ] := f ( n ) = 0 for all n < 0 , with the norm inherited from X

For n ∈ Z and t ∈ T , let χ n ( t ) := t n . The Fourier coefficients of a function f ∈ L 1 are given by � f ( n ) := � f , χ n � , n ∈ Z , where � � f , g � := f ( t ) g ( t ) dm ( t ) . T Hardy spaces Let X be a r.i. q-B.f.s. such that X ⊂ L 1 . Hardy space H [ X ] is defined as � � f ∈ X : � H [ X ] := f ( n ) = 0 for all n < 0 , with the norm inherited from X

Riesz projection P : L 2 ( T ) → H 2 ( T ) ( P is bounded on r.i. Y when it has nontrivial Boyd indices) � ∞ � ∞ f ( n ) t n �→ � � f ( n ) t n . P : n = −∞ k = 0 Toeplitz operator a ∈ L ∞ - algebraic case a ∈ M ( X , Y ) - non-algebraic case T a : H 2 → H 2 T a : f �→ PM a f T a : H [ X ] → H [ Y ] Flip operator J : L 1 → L 1 ( J is an isometry on r.i. X ) Jf ( t ) = t − 1 f ( t − 1 ) Hankel operator a ∈ L ∞ - algebraic case a ∈ M ( X , Y ) - non-algebraic case H a : H 2 → H 2 H a : f �→ PM a Jf H a : H [ X ] → H [ Y ]

Riesz projection P : L 2 ( T ) → H 2 ( T ) ( P is bounded on r.i. Y when it has nontrivial Boyd indices) � ∞ � ∞ f ( n ) t n �→ � � f ( n ) t n . P : n = −∞ k = 0 Toeplitz operator a ∈ L ∞ - algebraic case a ∈ M ( X , Y ) - non-algebraic case T a : H 2 → H 2 T a : f �→ PM a f T a : H [ X ] → H [ Y ] Flip operator J : L 1 → L 1 ( J is an isometry on r.i. X ) Jf ( t ) = t − 1 f ( t − 1 ) Hankel operator a ∈ L ∞ - algebraic case a ∈ M ( X , Y ) - non-algebraic case H a : H 2 → H 2 H a : f �→ PM a Jf H a : H [ X ] → H [ Y ]