The essential norms of Toeplitz operators with symbols in $C+H^\infty$ on weighted Hardy spaces are independent of the weights
Oleksiy Karlovych Eugene Shargorodsky

§ ABSTRACT 
Let 1<p<∞, let H^p be the Hardy space on the unit circle, and let 
H^p(w) be the Hardy space with a Muckenhoupt weight w∈ A_p on the unit 
circle. In 1988, Böttcher, Krupnik and Silbermann proved that
the essential norm of the Toeplitz operator T(a) with a∈ C
on the weighted Hardy space H^2(ϱ) with a power weight ϱ∈ A_2
is equal to a_L^∞. This implies that the essential norm of
T(a) on H^2(ϱ) does not depend on ϱ. We extend this result 
and show that if a∈ C+H^∞, then, for 1<p<∞, the essential 
norms of the Toeplitz operator T(a) on H^p and on H^p(w) are the same 
for all w∈ A_p. In particular, if w∈ A_2, then the essential norm 
of the Toeplitz operator T(a) with a∈ C+H^∞ on the weighted 
Hardy space H^2(w) is equal to a_L^∞.


Primary 47B35, 46E30

[
    Diana A. Bistrian, Gabriel Dimitriu, Ionel M. Navon
    5 September 2024
=======================================================




§ INTRODUCTION

For Banach spaces 𝒳,𝒴, let 
ℬ(𝒳,𝒴) and 
𝒦(𝒳,𝒴) 
denote the sets of bounded linear and compact linear operators from 
𝒳 to 𝒴, respectively. The norm of an operator 
A∈ℬ(𝒳,𝒴) is denoted
by A_ℬ(𝒳,𝒴). The essential norm of 
A ∈ℬ(𝒳,𝒴) 
is defined as follows:

    A_ℬ(𝒳,𝒴),e 
    := 
    inf{A - K_ℬ(𝒳,𝒴) :   K ∈𝒦(𝒳,𝒴)}.

As usual, we abbreviate ℬ(𝒳,𝒳) and 
𝒦(𝒳,𝒳) to
ℬ(𝒳) and 𝒦(𝒳), respectively.

Let 𝕋:={z∈ℂ:|z|=1} be the unit circle in the complex
plane. We equip 𝕋 with  the Lebesgue measure m normalised so 
that m(𝕋)=1. In this paper, all function spaces will be 
considered over 𝕋.
A measurable function w:𝕋→[0,∞] is said to be a weight
if 0<w<∞ a.e. on 𝕋. Let 1<p<∞ and w be a weight.
Weighted Lebesgue spaces L^p(w) consist of all measurable functions
f:𝕋→ℂ such that fw∈ L^p. The norm in L^p(w)
is defined by

    f_L^p(w):=fw_L^p
    =
    (∫_𝕋|f(t)|^pw^p(t) dm(t))^1/p.

Consider the operators S and P, defined for a function 
f∈ L^1 and a.e. point t∈𝕋 by

    (Sf)(t):=1/π i ∫_𝕋f(τ)/τ-t dτ,
      
    (Pf)(t):=1/2(f(t)+(Sf)(t)),

respectively, where the integral is understood in the Cauchy principal 
value sense. The operator S is called the Cauchy singular 
integral operator and the operator P is called the Riesz projection.
It is well known (see <cit.>
and also <cit.>, <cit.>) that the 
Riesz projection P is bounded on L^p(w) if and only if w 
belongs to the Muckenhoupt class A_p, that is,

    sup_γ⊂𝕋(1/m(γ)∫_γ w^p(t)  dm(t))^1/p(1/m(γ)∫_γ w^-p'(t)  dm(t))^1/p'
    <∞,

where 1/p+1/p'=1 and the supremum is taken over all arcs 
γ⊂𝕋. If w∈ A_p, then 
L^∞↪ L^p(w)↪ L^1.

For a function f ∈ L^1, let

    f(n) 
    = 
    1/2π∫_-π^π 
    f(e^iθ) e^-i nθ  dθ , 
       
    n ∈ℤ

be the Fourier coefficients of f. For 1<p<∞ and w∈ A_p, let

    H^p(w) = H[L^p(w)] := 
    {
    f∈ L^p(w) : f(n)=0     n<0
    }

be the weighted Hardy space. The classical Hardy spaces  H^p, 
1≤ p≤∞, are defined similarly if one replaces L^p(w) by L^p 
in the above definition. It is well known that if w∈ A_p, then P maps 
L^p(w) onto H^p(w).

Let 1<p<∞ and w∈ A_p. For a∈ L^∞, the Toeplitz operator 
with symbol a is defined by

    T(a)f=P(af),
       
    f∈ H^p(w).

It is clear that T(a)∈ℬ(H^p(w)) and 

    T(a)_ℬ(H^p(w)), e≤T(a)_ℬ(H^p(w))≤P_ℬ(L^p(w))a_L^∞.


I. Gohberg and N. Krupnik <cit.> proved that 
P_ℬ(L^p),e≥ 1/sin(π/p) and conjectured that
P_ℬ(L^p)=1/sin(π/p). This conjecture was confirmed
by B. Hollenbeck and I. Verbitsky in <cit.>. Thus


    T(a)_ℬ(H^p),e≤ 
    1/sin(π/p)
    a_L^∞,
      
    a∈ L^∞.



Let C denote the Banach space of all complex-valued continuous functions
on 𝕋 with the supremum norm and let 

    C+H^∞:={f∈ L^∞ : f=g+h, g∈ C, h∈ H^∞}.

In 1967, Sarason observed that C+H^∞ is a closed subalgebra
of L^∞ (see, e.g., <cit.> for the proof 
of this fact).

Consider power weights of the form


    ϱ(t):=∏_j=1^k |t-t_j|^λ_j


where t_1,…,t_k∈𝕋 are pairwise disjoint and 
λ_1,…,λ_k∈ℝ. These weights are usually called
Khvedelidze weights. It is well known that ϱ∈ A_p with
1<p<∞ if and only if -1/p<λ_j<1-1/p for all j∈{1,…,k}
(see, e.g., <cit.>).

In 1988, A. Böttcher, N. Krupnik, and B. Silbermann proved the following
(see <cit.>).


If ϱ is a power weight of the form (<ref>) satisfying

    -1/2<λ_j<1/2 
        
    j∈{1,…,k},

then for all a∈ C
one has T(a)_ℬ(H^2(ϱ)),e=a_L^∞.


In particular, the essential norm of T(a) on H^2(ϱ) does not
depend on the choice of a Khvedelidze weight
ϱ∈ A_2. Further they asked in <cit.> 
whether the essential norm of Toeplitz operators T(a) with 
a∈ C acting on Hardy spaces H^p depend on p∈(1,∞). 
The second author answered this question in the negative <cit.>. 
More precisely, it was shown that the equality 
T(a)_ℬ(H^p),e=a_L^∞ 
holds for all a∈ C if and only if p=2. Nevertheless, the following 
estimates for T(a)_ℬ(H^p),e hold
(see (<ref>) and <cit.>).


Let 1<p<∞ and a∈ C+H^∞. Then

    a_L^∞≤T(a)_ℬ(H^p),e≤min{2^|1-2/p|,1/sin(π/p)}a_L^∞.




The aim of this paper it to study the relations between the essential norms of 
Toeplitz operators with symbols in C+H^∞on unweighted and weighted 
Hardy spaces. The following is our main result.


Let 1<p<∞ and w∈ A_p. If a∈ C+H^∞, then

    T(a)_ℬ(H^p),e
    =
    T(a)_ℬ(H^p(w)),e.



It is instructive to compare this result with the behaviour of the essential 
norm S_ℬ(L^p(ϱ)),e of the Cauchy singular 
integral operator with ϱ∈ A_p of the form (<ref>), 
which depends not only on p∈ (1,∞), but also on ϱ 
(see <cit.> and Krupnik's survey <cit.>).

In fact, we prove an abstract form of Theorem <ref> 
(see Theorem <ref>), where we replace the Hardy 
space H^p with 1<p<∞ by the abstract Hardy space H[X] built
upon a Banach function space X such that P∈ℬ(X)
and replace the weighted Hardy space H^p(w) with w∈ A_p by
the abstract Hardy space H[X(w)] built upon 
X(w)={f:𝕋→ℂ:fw∈ X} with a weight w such that
P∈ℬ(X(w)). Our proof is fairly elementary and is completely 
different from that of Theorem <ref>.

It is clear that Theorems <ref> and <ref>
immediately imply the following extension of Theorem <ref>.


Suppose that 1<p<∞ and w∈ A_p. If a∈ C+H^∞, then


    a_L^∞≤T(a)_ℬ(H^p(w)),e≤{2^|1-2/p|,1/sin(π/p)}a_L^∞.

In particular, if w∈ A_2 and a∈ C+H^∞, then 
T(a)_ℬ(H^2(w)),e=a_L^∞.



We will use the following notation:

    𝐞_m(z) := z^m, 
       z ∈ℂ, 
       m ∈ℤ.


The paper is organised as follows. In Section <ref>, we 
collect definitions of a Banach function space, its associate space X', and 
a weighted Banach function space X(w). Note that if w∈ X and 1/w∈ X', 
then X(w) is a Banach function space itself. This allows us to define
the abstract Hardy space H[X(w)] built upon X(w). We recall that if
w∈ X and 1/w∈ X', then then the abstract Hardy spaces H[X] and
H[X(w)] are isometrically isomorphic. We conclude the preliminaries with
some useful properties of the Riesz projection P and the definition of
a Toeplitz operator T(a) on the abstract Hardy space built upon a Banach
function space X such that P∈ℬ(X).

Section <ref> is devoted to the proof of Theorem <ref>.
First we recall an observation from <cit.> that the Toeplitz operator
T(𝐞_-nh) with n∈ℕ and h∈ H^∞
is bounded on the abstract Hardy space H[X] for an arbitrary Banach function
space, i.e. even without the assumption that P∈ℬ(X).
Further, we show that the essential norms of T(𝐞_-nh)
on H[X] and H[X(w)] coincide if w∈ X and 1/w∈ X'.
Here we essentially use the isomorphic isomorphism of H[X] and H[X(w)].
Finally, taking into account that the set 
{𝐞_-nh:n∈ℕ,h∈ H^∞} is dense in C+H^∞, 
we prove Theorem <ref>. In turn, 
Theorem <ref> yield Theorem <ref>.



§ PRELIMINARIES



 §.§ Banach function spaces

Let ℳ be the set of all measurable complex-valued functions on 
𝕋 equipped with the normalized Lebesgue measure m and let 
ℳ^+ be the subset of functions in ℳ whose values lie 
in [0,∞]. Following <cit.>, a mapping 
ρ: ℳ^+→ [0,∞] is called a Banach function norm if, 
for all functions f,g, f_n∈ℳ^+ with n∈ℕ, and for 
all constants a≥ 0, the following  properties hold:


    (A1)      ρ(f)=0  ⇔  f=0 ,
    ρ(af)=aρ(f),
    ρ(f+g) ≤ρ(f)+ρ(g),
     (A2)      0≤ g ≤ f   ⇒ ρ(g) ≤ρ(f)
      ,
     (A3)      0≤ f_n ↑ f   ⇒ρ(f_n) ↑ρ(f)  ,
     (A4)      ρ(1) <∞,
     (A5)      ∫_𝕋 f(t) dm(t) ≤ Cρ(f)


with a constant C ∈ (0,∞) that may depend on ρ,  but is 
independent of f. When functions differing only on a set of measure zero 
are identified, the set X of all functions f∈ℳ for which 
ρ(|f|)<∞ is called a Banach function space. For each f∈ X, 
the norm of f is defined by f_X :=ρ(|f|). The set X equipped 
with the natural vector space operations and this norm becomes a Banach 
space (see <cit.>). If ρ is a Banach 
function norm, its associate norm ρ' is defined on ℳ^+ by

    ρ'(g):=sup{∫_𝕋 f(t)g(t) dm(t)  :  
    f∈ℳ^+,  ρ(f) ≤ 1
    },    g∈ℳ^+.

It is a Banach function norm itself <cit.>.
The Banach function space X' determined by the Banach function 
norm ρ' is called the associate space (Köthe dual) of X. 
The associate space X' can be viewed as a subspace of the 
Banach dual space X^*. 



 §.§ Weighted Banach function spaces


For a weight w and a Banach function space X, the weighted space X(w) 
consists of all measurable functions f:𝕋→ℂ such that 
fw∈ X. We equip it with the norm f_X(w)=fw_X.


Let X be a Banach function space with the associate spaces X' and 
let w be a weight.



  (a)
If w∈ X and 1/w∈ X', then X(w) is a Banach function space,
whose associate space is X'(1/w).


  (b)
If P is bounded on the normed space X(w), then w∈ X and 
1/w∈ X'.



Part (a) was proved in <cit.>, part (b) follows from 
<cit.>. 



 §.§ Isometric isomorphism of weighted and nonweighted abstract
Hardy spaces

Let X be a Banach function space and let X' be its associate space.
Lem­ma <ref>(a) implies that if w is a weight such that
w∈ X and 1/w∈ X', then X(w) is a Banach function space. Then 
L^∞↪ X(w)↪ L^1
and the abstract Hardy space H[X(w)] built upon X(w) is well defined
by

    H[X(w)]:=
    {
    f∈ X(w) : f(n)=0     n<0
    }.

For simplicity, we will write H[X] if w=1.



Let X be a Banach function space with the associate space X'
and let w be a weight such that w∈ X and 1/w∈ X'.
Consider

    W(z) := exp(1/2π∫_-π^πe^it + z/e^it - z log w(e^it)  dt), 
       
    |z|<1.

Then the mapping f↦ M_Wf:=W· f
is an isometric isomorphism of H[X(w)] onto H[X] and
the mapping g↦ M_W^-1g :=W^-1· g is an isometric
isomorphism of H[X] onto H[X(w)].




 §.§ Auxiliary lemmas on the Riesz projection

We will need the following auxiliary lemmas.



Let f ∈ L^1. Suppose there exists g ∈ H^1 such that 
f(n) = g(n) for all n ≥ 0. Then Pf = g.


The following lemma justifies the terminology about the operator P.



If X is a Banach function space on which the operator P is bounded,
then P maps the space X onto the abstract Hardy space H[X].




 §.§ Toeplitz operators on abstract Hardy spaces

Let X be a Banach function space. If a∈ L^∞, then the multiplication
operator by a defined by

    M_af:=af,    f∈ X,

is bounded on X and M_a_ℬ(X)≤a_L^∞.
Further, if the Riesz projection is bounded on X, then the Toeplitz
operator defined by

    T(a)f:=P(M_af)=P(af),
      
    f∈ H[X],

is bounded on H[X] and


    T(a)_ℬ(H[X])≤P_ℬ(X)a_L^∞.




§ PROOF OF THE MAIN RESULT




 §.§ Operator P_n and a special Toeplitz operator

We will need auxiliary results on a representation of a Toeplitz
operator T(𝐞_-nh) with n∈ℕ and h∈ H^∞
in terms of the projection P_n from X onto the subspace of H[X] of 
analytic polynomials of order n-1 obtained in our recent paper <cit.>.
We would like to underline that we do not require here and in
Subsection <ref> 
that the Riesz projection P is bounded on X.



For every n∈ℕ and f∈ L^1, let

    P_n f := ∑_k = 0^n-1f(k) 𝐞_k.

Then the operator P_n is bounded from L^1 to H^∞ and
from a Banach function space X to the abstract Hardy space H[X].




Let X be a Banach function space. If n ∈ℤ_+ and h ∈ H^∞, 
then the Toeplitz operator T(𝐞_-n h) : H[X] → H[X] is bounded
and


    T(𝐞_-n h) f 
    = 
    P(𝐞_-n h f) = 𝐞_-n (I - P_n)(hf),
      
    f∈ H[X].
 





 §.§ The essential norm of a special Toeplitz operator on a weighted 
Hardy space is independent of the weight


In this subsection we show that the essential norms of the special
Toepllitz operators with symbols of the form 
𝐞_-nh, where n∈ℕ and h∈ H^∞,
are independent of the weight.

 
Let X be a Banach function space with the associate space X'
and let w be a weight satisfying w∈ X and 1/w∈ X'. 
If n ∈ℕ and h ∈ H^∞, then

    T(𝐞_-n h)_ℬ(H[X(w)]),e 
    = 
    T(𝐞_-n h)_ℬ(H[X]),e .




Let the operators M_W and M_W^-1 be as in 
Lemma <ref>. Since W is an outer function 
(see <cit.>), one has |W|=w 
a.e. on 𝕋, and W∈ H[X], W^-1∈ H[X'].
 
By Lemma <ref>,
the operator T(𝐞_-nh) is bounded on H[X] and H[X(w)].
For any f ∈ H[X], one gets from (<ref>)
and Lemma <ref> that


    M_W T(𝐞_-n h) M_W^-1 f 
    = 
    W 𝐞_-n (I - P_n)(hW^-1 f) 
    
          = 
    P(W 𝐞_-n (I - P_n)(hW^-1 f)) 
    
          = 
    P(W 𝐞_-n (hW^-1 f) 
    - 
    W 𝐞_-n P_n(hW^-1 f)) 
    
          = 
    P(𝐞_-n (h f) 
    - 
    W 𝐞_-n P_n (hW^-1 f)) 
    
          = 
    P(𝐞_-n  (I - P_n)(h f) 
    + 
    𝐞_-n P_n(h f) 
    - 
    W 𝐞_-n P_n(hW^-1 f)) 
    
          = 
    P(T(𝐞_-n h) f 
    +  
    𝐞_-n P_n (h f) 
    - 
    W 𝐞_-n P_n (hW^-1 f)) 
    
          = 
    T(𝐞_-n h) f 
    + 
    T(𝐞_-n) P_n  M_h f 
    - 
    T(𝐞_-n) M_W P_n M_hW^-1 f .


Since W^-1∈ X' and h∈ H^∞, it follows from the Hölder 
inequality for Banach function spaces (see <cit.>)
that M_hW^-1∈ℬ(H[X], L^1). 

On the other hand, taking into account 
<cit.>, since 
W∈ H[X]⊂ H^1, one has Wg∈ H^1∩ X=H[X]
for every g∈ H^∞.
Hence M_W ∈ℬ(H^∞, H[X]). 

Then Lemmas <ref> and  <ref> 
(with h = 1) imply that

    T(𝐞_-n) M_W P_n M_hW^-1∈ℬ(H[X]),
      
    T(𝐞_-n) P_n M_h ∈ℬ(H[X]) .

So,

    K_0  :=  
    T(𝐞_-n) P_n M_h 
    - 
    T(𝐞_-n) M_W P_n M_hW^-1

is a bounded finite-rank operator on H[X], and


    M_W T(𝐞_-n h) M_W^-1 = T(𝐞_-n h) + K_0 .


Since K_0 ∈𝒦(H[X]), one has

    M_W K M_W^-1 - K_0 ∈𝒦(H[X])

for every K ∈𝒦(H[X(w)]). Moreover, for every 
T ∈𝒦(H[X]), there exists K ∈𝒦(H[X(w)]) such 
that

    T = M_W K M_W^-1 - K_0 .

Indeed, this K is given by the formula

    K = M_W^-1(T + K_0) M_W∈𝒦(H[X(w)]) .

Using (<ref>), one gets


    T(𝐞_-n h)_ℬ(H[X(w)]),e 
    = 
    inf_K ∈𝒦(H[X(w)])T(𝐞_-n h) - K_ℬ(H[X(w)])
          = 
    inf_K ∈𝒦(H[X(w)])M_W(T(𝐞_-n h) - K)M_W^-1_ℬ(H[X])
          = 
    inf_K ∈𝒦(H[X(w)])T(𝐞_-n h) - (M_W K M_W^-1 - K_0)_ℬ(H[X])
          = 
    inf_T ∈𝒦(H[X])T(𝐞_-n h) - T_ℬ(H[X]) 
    = 
    T(𝐞_-n h)_ℬ(H[X]),e,


which completes the proof.




 §.§ The essential norm of a Toeplitz operator on a weighted
Hardy space is independent of the weight


Now we are in a position to prove the main result of this section.


Let X be a Banach function space and let w be a weight such that 
the Riesz projection P is bounded on the spaces X and X(w).
If a∈ C+H^∞, then 


    T(a)_ℬ(H[X(w)]),e 
    = 
    T(a)_ℬ(H[X]),e.




It follows from <cit.> that there is a sequence
{a_m} of functions of the form 𝐞_-nh with h ∈ H^∞
and n ∈ℕ such that a-a_m_L^∞→ 0 as m→∞.
Since P∈ℬ(X(w)), Lemma <ref>(b) implies that
w∈ X and 1/w∈ X'. Then Lemma <ref> 
yields 


    T(a_m)_ℬ(H[X(w)]),e 
    = 
    T(a_m)_ℬ(H[X]),e,
      
    m∈ℕ.


Since P∈ℬ(X), we obtain from (<ref>)
that


    |
    T(a)_ℬ(H[X]),e
    -
    T(a_m)_ℬ(H[X]),e|
       ≤T(a-a_m)_ℬ(H[X]),e
       ≤P_ℬ(X)a-a_m_L^∞,


whence


    lim_m→∞T(a_m)_ℬ(H[X]),e
    =
    T(a)_ℬ(H[X]),e.


Similarly, since P∈ℬ(X(w)), we get


    lim_m→∞T(a_m)_ℬ(H[X(w)]),e
    =
    T(a)_ℬ(H[X(w)]),e.


Combining 
(<ref>)–(<ref>),
we arrive at (<ref>).




 §.§ Proof of Theorem <ref>

If 1<p<∞ and w∈ A_p, then the Riesz projection P is bounded
on the Lebesgue space L^p and on its weighted counterpart L^p(w).
In this case, Theorem <ref> follows immediately from 
Theorem <ref>.



§ DECLARATIONS



 §.§ Funding

This work is funded by national funds through the FCT - Fundação para a 
Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 
(<https://doi.org/10.54499/UIDB/00297/2020>)
and UIDP/ 00297/2020

(<https://doi.org/10.54499/UIDP/00297/2020>)
(Center for Mathematics and Applications).



 §.§ Data availability

Data sharing not applicable to this article as no datasets
were generated or analysed during the current study.



 §.§ Conflict of interest

All authors certify that they have no affiliations with
or involvement in any organisation or entity with any financial interest or
non-financial interest in the subject matter or materials discussed in this
manuscript.

abbrv

