SLIDE 1 Basic properties of Toeplitz and Hankel operators in non-algebraic setting
Karol Leśnik
Poznań University of Technology
Paweł Domański Memorial Conference, Będlewo 2018
SLIDE 2 L0 := L0(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 ⊂ L0 such that
◮ if f ∈ X, g ∈ L0 and |g| |f | -a.e., then g ∈ X and gX f X, ◮ χE ∈ 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.
SLIDE 3 K¨
For a q-B.f.s. X, its K¨
- the dual X ′ is defined as the space of functions
g ∈ L0 satisfying gX ′ = sup
|f (t)g(t)| dm(t) : f X 1
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 (fn)n∈N ⊂ X, 0 fn |f | with fn → 0 a.e., there holds fnX → 0. The subspace of order continuous elements of X is denoted by Xo.
SLIDE 4 K¨
For a q-B.f.s. X, its K¨
- the dual X ′ is defined as the space of functions
g ∈ L0 satisfying gX ′ = sup
|f (t)g(t)| dm(t) : f X 1
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 (fn)n∈N ⊂ X, 0 fn |f | with fn → 0 a.e., there holds fnX → 0. The subspace of order continuous elements of X is denoted by Xo.
SLIDE 5 K¨
For a q-B.f.s. X, its K¨
- the dual X ′ is defined as the space of functions
g ∈ L0 satisfying gX ′ = sup
|f (t)g(t)| dm(t) : f X 1
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 (fn)n∈N ⊂ X, 0 fn |f | with fn → 0 a.e., there holds fnX → 0. The subspace of order continuous elements of X is denoted by Xo.
SLIDE 6
The distribution function µf of f ∈ L0 is given by µf (λ) = m{t ∈ T : |f (t)| > λ}, λ 0. f , g ∈ L0 are equimeasurable if µf ≡ µg The non-increasing rearrangement f ∗ of f ∈ L0 is defined by f ∗(x) = inf{λ : µf (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 ∈ L0 f ∈ X ⇒ g ∈ X and f X = gX.
SLIDE 7
The distribution function µf of f ∈ L0 is given by µf (λ) = m{t ∈ T : |f (t)| > λ}, λ 0. f , g ∈ L0 are equimeasurable if µf ≡ µg The non-increasing rearrangement f ∗ of f ∈ L0 is defined by f ∗(x) = inf{λ : µf (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 ∈ L0 f ∈ X ⇒ g ∈ X and f X = gX.
SLIDE 8
The distribution function µf of f ∈ L0 is given by µf (λ) = m{t ∈ T : |f (t)| > λ}, λ 0. f , g ∈ L0 are equimeasurable if µf ≡ µg The non-increasing rearrangement f ∗ of f ∈ L0 is defined by f ∗(x) = inf{λ : µf (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 ∈ L0 f ∈ X ⇒ g ∈ X and f X = gX.
SLIDE 9
The distribution function µf of f ∈ L0 is given by µf (λ) = m{t ∈ T : |f (t)| > λ}, λ 0. f , g ∈ L0 are equimeasurable if µf ≡ µg The non-increasing rearrangement f ∗ of f ∈ L0 is defined by f ∗(x) = inf{λ : µf (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 ∈ L0 f ∈ X ⇒ g ∈ X and f X = gX.
SLIDE 10 Dilation operator
Let X be a r.i. q-B.f. space. For each s ∈ R+ the dilation operator Ds is defined as (Dsf )(eiθ) =
θs ∈ [0, 2π), 0, θs ∈ [0, 2π), θ ∈ [0, 2π).
Boyd indices
The limits αX = lim
s→0+
log D1/sX→X log s , βX = lim
s→∞
log D1/sX→X log s are called the lower and upper Boyd indices of X, respectively. We say that the Boyd indices are nontrivial if αX, βX ∈ (0, 1).
SLIDE 11 Dilation operator
Let X be a r.i. q-B.f. space. For each s ∈ R+ the dilation operator Ds is defined as (Dsf )(eiθ) =
θs ∈ [0, 2π), 0, θs ∈ [0, 2π), θ ∈ [0, 2π).
Boyd indices
The limits αX = lim
s→0+
log D1/sX→X log s , βX = lim
s→∞
log D1/sX→X log s are called the lower and upper Boyd indices of X, respectively. We say that the Boyd indices are nontrivial if αX, βX ∈ (0, 1).
SLIDE 12
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 ∈ L0 : fg ∈ Y for all g ∈ X} (1) with the norm f M(X,Y ) = sup{fgY : g ∈ X, gX 1}. (2) Each f ∈ M(X, Y ) is the symbol of multiplication operator Mf : g → fg, Mf : X → Y .
SLIDE 13
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 ∈ L0 : fg ∈ Y for all g ∈ X} (1) with the norm f M(X,Y ) = sup{fgY : g ∈ X, gX 1}. (2) Each f ∈ M(X, Y ) is the symbol of multiplication operator Mf : g → fg, Mf : X → Y .
SLIDE 14 Examples
◮ M
≡ E ′ - K¨
◮ If 1 q < p < ∞, 1/r = 1/q − 1/p, then
M(Lp, Lq) ≡ Lr.
◮ Let 1 p < q < ∞, then M(Lp, Lq) = {0}. ◮
M(Lϕ1, Lϕ) = Lϕ2, where ϕ2 (u) = sup
v>0
{ϕ (uv) − ϕ1 (v)} .
SLIDE 15 Examples
◮ M
≡ E ′ - K¨
◮ If 1 q < p < ∞, 1/r = 1/q − 1/p, then
M(Lp, Lq) ≡ Lr.
◮ Let 1 p < q < ∞, then M(Lp, Lq) = {0}. ◮
M(Lϕ1, Lϕ) = Lϕ2, where ϕ2 (u) = sup
v>0
{ϕ (uv) − ϕ1 (v)} .
SLIDE 16 Examples
◮ M
≡ E ′ - K¨
◮ If 1 q < p < ∞, 1/r = 1/q − 1/p, then
M(Lp, Lq) ≡ Lr.
◮ Let 1 p < q < ∞, then M(Lp, Lq) = {0}. ◮
M(Lϕ1, Lϕ) = Lϕ2, where ϕ2 (u) = sup
v>0
{ϕ (uv) − ϕ1 (v)} .
SLIDE 17 Examples
◮ M
≡ E ′ - K¨
◮ If 1 q < p < ∞, 1/r = 1/q − 1/p, then
M(Lp, Lq) ≡ Lr.
◮ Let 1 p < q < ∞, then M(Lp, Lq) = {0}. ◮
M(Lϕ1, Lϕ) = Lϕ2, where ϕ2 (u) = sup
v>0
{ϕ (uv) − ϕ1 (v)} .
SLIDE 18
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 zX⊙Y = inf {xX yY : z = xy, x ∈ X and y ∈ Y } . (4)
SLIDE 19 For n ∈ Z and t ∈ T, let χn(t) := tn. The Fourier coefficients of a function f ∈ L1 are given by
n ∈ Z, where f , g :=
f (t)g(t) dm(t).
Hardy spaces
Let X be a r.i. q-B.f.s. such that X ⊂ L1. Hardy space H[X] is defined as H[X] :=
f (n) = 0 for all n < 0
with the norm inherited from X
SLIDE 20 For n ∈ Z and t ∈ T, let χn(t) := tn. The Fourier coefficients of a function f ∈ L1 are given by
n ∈ Z, where f , g :=
f (t)g(t) dm(t).
Hardy spaces
Let X be a r.i. q-B.f.s. such that X ⊂ L1. Hardy space H[X] is defined as H[X] :=
f (n) = 0 for all n < 0
with the norm inherited from X
SLIDE 21 Riesz projection
P : L2(T) → H2(T) (P is bounded on r.i. Y when it has nontrivial Boyd indices) P :
∞
∞
Toeplitz operator
a ∈ L∞ - algebraic case a ∈ M(X, Y ) - non-algebraic case Ta : f → PMaf Ta : H2 → H2 Ta : H[X] → H[Y ]
Flip operator
J : L1 → L1 (J is an isometry on r.i. X) Jf (t) = t−1f (t−1)
Hankel operator
a ∈ L∞
- algebraic case a ∈ M(X, Y ) - non-algebraic case
Ha : f → PMaJf Ha : H2 → H2 Ha : H[X] → H[Y ]
SLIDE 22 Riesz projection
P : L2(T) → H2(T) (P is bounded on r.i. Y when it has nontrivial Boyd indices) P :
∞
∞
Toeplitz operator
a ∈ L∞ - algebraic case a ∈ M(X, Y ) - non-algebraic case Ta : f → PMaf Ta : H2 → H2 Ta : H[X] → H[Y ]
Flip operator
J : L1 → L1 (J is an isometry on r.i. X) Jf (t) = t−1f (t−1)
Hankel operator
a ∈ L∞
- algebraic case a ∈ M(X, Y ) - non-algebraic case
Ha : f → PMaJf Ha : H2 → H2 Ha : H[X] → H[Y ]
SLIDE 23 Riesz projection
P : L2(T) → H2(T) (P is bounded on r.i. Y when it has nontrivial Boyd indices) P :
∞
∞
Toeplitz operator
a ∈ L∞ - algebraic case a ∈ M(X, Y ) - non-algebraic case Ta : f → PMaf Ta : H2 → H2 Ta : H[X] → H[Y ]
Flip operator
J : L1 → L1 (J is an isometry on r.i. X) Jf (t) = t−1f (t−1)
Hankel operator
a ∈ L∞
- algebraic case a ∈ M(X, Y ) - non-algebraic case
Ha : f → PMaJf Ha : H2 → H2 Ha : H[X] → H[Y ]
SLIDE 24 Riesz projection
P : L2(T) → H2(T) (P is bounded on r.i. Y when it has nontrivial Boyd indices) P :
∞
∞
Toeplitz operator
a ∈ L∞ - algebraic case a ∈ M(X, Y ) - non-algebraic case Ta : f → PMaf Ta : H2 → H2 Ta : H[X] → H[Y ]
Flip operator
J : L1 → L1 (J is an isometry on r.i. X) Jf (t) = t−1f (t−1)
Hankel operator
a ∈ L∞
- algebraic case a ∈ M(X, Y ) - non-algebraic case
Ha : f → PMaJf Ha : H2 → H2 Ha : H[X] → H[Y ]
SLIDE 25 Riesz projection
P : L2(T) → H2(T) (P is bounded on r.i. Y when it has nontrivial Boyd indices) P :
∞
∞
Toeplitz operator
a ∈ L∞ - algebraic case a ∈ M(X, Y ) - non-algebraic case Ta : f → PMaf Ta : H2 → H2 Ta : H[X] → H[Y ]
Flip operator
J : L1 → L1 (J is an isometry on r.i. X) Jf (t) = t−1f (t−1)
Hankel operator
a ∈ L∞
- algebraic case a ∈ M(X, Y ) - non-algebraic case
Ha : f → PMaJf Ha : H2 → H2 Ha : H[X] → H[Y ]
SLIDE 26 Riesz projection
P : L2(T) → H2(T) (P is bounded on r.i. Y when it has nontrivial Boyd indices) P :
∞
∞
Toeplitz operator
a ∈ L∞ - algebraic case a ∈ M(X, Y ) - non-algebraic case Ta : f → PMaf Ta : H2 → H2 Ta : H[X] → H[Y ]
Flip operator
J : L1 → L1 (J is an isometry on r.i. X) Jf (t) = t−1f (t−1)
Hankel operator
a ∈ L∞
- algebraic case a ∈ M(X, Y ) - non-algebraic case
Ha : f → PMaJf Ha : H2 → H2 Ha : H[X] → H[Y ]
SLIDE 27 Riesz projection
P : L2(T) → H2(T) (P is bounded on r.i. Y when it has nontrivial Boyd indices) P :
∞
∞
Toeplitz operator
a ∈ L∞ - algebraic case a ∈ M(X, Y ) - non-algebraic case Ta : f → PMaf Ta : H2 → H2 Ta : H[X] → H[Y ]
Flip operator
J : L1 → L1 (J is an isometry on r.i. X) Jf (t) = t−1f (t−1)
Hankel operator
a ∈ L∞
- algebraic case a ∈ M(X, Y ) - non-algebraic case
Ha : f → PMaJf Ha : H2 → H2 Ha : H[X] → H[Y ]
SLIDE 28 Toeplitz matrix
(an)∞
n=−∞ - sequence of complex numbers
a0 a−1 a−2 . . . a1 a0 a−1 . . . a2 a1 a0 . . . . . . . . . . . . ...
Hankel matrix
(an)∞
n=1 - sequence of complex numbers
a1 a2 a3 . . . a2 a3 a4 . . . a3 a4 a5 . . . . . . . . . . . . ...
SLIDE 29 Toeplitz matrix
(an)∞
n=−∞ - sequence of complex numbers
a0 a−1 a−2 . . . a1 a0 a−1 . . . a2 a1 a0 . . . . . . . . . . . . ...
Hankel matrix
(an)∞
n=1 - sequence of complex numbers
a1 a2 a3 . . . a2 a3 a4 . . . a3 a4 a5 . . . . . . . . . . . . ...
SLIDE 30
General Nehari Theorem
Let X, Y be two r.i. B.f. spaces, such that X is separable, X ⊂ Y , Y has nontrivial Boyd indices and one of the following conditions holds: i) X ⊙ M(X, Y ) = Y and X, Y ∈ (FP), ii) βX < αY . If a continuous linear operator A : H[X] → H[Y ] is such that Aχj, χk = ak+j+1 for j, k 0 and some sequence (ak)k>0, then there exists a ∈ M(X, Y ) such that ˆ a(n) = an for n > 0 and A = Ha, i.e. A : f → PaJf . Moreover, c distM(X,Y )(a, H[M(X, Y )]) HaH[X]→H[Y ] PY →Y distM(X,Y )(a, H[M(X, Y )]).
SLIDE 31
General Nehari Theorem
Let X, Y be two r.i. B.f. spaces, such that X is separable, X ⊂ Y , Y has nontrivial Boyd indices and one of the following conditions holds: i) X ⊙ M(X, Y ) = Y and X, Y ∈ (FP), ii) βX < αY . If a continuous linear operator A : H[X] → H[Y ] is such that Aχj, χk = ak+j+1 for j, k 0 and some sequence (ak)k>0, then there exists a ∈ M(X, Y ) such that ˆ a(n) = an for n > 0 and A = Ha, i.e. A : f → PaJf . Moreover, c distM(X,Y )(a, H[M(X, Y )]) HaH[X]→H[Y ] PY →Y distM(X,Y )(a, H[M(X, Y )]).
SLIDE 32
General Nehari Theorem
Let X, Y be two r.i. B.f. spaces, such that X is separable, X ⊂ Y , Y has nontrivial Boyd indices and one of the following conditions holds: i) X ⊙ M(X, Y ) = Y and X, Y ∈ (FP), ii) βX < αY . If a continuous linear operator A : H[X] → H[Y ] is such that Aχj, χk = ak+j+1 for j, k 0 and some sequence (ak)k>0, then there exists a ∈ M(X, Y ) such that ˆ a(n) = an for n > 0 and A = Ha, i.e. A : f → PaJf . Moreover, c distM(X,Y )(a, H[M(X, Y )]) HaH[X]→H[Y ] PY →Y distM(X,Y )(a, H[M(X, Y )]).
SLIDE 33
General Nehari Theorem
Let X, Y be two r.i. B.f. spaces, such that X is separable, X ⊂ Y , Y has nontrivial Boyd indices and one of the following conditions holds: i) X ⊙ M(X, Y ) = Y and X, Y ∈ (FP), ii) βX < αY . If a continuous linear operator A : H[X] → H[Y ] is such that Aχj, χk = ak+j+1 for j, k 0 and some sequence (ak)k>0, then there exists a ∈ M(X, Y ) such that ˆ a(n) = an for n > 0 and A = Ha, i.e. A : f → PaJf . Moreover, c distM(X,Y )(a, H[M(X, Y )]) HaH[X]→H[Y ] PY →Y distM(X,Y )(a, H[M(X, Y )]).
SLIDE 34
General Nehari Theorem
Let X, Y be two r.i. B.f. spaces, such that X is separable, X ⊂ Y , Y has nontrivial Boyd indices and one of the following conditions holds: i) X ⊙ M(X, Y ) = Y and X, Y ∈ (FP), ii) βX < αY . If a continuous linear operator A : H[X] → H[Y ] is such that Aχj, χk = ak+j+1 for j, k 0 and some sequence (ak)k>0, then there exists a ∈ M(X, Y ) such that ˆ a(n) = an for n > 0 and A = Ha, i.e. A : f → PaJf . Moreover, c distM(X,Y )(a, H[M(X, Y )]) HaH[X]→H[Y ] PY →Y distM(X,Y )(a, H[M(X, Y )]).
SLIDE 35 Measuring compactness/noncompactness
◮ Given a set A in a Banach space X, its Kuratowski measure of
noncompactness α(A) is defined as α(A) = inf{δ > 0 : A ⊂
N
Bk, diam(Bk) δ and N < ∞}.
◮ For a bounded operator T : X → Y , its Kuratowski measure of
noncompactness α(T) is just the measure of noncompactness of the set T(B(X)) in Y , where B(X) is the unit ball of X, i.e. α(T) := α(T(B(X))).
◮ For T : X → Y the essential norm is given by
Te := inf{T − K : K : X → Y is compact}
SLIDE 36 Measuring compactness/noncompactness
◮ Given a set A in a Banach space X, its Kuratowski measure of
noncompactness α(A) is defined as α(A) = inf{δ > 0 : A ⊂
N
Bk, diam(Bk) δ and N < ∞}.
◮ For a bounded operator T : X → Y , its Kuratowski measure of
noncompactness α(T) is just the measure of noncompactness of the set T(B(X)) in Y , where B(X) is the unit ball of X, i.e. α(T) := α(T(B(X))).
◮ For T : X → Y the essential norm is given by
Te := inf{T − K : K : X → Y is compact}
SLIDE 37 Measuring compactness/noncompactness
◮ Given a set A in a Banach space X, its Kuratowski measure of
noncompactness α(A) is defined as α(A) = inf{δ > 0 : A ⊂
N
Bk, diam(Bk) δ and N < ∞}.
◮ For a bounded operator T : X → Y , its Kuratowski measure of
noncompactness α(T) is just the measure of noncompactness of the set T(B(X)) in Y , where B(X) is the unit ball of X, i.e. α(T) := α(T(B(X))).
◮ For T : X → Y the essential norm is given by
Te := inf{T − K : K : X → Y is compact}
SLIDE 38 Noncompactness of Toeplitz operators
Theorem
Let X, Y be r.i. B.f. spaces such that X ⊂ Y and Y has nontrivial Boyd
- indices. Suppose a ∈ M(X, Y ). Then the Kuratowski measure of
noncompactness of the Toeplitz operator Ta satisfies α(Ta) (γβ)−1 max
n∈Z |
a(n)|, (5) for γ and β being constants of inclusions Y ⊂γ L1 and L∞ ⊂β X.
SLIDE 39
Compactness of Hankel operators
Hartman Theorem
A Hankel operator Ha : H2 → H2 is compact if and only if a ∈ H∞ + C.
Theorem
Let X, Y be r.i. B.f. spaces with the Fatou property such that X ⊂ Y (X = Y ) and Y has nontrivial Boyd indices. For a ∈ M := M(X, Y ) and the Hankel operator Ha : H[X] → H[Y ] there holds: (a) Hae PY →Y distM(a, Mo + H[M])). In particular, if a ∈ Mo + H[M]) then Ha is compact. (b) If spaces X, Y satisfy assumptions of the General Nehari Theorem and X is reflexive, then also Hae c distM(a, Mo + H[M])), for some constant c > 0 depending only on spaces X, Y .
SLIDE 40
Compactness of Hankel operators
Hartman Theorem
A Hankel operator Ha : H2 → H2 is compact if and only if a ∈ H∞ + C.
Theorem
Let X, Y be r.i. B.f. spaces with the Fatou property such that X ⊂ Y (X = Y ) and Y has nontrivial Boyd indices. For a ∈ M := M(X, Y ) and the Hankel operator Ha : H[X] → H[Y ] there holds: (a) Hae PY →Y distM(a, Mo + H[M])). In particular, if a ∈ Mo + H[M]) then Ha is compact. (b) If spaces X, Y satisfy assumptions of the General Nehari Theorem and X is reflexive, then also Hae c distM(a, Mo + H[M])), for some constant c > 0 depending only on spaces X, Y .
SLIDE 41
Compactness of Hankel operators
Hartman Theorem
A Hankel operator Ha : H2 → H2 is compact if and only if a ∈ H∞ + C.
Theorem
Let X, Y be r.i. B.f. spaces with the Fatou property such that X ⊂ Y (X = Y ) and Y has nontrivial Boyd indices. For a ∈ M := M(X, Y ) and the Hankel operator Ha : H[X] → H[Y ] there holds: (a) Hae PY →Y distM(a, Mo + H[M])). In particular, if a ∈ Mo + H[M]) then Ha is compact. (b) If spaces X, Y satisfy assumptions of the General Nehari Theorem and X is reflexive, then also Hae c distM(a, Mo + H[M])), for some constant c > 0 depending only on spaces X, Y .
SLIDE 42
Compactness of Hankel operators
Hartman Theorem
A Hankel operator Ha : H2 → H2 is compact if and only if a ∈ H∞ + C.
Theorem
Let X, Y be r.i. B.f. spaces with the Fatou property such that X ⊂ Y (X = Y ) and Y has nontrivial Boyd indices. For a ∈ M := M(X, Y ) and the Hankel operator Ha : H[X] → H[Y ] there holds: (a) Hae PY →Y distM(a, Mo + H[M])). In particular, if a ∈ Mo + H[M]) then Ha is compact. (b) If spaces X, Y satisfy assumptions of the General Nehari Theorem and X is reflexive, then also Hae c distM(a, Mo + H[M])), for some constant c > 0 depending only on spaces X, Y .
SLIDE 43 Compactness of Hankel operators
Corollary
(a) X, Y - r.i. B.f. spaces, X ⊂ Y and Y has nontrivial Boyd indices. If M(X, Y ) is separable, then for each a ∈ M(X, Y ) the Hankel
- perator Ha : H[X] → H[Y ] is compact.
(b) If 1 < q < p < ∞ then all Hankel operators from Hp to Hq are compact. (c) Let 1 < p2 < p1 < ∞.
(c1) If 1 q1 < q2 ∞ then each Hankel operator Ha : Hp1,q1 → Hp2,q2 is compact. (c2) If 1 q2 q1 < ∞, then a Hankel operator Ha : Hp1,q1 → Hp2,q2 is compact if and only if Pa ∈ H[Lp,∞
p = 1 p2 − 1 p1 .
SLIDE 44 Compactness of Hankel operators
Corollary
(a) X, Y - r.i. B.f. spaces, X ⊂ Y and Y has nontrivial Boyd indices. If M(X, Y ) is separable, then for each a ∈ M(X, Y ) the Hankel
- perator Ha : H[X] → H[Y ] is compact.
(b) If 1 < q < p < ∞ then all Hankel operators from Hp to Hq are compact. (c) Let 1 < p2 < p1 < ∞.
(c1) If 1 q1 < q2 ∞ then each Hankel operator Ha : Hp1,q1 → Hp2,q2 is compact. (c2) If 1 q2 q1 < ∞, then a Hankel operator Ha : Hp1,q1 → Hp2,q2 is compact if and only if Pa ∈ H[Lp,∞
p = 1 p2 − 1 p1 .
SLIDE 45 Compactness of Hankel operators
Corollary
(a) X, Y - r.i. B.f. spaces, X ⊂ Y and Y has nontrivial Boyd indices. If M(X, Y ) is separable, then for each a ∈ M(X, Y ) the Hankel
- perator Ha : H[X] → H[Y ] is compact.
(b) If 1 < q < p < ∞ then all Hankel operators from Hp to Hq are compact. (c) Let 1 < p2 < p1 < ∞.
(c1) If 1 q1 < q2 ∞ then each Hankel operator Ha : Hp1,q1 → Hp2,q2 is compact. (c2) If 1 q2 q1 < ∞, then a Hankel operator Ha : Hp1,q1 → Hp2,q2 is compact if and only if Pa ∈ H[Lp,∞
p = 1 p2 − 1 p1 .
SLIDE 46 Compactness of Hankel operators
Corollary
(a) X, Y - r.i. B.f. spaces, X ⊂ Y and Y has nontrivial Boyd indices. If M(X, Y ) is separable, then for each a ∈ M(X, Y ) the Hankel
- perator Ha : H[X] → H[Y ] is compact.
(b) If 1 < q < p < ∞ then all Hankel operators from Hp to Hq are compact. (c) Let 1 < p2 < p1 < ∞.
(c1) If 1 q1 < q2 ∞ then each Hankel operator Ha : Hp1,q1 → Hp2,q2 is compact. (c2) If 1 q2 q1 < ∞, then a Hankel operator Ha : Hp1,q1 → Hp2,q2 is compact if and only if Pa ∈ H[Lp,∞
p = 1 p2 − 1 p1 .
SLIDE 47
Commutators - does it make sense?
Commutator
If T, S : X → X then the commutator is defined as [T, S] = TS − ST.
Commutator of Toeplitz operators
Let Ta, Tb be two Toeplitz operators. [Ta, Tb] = TaTb − TbTa.
Semi-commutator of Toeplitz operators
Let Ta, Tb be two Toeplitz operators. [Ta, Tb) = TaTb − Tab.
SLIDE 48
Commutators - does it make sense?
Commutator
If T, S : X → X then the commutator is defined as [T, S] = TS − ST.
Commutator of Toeplitz operators
Let Ta, Tb be two Toeplitz operators. [Ta, Tb] = TaTb − TbTa.
Semi-commutator of Toeplitz operators
Let Ta, Tb be two Toeplitz operators. [Ta, Tb) = TaTb − Tab.
SLIDE 49
Commutators - does it make sense?
Commutator
If T, S : X → X then the commutator is defined as [T, S] = TS − ST.
Commutator of Toeplitz operators
Let Ta, Tb be two Toeplitz operators. [Ta, Tb] = TaTb − TbTa.
Semi-commutator of Toeplitz operators
Let Ta, Tb be two Toeplitz operators. [Ta, Tb) = TaTb − Tab.
SLIDE 50 Commutators - example
◮ Let 1 < q < r < p < ∞ and a ∈ M(Lp, Lr), b ∈ M(Lr, Lq). ◮ Then a ∈ Lh0 and b ∈ Lh1, where 1/h0 = 1/r − 1/p and
1/h1 = 1/q − 1/r.
◮ Also a ∈ M(Ls, Lq) = Lh0 and b ∈ M(Lp, Ls) = Lh1, where
1/s − 1/p = 1/q − 1/r. Hp Hr Hs Hq
Ta Tb Tab Tb Ta
(6)
SLIDE 51 Commutators - example
◮ Let 1 < q < r < p < ∞ and a ∈ M(Lp, Lr), b ∈ M(Lr, Lq). ◮ Then a ∈ Lh0 and b ∈ Lh1, where 1/h0 = 1/r − 1/p and
1/h1 = 1/q − 1/r.
◮ Also a ∈ M(Ls, Lq) = Lh0 and b ∈ M(Lp, Ls) = Lh1, where
1/s − 1/p = 1/q − 1/r. Hp Hr Hs Hq
Ta Tb Tab Tb Ta
(6)
SLIDE 52 Commutators - example
◮ Let 1 < q < r < p < ∞ and a ∈ M(Lp, Lr), b ∈ M(Lr, Lq). ◮ Then a ∈ Lh0 and b ∈ Lh1, where 1/h0 = 1/r − 1/p and
1/h1 = 1/q − 1/r.
◮ Also a ∈ M(Ls, Lq) = Lh0 and b ∈ M(Lp, Ls) = Lh1, where
1/s − 1/p = 1/q − 1/r. Hp Hr Hs Hq
Ta Tb Tab Tb Ta
(6)
SLIDE 53 Commutators - example
◮ Let 1 < q < r < p < ∞ and a ∈ M(Lp, Lr), b ∈ M(Lr, Lq). ◮ Then a ∈ Lh0 and b ∈ Lh1, where 1/h0 = 1/r − 1/p and
1/h1 = 1/q − 1/r.
◮ Also a ∈ M(Ls, Lq) = Lh0 and b ∈ M(Lp, Ls) = Lh1, where
1/s − 1/p = 1/q − 1/r. Hp Hr Hs Hq
Ta Tb Tab Tb Ta
(6)
SLIDE 54 Commutators - general situation
H[X] H[Y ] H[W =?] H[Z]
Ta Tb Tb Ta
(7)
Theorem
Let X ⊂ Y ⊂ Z be r.i. B.f. spaces. If Ma : X → Y and Mb : Y → Z and W = X ⊙ M(Y , Z) then the following diagram commutes X Y W Z
Ma Mb Mb Ma
(8)
SLIDE 55 Commutators - general situation
H[X] H[Y ] H[W =?] H[Z]
Ta Tb Tb Ta
(7)
Theorem
Let X ⊂ Y ⊂ Z be r.i. B.f. spaces. If Ma : X → Y and Mb : Y → Z and W = X ⊙ M(Y , Z) then the following diagram commutes X Y W Z
Ma Mb Mb Ma
(8)
SLIDE 56 Compact commutators
Axler–Chang–Sarason–Volberg
[Ta, Tb) is compact on H2 if and only if H∞[a] ∩ H∞[b] ⊂ H∞ + C.
Theorem
Let X ⊂ Y ⊂ Z be r.i. B.f. spaces such that Y , Z and W = X ⊙ M(Y , Z) have nontrivial Boyd indices and W is a Banach
- space. Assume that a ∈ M(X, Y ), b ∈ M(Y , Z).
If a ∈ M(X, Y )o or b ∈ M(Y , Z)o, then the commutator [Ta, Tb] and the semi-commutator [Ta, Tb) are compact.
SLIDE 57 Compact commutators
Axler–Chang–Sarason–Volberg
[Ta, Tb) is compact on H2 if and only if H∞[a] ∩ H∞[b] ⊂ H∞ + C.
Theorem
Let X ⊂ Y ⊂ Z be r.i. B.f. spaces such that Y , Z and W = X ⊙ M(Y , Z) have nontrivial Boyd indices and W is a Banach
- space. Assume that a ∈ M(X, Y ), b ∈ M(Y , Z).
If a ∈ M(X, Y )o or b ∈ M(Y , Z)o, then the commutator [Ta, Tb] and the semi-commutator [Ta, Tb) are compact.
SLIDE 58
Regularization
Let f ∈ Z := X ⊙ Y i.e. f = gh for some g ∈ X, h ∈ Y and f Z ≈ gXgY .
Question:
Suppose that f has some additional property (is analytic, smooth, simple, etc.). Can we choose g ∈ X and h ∈ Y with the same property in such a way that f = gh and f Z ≈ gXgY ?
SLIDE 59
Regularization
Let f ∈ Z := X ⊙ Y i.e. f = gh for some g ∈ X, h ∈ Y and f Z ≈ gXgY .
Question:
Suppose that f has some additional property (is analytic, smooth, simple, etc.). Can we choose g ∈ X and h ∈ Y with the same property in such a way that f = gh and f Z ≈ gXgY ?
SLIDE 60 Regularization for step functions
Theorem
Let X and Y be two r.i.B.f. spaces. Suppose that z ∈ X ⊙ Y is of the form z =
∞
cnχAn, where (An) is any sequence of pairwise disjoint sets. Then zX⊙Y is attained on elements of the same form. In other words, zX⊙Y = inf
z = xy, x =
∞
anχAn ∈ X, y =
∞
bnχAn ∈ Y
SLIDE 61 Regularization for analytic functions
Theorem
Let X and Y be two L-convex q-B.f. spaces and X ⊂ Y . Then:
- 1. H[M(X, Y )] = M(H[X], H[Y ]),
- 2. M(X, Y ) = M(H[X], Y ).
Moreover, when X, Y are B.f.s., then the above spaces have also equal norms.
Theorem
Let X, Y be two q-B.f. spaces. Then
- 1. H[X] ⊙ H[Y ] = H[X ⊙ Y ];
- 2. H[X] ⊙ Y = X ⊙ Y
with equality of semi-norms.
SLIDE 62 Regularization for analytic functions
Theorem
Let X and Y be two L-convex q-B.f. spaces and X ⊂ Y . Then:
- 1. H[M(X, Y )] = M(H[X], H[Y ]),
- 2. M(X, Y ) = M(H[X], Y ).
Moreover, when X, Y are B.f.s., then the above spaces have also equal norms.
Theorem
Let X, Y be two q-B.f. spaces. Then
- 1. H[X] ⊙ H[Y ] = H[X ⊙ Y ];
- 2. H[X] ⊙ Y = X ⊙ Y
with equality of semi-norms.
SLIDE 63 Applications
p-convexity
A B.f.s. X is called p-convex when there is C > 0 such that for arbitrary (xk)n
k=1 ∈ X
(
n
|xk|p)1/pX C(
n
xkp
X)1/p.
p-concavity
A B.f.s. X is called p-concave when there is C > 0 such that for arbitrary (xk)n
k=1 ∈ X
(
n
xkp
X)1/p C( n
|xk|p)1/pX.
SLIDE 64 Applications
p-convexity
A B.f.s. X is called p-convex when there is C > 0 such that for arbitrary (xk)n
k=1 ∈ X
(
n
|xk|p)1/pX C(
n
xkp
X)1/p.
p-concavity
A B.f.s. X is called p-concave when there is C > 0 such that for arbitrary (xk)n
k=1 ∈ X
(
n
xkp
X)1/p C( n
|xk|p)1/pX.
SLIDE 65 Applications
Factorization Theorem
Let X, Y be two r.i. B.f.spaces such that X ⊂ Y and Y has nontrivial Boyd indices. If X is p-convex and Y is p-concave for some 1 < p < ∞, then each Toeplitz operator Ta : H[X] → H[Y ] factorizes strongly through Hp, i.e. there are b ∈ M(Lp, Y ) and φ ∈ H[M(X, Lp)] such that a = bφ and Ta = TbMφ, i.e. the diagram commutes H[X] H[Y ] Hp
Ta Mφ Tb
SLIDE 66
Thank you!
