De Branges–Rovnyak spaces and local Dirichlet spaces of higher order

Bartosz Łanucha; Małgorzata Michalska; Maria Nowak; Andrzej Sołtysiak

doi:10.1515/forum-2023-0234

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De Branges–Rovnyak spaces and local Dirichlet spaces of higher order

Bartosz Łanucha , Małgorzata Michalska , Maria Nowak and Andrzej Sołtysiak

Published/Copyright: October 4, 2023

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From the journal Forum Mathematicum Volume 36 Issue 1

Abstract

We discuss de Branges–Rovnyak spaces ℋ ⁢ ( b ) generated by nonextreme and rational functions b and local Dirichlet spaces of order m introduced in [S. Luo, C. Gu and S. Richter, Higher order local Dirichlet integrals and de Branges–Rovnyak spaces, Adv. Math. 385 2021, Paper No. 107748]. In that paper, the authors characterized nonextreme b for which the operator Y = S | ℋ ⁢ ( b ) , the restriction of the shift operator S on H 2 to ℋ ⁢ ( b ) , is a strict 2 ⁢ m -isometry and proved that such spaces ℋ ⁢ ( b ) are equal to local Dirichlet spaces of order m. Here we give a characterization of local Dirichlet spaces of order m in terms of the m-th derivatives that is a generalization of a known result on local Dirichlet spaces. We also find explicit formulas for b in the case when ℋ ⁢ ( b ) coincides with local Dirichlet space of order m with equality of norms. Finally, we prove a property of wandering vectors of Y analogous to the property of wandering vectors of the restriction of S to harmonically weighted Dirichlet spaces obtained in [D. Sarason, Harmonically weighted Dirichlet spaces associated with finitely atomic measures, Integral Equations Operator Theory 31 1998, 2, 186–213].

Keywords: Hardy space; de Branges–Rovnyak spaces; local Dirichlet space; wandering vector

MSC 2020: 47B35; 30H10

1 Introduction

Let 𝔻 denote the open unit disc in the complex plane ℂ , and let 𝕋 = ∂ ⁡ 𝔻 . For φ ∈ L ∞ ⁢ ( 𝕋 ) the Toeplitz operator on the Hardy space H 2 of the disc 𝔻 is defined by T φ ⁢ f = P ⁢ ( φ ⁢ f ) , where P is the orthogonal projection of L 2 ⁢ ( 𝕋 ) onto H 2 . In particular, denote S = T z .

For a function b in the closed unit ball of H ∞ , the de Branges–Rovnyak space H ⁢ ( b ) is the image of H 2 under the operator ( I - T b ⁢ T b ¯ ) 1 2 with the corresponding range norm ∥ ⋅ ∥ b (see the books [9, 3] and references given there). It is known that ℋ ⁢ ( b ) is a Hilbert space with reproducing kernel

k w b ( z ) = 1 - b ⁢ ( w ) ¯ ⁢ b ⁢ ( z ) 1 - w ¯ ⁢ z ( z , w ∈ 𝔻 ) .

In the case b is an inner function, ℋ ⁢ ( b ) = H 2 ⊖ b ⁢ H 2 is the so-called model space.

The theory of ℋ ⁢ ( b ) spaces divides into two cases, according to whether b is or is not an extreme point of the closed unit ball of H ∞ . An important property of ℋ ⁢ ( b ) spaces is that they are S * -invariant and in the case when b is nonextreme they are also S-invariant. Moreover, for nonextreme b the operator Y = S | ℋ ( b ) is bounded and it expands the norm.

Here we will concentrate on the case when the function b is not an extreme point of the unit ball of H ∞ (equivalently, log ⁡ ( 1 - | b | 2 ) is integrable on 𝕋 ) . Then there exists an outer function a ∈ H ∞ for which | a | 2 + | b | 2 = 1 a.e. on 𝕋 . Moreover, if we suppose a ⁢ ( 0 ) > 0 , then a is uniquely determined, and we say that ( b , a ) is a pair. If ( b , a ) is a pair, then the quotient φ = b a is in the Smirnov class 𝒩 + . Conversely, every nonzero function φ ∈ 𝒩 + has a unique representation (the so-called canonical representation) φ = b a , where ( b , a ) is a pair. In this case T φ is an unbounded operator on H 2 with the domain 𝔇 ⁢ ( T φ ) = { f ∈ H 2 : φ ⁢ f ∈ H 2 } = a ⁢ H 2 . Thus T φ is densely defined and closed. Consequently, its adjoint T φ ∗ = T φ ¯ is densely defined and closed. Moreover, 𝔇 ⁢ ( T φ ¯ ) = ℋ ⁢ ( b ) and for f ∈ ℋ ⁢ ( b ) ,

(1.1) ∥ f ∥ b 2 = ∥ f ∥ 2 2 + ∥ T φ ¯ ⁢ f ∥ 2 2 .

For more details on de Branges–Rovnyak spaces connection with unbounded Toeplitz operators see [12].

In the papers [10, 1, 2, 5] relations between de Branges–Rovnyak spaces and harmonically weighted Dirichlet spaces have been studied.

For a finite measure μ on 𝕋 let P μ denote the Poisson integral of μ given by

P μ ⁢ ( z ) = ∫ 𝕋 1 - | z | 2 | ζ - z | 2 ⁢ 𝑑 μ ⁢ ( ζ ) , z ∈ 𝔻 .

The associated harmonically weighted Dirichlet space 𝒟 ⁢ ( μ ) consists of functions f analytic in 𝔻 for which

(1.2) 𝒟 μ ⁢ ( f ) = ∫ 𝔻 | f ′ ⁢ ( z ) | 2 ⁢ P μ ⁢ ( z ) ⁢ 𝑑 A ⁢ ( z ) < ∞ ,

where A denotes the normalized area measure on 𝔻 .

The spaces 𝒟 ⁢ ( μ ) were introduced by S. Richter in [7], where it was proved that certain two-isometries on a Hilbert space can be represented as multiplication by z on a space 𝒟 ⁢ ( μ ) . The results on 𝒟 ⁢ ( μ ) stated below were also obtained in [7].

If μ is a finite measure on 𝕋 such that μ ⁢ ( 𝕋 ) > 0 , then 𝒟 ⁢ ( μ ) ⊂ H 2 and 𝒟 ⁢ ( μ ) is a Hilbert space with the norm ∥ ⋅ ∥ 𝒟 ⁢ ( μ ) given by

∥ f ∥ 𝒟 ⁢ ( μ ) 2 = ∥ f ∥ 2 2 + 𝒟 μ ⁢ ( f ) .

If μ = δ λ is the Dirac measure at λ ∈ 𝕋 , then

𝒟 δ λ ⁢ ( f ) = 𝒟 λ ⁢ ( f ) = ∫ 𝔻 | f ′ ⁢ ( z ) | 2 ⁢ 1 - | z | 2 | z - λ | 2 ⁢ 𝑑 A ⁢ ( z )

and 𝒟 λ ⁢ ( f ) is called the local Dirichlet integral of f at λ.

For f ∈ H 2 , by Fubini’s theorem, 𝒟 μ ⁢ ( f ) given by (1.2) can be expressed as

𝒟 μ ⁢ ( f ) = ∫ 𝕋 𝒟 λ ⁢ ( f ) ⁢ 𝑑 μ ⁢ ( λ ) .

Moreover, if λ ∈ 𝕋 is such that 𝒟 λ ⁢ ( f ) < ∞ , then the nontangential limit f ⁢ ( λ ) exists.

The following local Douglas formula for 𝒟 λ ⁢ ( f ) was proved by Richter and Sundberg [8]: if f ∈ H 2 and f ⁢ ( λ ) exists, then

𝒟 λ ⁢ ( f ) = 1 2 ⁢ π ⁢ ∫ 𝕋 | f ⁢ ( z ) - f ⁢ ( λ ) z - λ | 2 ⁢ | d ⁢ z | .

If f ⁢ ( λ ) does not exist, then we set 𝒟 λ ⁢ ( f ) = ∞ . The space 𝒟 ⁢ ( δ λ ) = 𝒟 λ is called the local Dirichlet space at λ. It has also been proved in [8] that

𝒟 λ = { f ∈ Hol ⁢ ( 𝔻 ) : f = c + ( z - λ ) ⁢ g : c ∈ ℂ , g ∈ H 2 } .

In [6] S. Luo, C. Gu and S. Richter characterized nonextreme b for which Y is a strict 2 ⁢ m -isometry, m ∈ ℕ . It turns out that the corresponding spaces ℋ ⁢ ( b ) are equal to the so-called local Dirichlet spaces of order m , 𝒟 λ m , λ ∈ 𝕋 , defined as follows:

(1.3) 𝒟 λ m = { f ∈ Hol ⁢ ( 𝔻 ) : f = p + ( z - λ ) m ⁢ g , p ⁢ is a polynomial of degree < m , g ∈ H 2 } .

They also showed that if f ∈ H 2 , then f ∈ 𝒟 λ m if and only if for each j = 0 , 1 , … , m - 1 the function f ( j ) has a nontangential limit at λ and

𝒟 λ m ⁢ ( f ) = 1 2 ⁢ π ⁢ ∫ 𝕋 | f ⁢ ( z ) - T m - 1 ⁢ ( f , λ ) ⁢ ( z ) ( z - λ ) m | 2 ⁢ | d ⁢ z | < ∞ ,

where

T m - 1 ⁢ ( f , λ ) ⁢ ( z ) = f ⁢ ( λ ) + f ′ ⁢ ( λ ) ⁢ ( z - λ ) + … + f ( m - 1 ) ⁢ ( λ ) ( m - 1 ) ! ⁢ ( z - λ ) m - 1 .

In the space 𝒟 λ m the norm is given by

(1.4) ∥ f ∥ 𝒟 λ m 2 = ∥ f ∥ 2 2 + 𝒟 λ m ⁢ ( f ) .

In the next section we obtain the following characterization of the space 𝒟 λ m .

Theorem 1.

A function f holomorphic in D is in the space D λ m if and only if

I λ m ⁢ ( f ) = 1 ⁢ ∫ 𝔻 | f ( m ) ⁢ ( z ) ( z - λ ) m | 2 ⁢ ( 1 - | z | 2 ) 2 ⁢ m - 1 ⁢ 𝑑 A ⁢ ( z ) < ∞ .

Moreover, for f ∈ Hol ⁡ ( D ) ,

I λ m ⁢ ( f ) = ( 2 ⁢ m - 1 ) ! ⁢ 𝒟 λ m ⁢ ( f ) .

In [10] D. Sarason proved that if for λ ∈ 𝕋 ,

(1.5) b λ ( z ) = ( 1 - α ) ⁢ λ ¯ ⁢ z 1 - α ⁢ λ ¯ ⁢ z , α = 3 - 5 2 ( a λ ( z ) = ( 1 - α ) ⁢ ( 1 - λ ¯ ⁢ z ) 1 - α ⁢ λ ¯ ⁢ z ) ,

then the space 𝒟 λ coincides with ℋ ⁢ ( b λ ) with equality of norms. It follows from the paper [1] that ℋ ⁢ ( b λ ) = 𝒟 λ with equality of norms only for b λ given by (1.5).

In [6] the authors also described a relation between local Dirichlet space of higher order 𝒟 λ m and ℋ ⁢ ( b ) . In particular, they obtained a necessary and sufficient condition for ℋ ⁢ ( b ) = 𝒟 λ m with equality of norms. In Section 3, we derive explicit formulas for such functions b analogous to (1.5).

A nonzero vector in a Hilbert space is called a wandering vector of a given operator if it is orthogonal to its orbit under the positive powers of the operator. In [11] the author described the wandering vector of the shift operator S μ = S | 𝒟 ( μ ) on the harmonically weighted Dirichlet space 𝒟 ⁢ ( μ ) associated with a finitely atomic measure μ = ∑ i = 1 n μ i ⁢ δ λ i where λ 1 , … , λ n , are distinct points on 𝕋 and μ 1 , … , μ n are positive numbers. One of his results states that the outer part of the wandering vector of S μ lies in the model space generated by a certain finite Blaschke product. In the last section we consider the spaces ℋ ⁢ ( b ) when b is nonextreme and rational, and show that in this case the outer part of a wandering vector of the operator Y = S | ℋ ( b ) has similar property.

2 Local Dirichlet space of order m

In the proof of Theorem 1 we will need the following technical lemma.

Lemma 1.

For a positive integer m and z , w ∈ D ,

(2.1) ∂ 2 ⁢ m ∂ ⁡ w ¯ m ⁢ ∂ ⁡ z m ⁢ ( ( w ¯ - 1 ) m ⁢ ( z - 1 ) m 1 - w ¯ ⁢ z ) = ( 2 ⁢ m ) ! ⁢ ( w ¯ - 1 ) m ⁢ ( z - 1 ) m ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 1 .

Proof.

We proceed by induction. For m = 1 the equality is obvious. Assume (2.1) holds true for an m. We will use the following Leibniz formula:

∂ 2 ⁢ m ∂ ⁡ w ¯ m ⁢ ∂ ⁡ z m ⁢ ( f ⁢ ( w ¯ , z ) ⋅ g ⁢ ( w ¯ , z ) ) = ∑ 0 ≤ k ≤ m 0 ≤ l ≤ m ( m k ) ⁢ ( m l ) ⁢ ∂ 2 ⁢ m - k - l ∂ ⁡ w ¯ m - k ⁢ ∂ ⁡ z m - l ⁢ f ⁢ ( w ¯ , z ) ⁢ ∂ k + l ∂ ⁡ w ¯ k ⁢ ∂ ⁡ z l ⁢ g ⁢ ( w ¯ , z ) .

Hence

∂ 2 ⁢ m + 2 ∂ ⁡ w ¯ m + 1 ⁢ ∂ ⁡ z m + 1 ⁢ ( ( w ¯ - 1 ) m + 1 ⁢ ( z - 1 ) m + 1 1 - w ¯ ⁢ z ) = ∂ 2 ∂ ⁡ w ¯ ⁢ ∂ ⁡ z ⁢ [ ∂ 2 ⁢ m ∂ ⁡ w ¯ m ⁢ ∂ ⁡ z m ⁢ ( ( w ¯ - 1 ) m ⁢ ( z - 1 ) m 1 - w ¯ ⁢ z ⋅ ( w ¯ - 1 ) ⁢ ( z - 1 ) ) ]

= ∂ 2 ∂ ⁡ w ¯ ⁢ ∂ ⁡ z [ ∑ 0 ≤ k ≤ m 0 ≤ l ≤ m ( m k ) ( m l ) ∂ 2 ⁢ m - k - l ∂ ⁡ w ¯ m - k ⁢ ∂ ⁡ z m - l ( ( w ¯ - 1 ) m ⁢ ( z - 1 ) m 1 - w ¯ ⁢ z )

⋅ ∂ k + l ∂ ⁡ w ¯ k ⁢ ∂ ⁡ z l ( ( w ¯ - 1 ) ( z - 1 ) ) ] .

Since for k ≥ 2 the derivatives ∂ k ∂ ⁡ w ¯ k ⁢ ( w ¯ - 1 ) = ∂ k ∂ ⁡ z k ⁢ ( z - 1 ) vanish, we have

∂ 2 ⁢ m + 2 ∂ ⁡ w ¯ m + 1 ⁢ ∂ ⁡ z m + 1 ( ( w ¯ - 1 ) m + 1 ⁢ ( z - 1 ) m + 1 1 - w ¯ ⁢ z ) = ∂ 2 ∂ ⁡ w ¯ ⁢ ∂ ⁡ z [ ( w ¯ - 1 ) ( z - 1 ) ∂ 2 ⁢ m ∂ ⁡ w ¯ m ⁢ ∂ ⁡ z m ( ( w ¯ - 1 ) m ⁢ ( z - 1 ) m 1 - w ¯ ⁢ z )

+ m ⁢ ( z - 1 ) ⁢ ∂ 2 ⁢ m - 1 ∂ ⁡ w ¯ m - 1 ⁢ ∂ ⁡ z m ⁢ ( ( w ¯ - 1 ) m ⁢ ( z - 1 ) m 1 - w ¯ ⁢ z )

+ m ⁢ ( w ¯ - 1 ) ⁢ ∂ 2 ⁢ m - 1 ∂ ⁡ w ¯ m ⁢ ∂ ⁡ z m - 1 ⁢ ( ( w ¯ - 1 ) m ⁢ ( z - 1 ) m 1 - w ¯ ⁢ z )

+ m 2 ∂ 2 ⁢ m - 2 ∂ ⁡ w ¯ m - 1 ⁢ ∂ ⁡ z m - 1 ( ( w ¯ - 1 ) m ⁢ ( z - 1 ) m 1 - w ¯ ⁢ z ) ] .

By the induction hypothesis,

∂ 2 ⁢ m + 2 ∂ ⁡ w ¯ m + 1 ⁢ ∂ ⁡ z m + 1 ( ( w ¯ - 1 ) m + 1 ⁢ ( z - 1 ) m + 1 1 - w ¯ ⁢ z ) = ( 2 m ) ! { ∂ 2 ∂ ⁡ w ¯ ⁢ ∂ ⁡ z ( ( w ¯ - 1 ) m + 1 ⁢ ( z - 1 ) m + 1 ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 1 ) + m 2 ( w ¯ - 1 ) m ⁢ ( z - 1 ) m ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 1

+ m ∂ ∂ ⁡ z ( ( w ¯ - 1 ) m ⁢ ( z - 1 ) m + 1 ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 1 ) + m ∂ ∂ ⁡ w ¯ ( ( w ¯ - 1 ) m + 1 ⁢ ( z - 1 ) m ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 1 ) } .

A calculation shows that

∂ 2 ∂ ⁡ w ¯ ⁢ ∂ ⁡ z ( ( w ¯ - 1 ) m + 1 ⁢ ( z - 1 ) m + 1 ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 1 ) = ( w ¯ - 1 ) m ⁢ ( z - 1 ) m ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 3 [ 2 + 4 m + m 2 - w ¯ ( 2 + 5 m + 2 m 2 )

- z ⁢ ( 2 + 5 ⁢ m + 2 ⁢ m 2 ) + w ¯ ⁢ z ⁢ ( 2 + 8 ⁢ m + 6 ⁢ m 2 ) - w ¯ 2 ⁢ z ⁢ ( m + 2 ⁢ m 2 )

- w ¯ z 2 ( m + 2 m 2 ) + m 2 w ¯ 2 z 2 ] ,

m ∂ ∂ ⁡ z ( ( w ¯ - 1 ) m ⁢ ( z - 1 ) m + 1 ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 1 ) = ( w ¯ - 1 ) m ⁢ ( z - 1 ) m ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 3 [ m + m 2 - w ¯ ( m + 2 m 2 ) - m w ¯ z

+ w ¯ 2 z ( m + 2 m 2 ) - m 2 w ¯ 2 z 2 ] ,

m ∂ ∂ ⁡ w ¯ ( ( w ¯ - 1 ) m + 1 ⁢ ( z - 1 ) m ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 1 ) = ( w ¯ - 1 ) m ⁢ ( z - 1 ) m ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 3 [ m + m 2 - z ( m + 2 m 2 ) - m w ¯ z

+ w ¯ z 2 ( m + 2 m 2 ) - m 2 w ¯ 2 z 2 ]

and

m 2 ⁢ ( w ¯ - 1 ) m ⁢ ( z - 1 ) m ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 1 = ( w ¯ - 1 ) m ⁢ ( z - 1 ) m ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 3 ⁢ [ m 2 - 2 ⁢ m 2 ⁢ w ¯ ⁢ z + m 2 ⁢ w ¯ 2 ⁢ z 2 ] .

Thus

∂ 2 ⁢ m + 2 ∂ ⁡ w ¯ m + 1 ⁢ ∂ ⁡ z m + 1 ⁢ ( ( w ¯ - 1 ) m + 1 ⁢ ( z - 1 ) m + 1 1 - w ¯ ⁢ z ) = ( 2 ⁢ m ) ! ⁢ ( 2 + 6 ⁢ m + 4 ⁢ m 2 ) ⁢ ( w ¯ - 1 ) m ⁢ ( z - 1 ) m ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 3 ⁢ [ 1 - w ¯ - z + w ¯ ⁢ z ]

= ( 2 ⁢ m + 2 ) ! ⁢ ( w ¯ - 1 ) m + 1 ⁢ ( z - 1 ) m + 1 ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 3 . ∎

Proof of Theorem 1.

We will use Sarason’s idea applied in [10, proof of Proposition 1]. Without loss of generality we may assume that λ = 1 . Let A m = T ( z - 1 ) m be the Toeplitz operator with the symbol ( z - 1 ) m on H 2 . Let ℳ ⁢ ( A m ) be the range of A m equipped with the Hilbert space structure that makes A m a coisometry from H 2 onto ℳ ⁢ ( A m ) . Since the reproducing kernel function for the space H 2 is ( 1 - w ¯ ⁢ z ) - 1 , for g ∈ ℳ ⁢ ( A m ) we get

g ⁢ ( w ) = 〈 g , ( z - 1 ) m ⁢ ( w ¯ - 1 ) m 1 - w ¯ ⁢ z 〉 ℳ ⁢ ( A m ) , w ∈ 𝔻 ,

where the last inner product is in the space ℳ ⁢ ( A m ) . This implies that

g ( m ) ⁢ ( w ) = 〈 g , ∂ m ∂ ⁡ w ¯ m ⁢ ( z - 1 ) m ⁢ ( w ¯ - 1 ) m 1 - w ¯ ⁢ z 〉 ℳ ⁢ ( A m )

= 〈 g ( m ) , ∂ 2 ⁢ m ∂ ⁡ z m ⁢ ∂ ⁡ w ¯ m ⁢ ( z - 1 ) m ⁢ ( w ¯ - 1 ) m 1 - w ¯ ⁢ z 〉 ℳ ( m ) ⁢ ( A m ) ,

where the last inner product is taken in the range space ℳ ( m ) ⁢ ( A m ) of the operator of differentiation of order m. By Lemma 1,

∂ 2 ⁢ m ∂ ⁡ z m ⁢ ∂ ⁡ w ¯ m ⁢ ( z - 1 ) m ⁢ ( w ¯ - 1 ) m 1 - w ¯ ⁢ z = ( 2 ⁢ m ) ! ⁢ ( z - 1 ) m ⁢ ( w ¯ - 1 ) m ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 1 .

Observe that by (1.3),

ℳ ( m ) ⁢ ( A m ) = { f ( m ) : f ∈ 𝒟 1 m } .

We now note that the reproducing kernel for the space

A 2 ⁢ ( ρ m ) = { h ∈ Hol ⁢ ( 𝔻 ) : ∥ h ∥ A 2 ⁢ ( ρ m ) 2 = 1 ( 2 ⁢ m - 1 ) ! ⁢ ∫ 𝔻 | h ⁢ ( z ) | 2 ⁢ ( 1 - | z | 2 ) 2 ⁢ m - 1 ⁢ 𝑑 A ⁢ ( z ) < ∞ }

equals

( 2 ⁢ m ) ! ( 1 - w ¯ ⁢ z ) 2 ⁢ m + 1 .

Thus the space { ( z - 1 ) m ⁢ h : h ∈ A 2 ⁢ ( ρ m ) } has the same reproducing kernel as the space { f ( m ) : f ∈ 𝒟 1 m } . This means that f ∈ 𝒟 1 m if and only if h = f ( m ) ( z - 1 ) m ∈ A 2 ⁢ ( ρ m ) .

In other words, 𝒟 1 m ⁢ ( f ) < + ∞ if and only if I 1 m ⁢ ( f ) < + ∞ . Moreover, in that case f = p + ( z - 1 ) m ⁢ g ( g ∈ H 2 and p is a polynomial of degree < m , see (1.3)) and

𝒟 1 m ⁢ ( f ) = ∥ g ∥ 2 2 = ∥ ( z - 1 ) m ⁢ g ∥ ℳ ⁢ ( A m ) 2 = ∥ ( ( z - 1 ) m ⁢ g ) ( m ) ∥ ℳ ( m ) ⁢ ( A m ) 2

= ∥ f ( m ) ∥ ℳ ( m ) ⁢ ( A m ) 2 = ∥ ( z - 1 ) m ⁢ h ∥ ℳ ( m ) ⁢ ( A m ) 2 = ∥ h ∥ A 2 ⁢ ( ρ m ) 2

= ∥ f ( m ) ( z - 1 ) m ∥ A 2 ⁢ ( ρ m ) 2 = 1 ( 2 ⁢ m - 1 ) ! ⁢ I 1 m ⁢ ( f ) . ∎

Corollary 1.

If a function f holomorphic in D is such that for a λ ∈ T ,

(2.2) ∫ 𝔻 | f ( m ) ⁢ ( z ) ( z - λ ) m | 2 ⁢ ( 1 - | z | 2 ) 2 ⁢ m - 1 ⁢ 𝑑 A ⁢ ( z ) < ∞ ,

then f ∈ H 2 and for each j = 0 , 1 , … , m - 1 the function f ( j ) has a nontangential limit at λ.

Proof.

We can clearly assume that λ = 1 . It follows from the proof of Theorem 1 that the operator T m given by

T m ⁢ ( g ) = ( ( z - 1 ) m ⁢ g ) ( m ) ( z - 1 ) m

is an isometry of H 2 onto A 2 ⁢ ( ρ m ) . If f satisfies condition (2.2), there exists g ∈ H 2 such that f ( m ) = ( ( z - 1 ) m ⁢ g ) ( m ) . This means that f ⁢ ( z ) = ( z - 1 ) m ⁢ g + p , where p is a polynomial of degree < m , and the reasoning used in [6, proof of Lemma 9.1] proves the claim. ∎

3 De Branges–Rovnyak spaces ℋ ⁢ ( b ) and local Dirichlet spaces of finite order

We first cite one of the main results contained in [6] using notation from the Introduction. Recall that a Hilbert space operator T is an m-isometry if

∑ k = 0 m ( - 1 ) m - k ⁢ ( m k ) ⁢ T * k ⁢ T k = 0 .

It is easy to check that every m-isometry is a k-isometry for every k ≥ m . An m-isometry that is not an ( m - 1 ) -isometry is called a strict m-isometry.

Theorem ([6]).

Let b be a non–extreme point of the unit ball of H ∞ with b ⁢ ( 0 ) = 0 , and let m ∈ N . Then Y is not a strict ( 2 ⁢ m + 1 ) -isometry, and the following are equivalent:

Y is a strict 2 ⁢ m - isometry.
( b , a ) is a rational pair such that a has a single zero of multiplicity m at a point λ ∈ 𝕋 .
There is a λ ∈ 𝕋 and a polynomial p ~ of degree < m with p ~ ⁢ ( λ ) ≠ 0 such that

∥ f ∥ b 2 = ∥ f ∥ 2 2 + 𝒟 λ m ⁢ ( p ~ ⁢ f ) .

If the three conditions hold, then there are polynomials p and r of degree ≤ m such that b = p r , a = ( z - λ ) m r , and p ~ ⁢ ( z ) = z m ⁢ p ⁢ ( 1 z ¯ ) ¯ for z ∈ D , and | r ⁢ ( z ) | 2 = | p ⁢ ( z ) | 2 + | z - λ | 2 ⁢ m for all z ∈ T . Furthermore, H ⁢ ( b ) = D λ m with equivalence of norms.

In particular, ℋ ⁢ ( b ) = 𝒟 λ m with equality of norms if and only if b ⁢ ( z ) = z m r ⁢ ( z ) , where r is a polynomial of degree m that has no zeros in 𝔻 ¯ and such that | r ⁢ ( z ) | 2 = 1 + | z - λ | 2 ⁢ m for all z ∈ 𝕋 .

Now we give another proof of the suffcient condition for ℋ ⁢ ( b ) = 𝒟 λ m with equality of norms. The following proposition is actually contained in the above cited result of [6]. Our proof is based on considering ℋ ⁢ ( b ) as a domain of an unbounded Toeplitz operator T φ , φ ∈ 𝒩 + (see the Introduction). We provide a formula for φ = b a such that ℋ ⁢ ( b ) = 𝒟 λ m with equality of norms (see (3.1)). Moreover, we essentially compute T φ ⁢ f for such φ (see (3.2)).

Proposition ([6]).

If for λ ∈ T the function φ λ m ∈ N + is defined by

(3.1) φ λ m ⁢ ( z ) = ( λ ¯ ⁢ z ) m ( 1 - λ ¯ ⁢ z ) m , z ∈ 𝔻 ,

and its canonical representation is φ λ m = b λ a λ , then H ⁢ ( b λ ) = D λ m with equality of norms.

Proof.

We first show that for f ∈ Hol ⁢ ( 𝔻 ¯ ) , the space of functions holomorphic on 𝔻 ¯ ,

(3.2) T φ λ m ¯ ⁢ f ⁢ ( z ) = λ m ⁢ f ⁢ ( z ) - T m - 1 ⁢ ( f , λ ) ⁢ ( z ) ( z - λ ) m .

Let for λ ∈ 𝕋 ,

k λ ⁢ ( z ) = 1 1 - λ ¯ ⁢ z , z ∈ 𝔻 .

It has been derived in [5] that for f ∈ Hol ⁢ ( 𝔻 ¯ ) ,

T k ¯ λ ⁢ f ⁢ ( z ) = f ⁢ ( z ) + λ ⁢ f ⁢ ( λ ) - f ⁢ ( z ) λ - z .

Since

φ λ 1 ⁢ ( z ) = λ ¯ ⁢ z 1 - λ ¯ ⁢ z ,

we get

T φ λ 1 ¯ ⁢ f ⁢ ( z ) = T k λ - 1 ¯ ⁢ f ⁢ ( z ) = λ ⁢ f ⁢ ( z ) - f ⁢ ( λ ) z - λ

which proves (3.2) for m = 1 . Assume now that (3.2) holds for m - 1 . Since for a nonextreme b the space Hol ⁢ ( 𝔻 ¯ ) is a subspace of ℋ ⁢ ( b ) and T φ ¯ ⁢ ( Hol ⁢ ( 𝔻 ¯ ) ) ⊂ Hol ⁢ ( 𝔻 ¯ ) , φ = b a (see [12, Lemma 6.1]), we get

T φ λ m ¯ ⁢ f ⁢ ( z ) = T φ λ 1 ¯ ⁢ T φ λ m - 1 ¯ ⁢ f ⁢ ( z )

= T φ λ 1 ¯ ⁢ ( λ m - 1 ⁢ f ⁢ ( z ) - T m - 2 ⁢ ( f , λ ) ⁢ ( z ) ( z - λ ) m - 1 ) = λ m ⁢ f ⁢ ( z ) - T m - 1 ⁢ ( f , λ ) ⁢ ( z ) ( z - λ ) m ,

where the last equality follows from the fact that for f ∈ Hol ⁢ ( 𝔻 ¯ ) ,

lim z → λ ⁡ f ⁢ ( z ) - T m - 2 ⁢ ( f , λ ) ⁢ ( z ) ( z - λ ) m - 1 = f ( m - 1 ) ⁢ ( λ ) ( m - 1 ) ! .

Note now that polynomials are a dense subset of both ℋ ⁢ ( b λ ) and 𝒟 λ m , and by (1.1), (1.4) and (3.2), for each polynomial f,

∥ f ∥ b λ = ∥ f ∥ 𝒟 λ m .

Since the norms ∥ ⋅ ∥ b λ and ∥ ⋅ ∥ 𝒟 λ m are equivalent, we get ℋ ⁢ ( b λ ) = 𝒟 λ m with equality of norms. ∎

In the following proposition we find the explicit formulas for b λ for which ℋ ⁢ ( b λ ) = 𝒟 λ m with equality of norms.

Proposition 1.

Let for λ ∈ T and m ∈ N the function φ λ m ∈ N + be defined by (3.1). Then the canonical representation ( b λ , a λ ) of φ λ m is given by

b λ ⁢ ( z ) = C m ⁢ ( λ ¯ ⁢ z ) m ∏ k = 1 m ( 1 - λ ¯ ⁢ z k ⁢ z ) , a λ ⁢ ( z ) = C m ⁢ ( 1 - λ ¯ ⁢ z ) m ∏ k = 1 m ( 1 - λ ¯ ⁢ z k ⁢ z )

with C m = ∏ k = 1 m ( 1 - z k ) , where z k are preimages of the m-th roots of 1 for odd m and m-th roots of -1 for even m under the Koebe function k ⁢ ( z ) = z ( 1 - z ) 2 .

Proof.

Assume that φ λ m = b λ a λ , where ( b λ , a λ ) is a pair. Then for | z | = 1 ,

1 + | 1 - λ ¯ ⁢ z | 2 ⁢ m = | 1 - λ ¯ ⁢ z | 2 ⁢ m | a λ ⁢ ( z ) | 2 .

By the Féjer–Riesz theorem there is a unique polynomial r of degree m without zeros in 𝔻 ¯ such that r ⁢ ( 0 ) > 0 and

(3.3) 1 + | 1 - λ ¯ ⁢ z | 2 ⁢ m = | r ⁢ ( z ) | 2 .

Now define a polynomial

w ⁢ ( z ) = z m + ( 2 ⁢ z - z 2 - 1 ) m = z m + ( - 1 ) m ⁢ ( 1 - z ) 2 ⁢ m

and observe that for | z | = 1 ,

| w ⁢ ( λ ¯ ⁢ z ) | = 1 + | 1 - λ ¯ ⁢ z | 2 ⁢ m .

Next note that w ⁢ ( z ) = 0 if and only if

( z ( 1 - z ) 2 ) m = ( - 1 ) m + 1 .

This implies that, in the case when m is odd, the zeros of w in 𝔻 are z k = g ⁢ ( e k ) , k = 0 , 1 , … , m - 1 , where g is the inverse of the Koebe function k ⁢ ( z ) = z ( 1 - z ) 2 and e k are the m-th roots of 1. In the case m is even, the zeros of w in 𝔻 are z k = g ⁢ ( e ~ k ) , k = 1 , … , m , where e ~ k are the m-th roots of -1. The other zeros of w are 1 z ¯ k , k = 1 , … , m . Consequently, the zeros of w ⁢ ( λ ¯ ⁢ z ) in 𝔻 are λ ⁢ z k , k = 1 , … , m . Thus

w ⁢ ( λ ¯ ⁢ z ) = c ⁢ ∏ k = 1 m ( z - λ ⁢ z k ) ⁢ ( z - 1 λ ⁢ z k ¯ ) .

Since for | z | = 1

| z - λ ⁢ z k | = | z ⁢ λ ⁢ z k ¯ - 1 | ,

the polynomial r ⁢ ( z ) is given by

r ⁢ ( z ) = c 0 ⁢ ∏ k = 1 m ( 1 - λ ¯ ⁢ z k ⁢ z ) ,

where in view of (3.3), c 0 = ( ∏ k = 1 m ( 1 - z k ) ) - 1 . We also mention that

min | z | = 1 ⁡ | r ⁢ ( z ) | = 1 = r ⁢ ( λ ) .

Consequently, we get

b λ ⁢ ( z ) = ( λ ¯ ⁢ z ) m r ⁢ ( z ) , a λ ⁢ ( z ) = ( 1 - λ ¯ ⁢ z ) m r ⁢ ( z ) ,

which proves the proposition. ∎

4 ℋ ⁢ ( b ) generated by nonextreme rational b

Let ( b , a ) be a rational pair and let λ 1 , λ 2 , … ⁢ λ m be all the zeros of a on 𝕋 , listed according to multiplicity. It has been proved in [2] that for every such rational pair

(4.1) ℋ ⁢ ( b ) = { ∏ j = 1 m ( z - λ j ) ⁢ g + s : g ∈ H 2 , s ∈ 𝒫 m - 1 } ,

where 𝒫 n denotes the set of polynomials of degree at most n. This means that the space ℋ ⁢ ( b ) is in fact determined by the zeros of a on 𝕋 . Moreover, in the case when λ 1 = λ 2 = … = λ m = λ , by (1.3), ℋ ⁢ ( b ) = 𝒟 λ m (as sets).

For fixed λ 1 , λ 2 , … ⁢ λ m ∈ 𝕋 and a polynomial p such that p ⁢ ( λ j ) ≠ 0 , j = 1 , 2 , … , m , consider the function φ ∈ 𝒩 + defined by

(4.2) φ ⁢ ( z ) = p ⁢ ( z ) ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) = b ⁢ ( z ) a ⁢ ( z ) ,

where ( b , a ) is a canonical pair. It is worth noting that the pair ( b , a ) is rational. Moreover, by the aforementioned result from [2], for each polynomial p, the space ℋ ⁢ ( b ) is described by (4.1) and the corresponding norms ∥ ⋅ ∥ b given by (1.1) are equivalent.

For a positive integer m set

φ m ⁢ ( z ) = z m ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) = b m ⁢ ( z ) a m ⁢ ( z ) .

The next proposition slightly extends the result contained in [6, Theorem 9.4].

Proposition 2.

Let φ = b a be defined by (4.2) with the polynomial p of degree ≤ m , and let q be a polynomial of degree ≤ m such that q ⁢ ( λ j ) ≠ 0 for j = 1 , … , m . Then

(4.3) ∥ f ∥ b 2 = ∥ f ∥ 2 2 + ∥ T φ ¯ m ⁢ ( q ⁢ f ) ∥ 2 2 for each ⁢ f ∈ ℋ ⁢ ( b )

if and only if, up to a multiplicative unimodular constant, q ⁢ ( z ) = p ~ ⁢ ( z ) = z m ⁢ p ⁢ ( 1 z ¯ ) ¯ .

Proof.

Assume that q = p ~ . Then, by [12, Proposition 6.5],

T φ ¯ ⁢ f = T 1 ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) ¯ ⁢ T p ¯ ⁢ f = T 1 ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) ¯ ⁢ T z ¯ m ⁢ ( p ~ ⁢ f ) = T φ ¯ m ⁢ ( p ~ ⁢ f ) .

Assume now that (4.3) holds with a polynomial q. Since

∥ f ∥ b 2 = ∥ f ∥ 2 2 + ∥ T φ ¯ ⁢ f ∥ 2 2 ,

we get

∥ T φ ¯ ⁢ f ∥ 2 2 = ∥ T φ ¯ m ⁢ ( q ⁢ f ) ∥ 2 2 .

On the other hand,

T q ~ ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) ¯ ⁢ f = T 1 ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) ¯ ⁢ T q ~ ¯ ⁢ f = T 1 ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) ¯ ⁢ T z ¯ m ⁢ ( q ⁢ f ) = T φ ¯ m ⁢ ( q ⁢ f ) ,

and so

∥ T φ ¯ ⁢ f ∥ 2 2 = ∥ T φ ¯ m ⁢ ( q ⁢ f ) ∥ 2 2 = ∥ T q ~ ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) ¯ ⁢ f ∥ 2 2 .

Consequently, if

φ * ⁢ ( z ) = q ~ ⁢ ( z ) ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) = b * ⁢ ( z ) a * ⁢ ( z ) ,

then ℋ ⁢ ( b ) = ℋ ⁢ ( b * ) with equality of norms. By [3, Vol. 2, Corollary 27.12], b = c * ⋅ b * for some unimodular constant c * . Since | a | = | a * | on 𝕋 and the modulus of an outer function on 𝕋 determines that function up to a multiplicative unimodular constant, we obtain that q = c ⋅ p ~ for some c ∈ 𝕋 . ∎

We now prove the following:

Theorem 2.

Let ( b , a ) be a rational pair such that the corresponding Smirnov function φ is given by (4.2) with p ⁢ ( z ) = α 0 + α 1 ⁢ z + … + α m ⁢ z m such that p ⁢ ( λ j ) ≠ 0 , j = 1 , 2 , … , m . Then there is a finite Blaschke product B m of degree m, such that the outer part of a wandering vector of Y lies in H ⁢ ( z ⁢ B m ) .

Proof.

We now use reasoning from the proof of Féjer–Riesz theorem (see, e.g., [4]). Set

v ⁢ ( e i ⁢ t ) = | ∏ j = 1 m ( 1 - λ ¯ j ⁢ e i ⁢ t ) | 2 + | p ⁢ ( e i ⁢ t ) | 2 = ∑ j = - m 0 m 0 c j ⁢ e i ⁢ j ⁢ t ,

where m 0 ≤ m and c ¯ j = c - j , j = 1 , 2 , … , m 0 . Then v ⁢ ( e i ⁢ t ) > 0 and

w ⁢ ( z ) = z m 0 ⁢ v ⁢ ( z ) = ∑ j = 0 2 ⁢ m 0 c j - m 0 ⁢ z j , z ∈ 𝔻 ,

is an analytic polynomial of degree 2 ⁢ m 0 with no zeros on 𝕋 and such that w ⁢ ( 0 ) ≠ 0 . Moreover, the zeros of w are w k and 1 w ¯ k , where w k ∈ ℂ ∖ 𝔻 ¯ , k = 1 , 2 , … , m 0 . Therefore

(4.4) w ⁢ ( z ) = c ⁢ ∏ j = 1 m 0 ( z - w j ) ⁢ ( 1 - w ¯ j ⁢ z ) , c > 0 .

With the notation above we define the function

W ⁢ ( z ) = ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) 2 + ( - 1 ) m ⁢ λ ¯ ⁢ p ⁢ ( z ) ⁢ p ~ ⁢ ( z ) ,

where λ = λ 1 ⋅ λ 2 ⁢ ⋯ ⁢ λ m . Observe that for | z | = 1 ,

W ⁢ ( z ) = ( - 1 ) m ⁢ λ ¯ ⁢ z m ⁢ ( ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) ⁢ ( 1 - λ j ⁢ z ¯ ) + z ¯ m ⁢ p ~ ⁢ ( z ) ⁢ p ⁢ ( z ) )

= ( - 1 ) m ⁢ λ ¯ ⁢ z m ⁢ ( ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) ⁢ ( 1 - λ j ⁢ z ¯ ) + p ⁢ ( z ) ¯ ⁢ p ⁢ ( z ) )

= ( - 1 ) m ⁢ λ ¯ ⁢ z m ⁢ v ⁢ ( z ) = ( - 1 ) m ⁢ λ ¯ ⁢ z m - m 0 ⁢ w ⁢ ( z ) .

Now we apply the idea from [11, proof of Lemma 3]. If we assume that h is a wandering vector of Y, then for k = 0 , 1 , … ,

0 = 〈 z k + 1 ⁢ ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) 2 ⁢ h , h 〉 b

= 〈 z k + 1 ⁢ ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) 2 ⁢ h , h 〉 2 + 〈 T φ ¯ ⁢ ( z k + 1 ⁢ ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) 2 ⁢ h ) , T φ ¯ ⁢ ( h ) 〉 2

= 〈 z k + 1 ⁢ ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) 2 ⁢ h , h 〉 2 + 〈 P ⁢ ( p ⁢ ( z ) ¯ ⁢ z k + 1 ⁢ ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) 2 ⁢ h ∏ j = 1 m ( 1 - λ j ⁢ z ¯ ) ) , T φ ¯ ⁢ ( h ) 〉 2

= 〈 z k + 1 ⁢ ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) 2 ⁢ h , h 〉 2 + 〈 ( - 1 ) m ⁢ λ ¯ ⁢ z m ⁢ p ⁢ ( z ) ¯ ⁢ z k + 1 ⁢ ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) ⁢ h , T φ ¯ ⁢ ( h ) 〉 2

= 〈 z k + 1 ⁢ ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) 2 ⁢ h , h 〉 2 + 〈 φ ⁢ ( z ) ⁢ ( - 1 ) m ⁢ λ ¯ ⁢ p ~ ⁢ ( z ) ⁢ z k + 1 ⁢ ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) ⁢ h , h 〉 2

= 〈 z k + 1 ⁢ ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) 2 ⁢ h , h 〉 2 + 〈 ( - 1 ) m ⁢ λ ¯ ⁢ p ⁢ ( z ) ⁢ p ~ ⁢ ( z ) ⁢ z k + 1 ⁢ h , h 〉 2

= 〈 z k + 1 ⁢ W ⁢ h 0 , h 0 〉 2 ,

where h 0 denotes the outer part of h. It follows that h 0 is orthogonal to the subspace of H 2 spanned by z k + 1 ⁢ W ⁢ h 0 , k = 1 , 2 , … . Since h 0 is an outer function, this subspace is z ⁢ B m ⁢ H 2 , where B m = z m - m 0 ⁢ B m 0 and B m 0 is a Blaschke product with zeros z k = 1 w ¯ k ∈ 𝔻 . Hence h ∈ ℋ ⁢ ( z ⁢ B m ) = H 2 ⊖ z ⁢ B m ⁢ H 2 . ∎

Remark.

Let b be a nonextreme function determined by φ = b a from Theorem 2. We remark that every function of the form ua, where u is inner, is a wandering vector of Y. This follows from the fact that multiplication by a is an isometry of H 2 into ℋ ⁢ ( b ) . We first note that by (4.4) we have

a ⁢ ( z ) = ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) r ⁢ ( z ) and b ⁢ ( z ) = p ⁢ ( z ) r ⁢ ( z ) ,

where r ⁢ ( z ) = c ⁢ ∏ j = 1 m ( z - w k ) . Consequently, for f ∈ H 2 ,

∥ a ⁢ f ∥ b 2 = ∥ a ⁢ f ∥ 2 2 + ∥ T φ ¯ ⁢ ( a ⁢ f ) ∥ 2 2 = ∥ a ⁢ f ∥ 2 2 + ∥ P ⁢ ( φ ¯ ⁢ a ⁢ f ) ∥ 2 2

= ∥ a f ∥ 2 2 + ∥ P ( p ⁢ ( z ) ¯ ∏ j = 1 m ( 1 - λ j ⁢ z ¯ ) ∏ j = 1 m ( 1 - λ ¯ j ⁢ z ) r ⁢ ( z ) f ) ∥ 2 2 = ∥ a f ∥ 2 2 + ∥ P ( z m ⁢ p ⁢ ( z ) ¯ r ⁢ ( z ) f ) ∥ 2 2

= ∥ a ⁢ f ∥ 2 2 + ∥ b ⁢ f ∥ 2 2 = ∥ f ∥ 2 2 .

It follows that for an inner function u and every k ≥ 0 ,

〈 Y k + 1 ⁢ ( u ⁢ a ) , u ⁢ a 〉 b = 〈 u ⁢ a ⁢ z k + 1 , u ⁢ a 〉 b = 〈 u ⁢ z k + 1 , u 〉 2 = 〈 z k + 1 , 1 〉 2 = 0 .

Communicated by Siegfried Echterhoff

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Received: 2023-06-23

Published Online: 2023-10-04

Published in Print: 2024-01-01

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Keywords for this article

Hardy space; de Branges–Rovnyak spaces; local Dirichlet space; wandering vector

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