Generalized Crofoot transform and applications

Rewayat Khan; Aamir Farooq

doi:10.1515/conop-2022-0138

Article Open Access

Generalized Crofoot transform and applications

Rewayat Khan and Aamir Farooq

Published/Copyright: February 18, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

From the journal Concrete Operators Volume 10 Issue 1

Abstract

Matrix-valued asymmetric truncated Toeplitz operators are compressions of multiplication operators acting between two model spaces. These are the generalization of matrix-valued truncated Toeplitz operators. In this article, we describe symbols of matrix-valued asymmetric truncated Toeplitz operators equal to the zero operator. We also use generalized Crofoot transform to find a connection between the symbols of matrix-valued asymmetric truncated Toeplitz operators T ( Θ 1 , Θ 2 ) and T ( Θ 1 ′ , Θ 2 ′ ) .

Keywords: generalized Crofoot transform; matrix-valued truncated Toeplitz operators; matrix-valued asymmetric truncated Toeplitz operators

MSC 2010: Primary 47B35; 47A45; Secondary 47B32; 30J05

1 Introduction

Truncated Toeplitz operators are compressions of multiplication operators to the backward shift invariant subspaces of the Hardy-Hilbert space H 2 . These subspaces are called model spaces and denoted by K θ = [ θ H 2 ] ⊥ , with θ a complex-valued inner function, i.e., ∣ θ ( e it ) ∣ = 1 a.e. on T . The study of truncated Toeplitz operators was initiated by Sarason in 2007 (see [14]).

In [9], matrix-valued truncated Toeplitz operators by considering E a finite d -dimensional Hilbert space and Θ an ℒ ( E ) -valued inner function have been introduced. The corresponding model space is denoted by K Θ = [ Θ H 2 ( E ) ] ⊥ . This is the main object of study in this article. In [9], a characterization of the symbol of the matrix-valued truncated Toeplitz operators equal to zero operator has been provided. Like the truncated Toeplitz operators, the matrix-valued truncated Toeplitz operators are not uniquely determined by their symbols (see [9, Theorem 6.3, Corollary 6.4]).

Recently in [10], a generalization of the Crofoot transform, called generalized Crofoot transform, has been introduced. The transform is a unitary operator and maps one vector-valued model space onto another. The study of truncated Toeplitz operators has a natural generalization to asymmetric truncated Toeplitz operators. This study was recently initiated in [1–5] (see also [6–8,11,12]). Similarly, the study of matrix-valued truncated Toeplitz operators can be extended to the study of matrix-valued asymmetric truncated Toeplitz operators which act between two different model spaces. In [2, Theorem 4.4], zero asymmetric truncated Toeplitz operators have been characterized in terms of their symbol for a special case when one of the inner functions divides the other. A proof for the general case can be found in [7, Theorem 2.1]. In this article, we generalize results from [7] and [9] to matrix-valued asymmetric truncated Toeplitz operators.

This article is organized as follows: In Section 2, we gather some properties of the model spaces, bounded truncated and matrix-valued truncated Toeplitz operators. In Section 3, we use generalized Crofoot transform to establish a connection between the symbols of matrix-valued asymmetric truncated Toeplitz operators on two different pairs of model spaces. In Section 4, we describe the symbols of matrix-valued asymmetric truncated Toeplitz operators equal to the zero operator.

2 Preliminaries

Let us consider the complex plane C , the unit open disc D = { z ∈ C : ∣ z ∣ < 1 } , and the unit circle T = { z ∈ C : ∣ z ∣ = 1 } . Throughout the article, E will denote d -dimensional complex Hilbert space, and ℒ ( E ) will denote the algebra of bounded linear operators on E , which may be identified with d × d matrices.

The space L 2 ( E ) is defined, as usual, by

L 2 ( E ) = f : T → E : f ( e it ) = ∑ − ∞ ∞ a n e int : a n ∈ E , ∑ − ∞ ∞ ‖ a n ‖ 2 < ∞ ,

endowed with the inner product

⟨ f , g ⟩ L 2 ( E ) = 1 2 π ∫ 0 2 π ⟨ f ( e it ) , g ( e it ) ⟩ E d t .

As usual, H 2 ( E ) is the space of E -valued analytic functions defined on the unit disc D , whose Taylor coefficients are square summable. It can also be seen as a closed subspace of L 2 ( E ) , which consists of functions from L 2 ( E ) such that their Fourier coefficients corresponding to negative indices vanish.

The unilateral shift S on H 2 ( E ) is the operator of multiplication by z , that is,

S f ( z ) = z f ( z ) .

The adjoint S ∗ is called the backward shift and is given by the formula

S ∗ f ( z ) = 1 z ( f ( z ) − f ( 0 ) ) .

The space of all bounded analytic functions in H 2 ( ℒ ( E ) ) is denoted by H ∞ ( ℒ ( E ) ) . A nonconstant function Θ ∈ H ∞ ( ℒ ( E ) ) is called an inner function if Θ is an isometry a.e. on T . In the sequel, we will always suppose that Θ is pure inner function, which means that ‖ Θ ( 0 ) ‖ < 1 . Beurling theorem provides a characterization of all backward shift nontrivial closed invariant subspaces of H 2 ( E ) . A closed nontrivial subspace of H 2 ( E ) is S ∗ -invariant if and only if it is of the form

K Θ = H 2 ( E ) ⊖ Θ H 2 ( E ) ,

for some inner function Θ ∈ H ∞ ( ℒ ( E ) ) . The space K Θ is called a model space.

A Toeplitz operator T Φ with symbol Φ ∈ L ∞ ( ℒ ( E ) ) is defined on H 2 ( E ) by

T Φ f = P ( Φ f ) ,

where P is an orthogonal projection from L 2 ( E ) onto H 2 ( E ) . The operators S and S ∗ are examples of Toeplitz operators with symbols z I E and z ¯ I E , respectively, where I E is the identity operator on E .

Recall that model spaces are reproducing kernel Hilbert spaces with reproducing kernel function k λ Θ x , i.e., for every f ∈ K Θ and each λ ∈ D , we have

⟨ f , k λ Θ x ⟩ = ⟨ f ( λ ) , x ⟩ ,

where the reproducing kernel k λ Θ ( z ) , which takes the values in ℒ ( E ) , is

k λ Θ ( z ) = 1 1 − λ ¯ z ( I − Θ ( z ) Θ ( λ ) ∗ ) .

It is well known that k λ Θ x ∈ H ∞ ( E ) and the set K Θ ∞ = K Θ ∩ H ∞ ( E ) is dense in K Θ .

In [9], matrix-valued truncated Toeplitz operators and their properties have been studied. Suppose Θ is a fixed pure inner function and that Φ ∈ L 2 ( ℒ ( E ) ) . Let P Θ be the orthogonal projection from H 2 ( E ) onto K Θ and consider the linear map f → P Θ ( Φ f ) defined on K Θ ∞ . In case it is bounded, it uniquely determines an operator in ℒ ( K Θ ) , denoted by A Φ Θ , and called a matrix-valued truncated Toeplitz operator; Φ is called the symbol of A Φ Θ . The space of all matrix-valued truncated Toeplitz operators on K Θ is denoted by T ( Θ ) . It is immediate that ( A Φ Θ ) ∗ = A Φ ∗ Θ .

Recently, the study of the class of standard asymmetric truncated Toeplitz operators has been initiated in [2–4]. Let Θ 1 and Θ 2 be two inner functions. A matrix-valued asymmetric truncated Toeplitz operator A Φ Θ 1 , Θ 2 with symbol Φ ∈ L 2 ( ℒ ( E ) ) is an operator from K Θ 1 into K Θ 2 densely defined by

A Φ Θ 1 , Θ 2 f = P Θ 2 ( Φ f ) ,

for all f in K Θ 1 ∞ .

The space of all matrix-valued asymmetric truncated Toeplitz operators is denoted by T ( Θ 1 , Θ 2 ) . These operators have also been studied in [7,8,11,12].

An example of matrix-valued asymmetric truncated Toeplitz operators is given as follows:

Example 2.1

Let Θ 1 ( z ) = z n I E , Θ 2 ( z ) = z m I E , such that m < n . Then the corresponding model spaces are as follows:

K Θ 1 = { a 0 + a 1 z + … + a n − 1 z n − 1 : a k ∈ E } , K Θ 2 = { b 0 + b 1 z + … + b m − 1 z m − 1 : b k ∈ E } ,

for Φ ( e it ) = ∑ s ∈ Z e i s t Δ s , with Δ s ∈ ℒ ( E ) , the operator A Φ Θ 1 , Θ 2 ∈ T ( Θ 1 , Θ 2 ) has the block matrix decomposition:

A Φ Θ 1 , Θ 2 = Δ 0 Δ 1 … Δ m − 1 … Δ n − 1 Δ − 1 Δ 0 … Δ m − 2 … Δ n − 2 ⋮ ⋮ ⋮ … ⋮ ⋮ Δ − ( m − 1 ) Δ − ( m − 2 ) … Δ 0 … Δ n − m .

3 Connection between T ( Θ 1 , Θ 2 ) and T ( Θ 1 ′ , Θ 2 ′ )

A bounded linear operator W ∈ ℒ ( E ) is called a contraction if ‖ W ‖ ≤ 1 and strict contraction if ‖ W ‖ < 1 . We will use the following standard notations. If W ∈ ℒ ( E ) is a contraction, then the operators D W = ( I − W ∗ W ) 1 2 and D W ∗ = ( I − W W ∗ ) 1 2 are called the defect operators of W .

Let W 1 , W 2 ∈ ℒ ( E ) be strict contractions and let Θ 1 , Θ 2 ∈ H ∞ ( ℒ ( E ) ) be pure inner functions. Then by [10], the function Θ 1 ′ defined in terms of W 1 and Θ 1 by

(3.1) Θ 1 ′ ( z ) = − W 1 + D W 1 ∗ ( I − Θ 1 ( z ) W 1 ∗ ) − 1 Θ 1 ( z ) D W 1

is a pure inner function. Similarly, the function

(3.2) Θ 2 ′ ( z ) = − W 2 + D W 2 ∗ ( I − Θ 2 ( z ) W 2 ∗ ) − 1 Θ 2 ( z ) D W 2

is also pure.

Let K Θ 1 and K Θ 2 be the model spaces corresponding to Θ 1 and Θ 2 , respectively, and let K Θ 1 ′ and K Θ 2 ′ be the model spaces corresponding to Θ 1 ′ and Θ 2 ′ , respectively.

The generalized Crofoot transform defined in [10, Theorem 3.3] is a unitary operator J W 1 : K Θ 1 ⟶ K Θ 1 ′ given by

J W 1 f ( z ) = D W 1 ∗ ( I − Θ 1 ( z ) W 1 ∗ ) − 1 f ( z ) ,

and its adjoint J W 1 ∗ : K Θ 1 ′ ⟶ K Θ 1 is given by

J W 1 ∗ g ( z ) = D W 1 ∗ ( I + Θ 1 ′ ( z ) W 1 ∗ ) − 1 g ( z ) .

We define J W 2 : K Θ 2 → K Θ 2 ′ and its adjoint J W 2 ∗ analogously.

The following formula [10, Proposition 3.4] shows the relation between J W 2 and the reproducing kernels in two spaces

(3.3) J W 2 ( k λ Θ 2 ( I − W 2 Θ 2 ( λ ) ∗ ) − 1 D W 2 ∗ y ) = k λ Θ 2 ′ y , y ∈ E .

Our next theorem is a vector-valued version of [7, Proposition 2.5].

Theorem 3.1

Let Θ 1 , Θ 2 ∈ H ∞ ( ℒ ( E ) ) be nonconstant inner functions and Φ ∈ L 2 ( ℒ ( E ) ) . Then the operator C : ℒ ( K Θ 1 , K Θ 2 ) → ℒ ( K Θ 1 ′ , K Θ 2 ′ ) defined by C ( A ) = J W 2 A J W 1 ∗ maps isometrically T ( Θ 1 , Θ 2 ) onto T ( Θ 1 ′ , Θ 2 ′ ) , and C ( A Φ Θ 1 , Θ 2 ) = A Ψ Θ 1 ′ , Θ 2 ′ , where Ψ is given by

(3.4) Ψ ( z ) = D W 2 ∗ ( I − Θ 2 ( z ) W 2 ∗ ) − 1 Φ ( z ) D W 1 ∗ ( I + Θ 1 ′ ( z ) W 1 ∗ ) − 1 .

Proof

Since J W 1 and J W 2 are unitary operators between the model spaces, it follows that C is isometric.

Let f ∈ K Θ 1 ′ . Then for any λ ∈ D and x ∈ E ,

⟨ C ( A Φ Θ 1 , Θ 2 ) f , k λ Θ 2 ′ x ⟩ = ⟨ J W 2 A Φ Θ 1 , Θ 2 J W 1 ∗ f , k λ Θ 2 ′ x ⟩ = ⟨ J W 2 A Φ Θ 1 , Θ 2 D W 1 ∗ ( I + Θ 1 ′ W 1 ∗ ) − 1 f , k λ Θ 2 ′ x ⟩ = ⟨ P Θ 2 ( Φ D W 1 ∗ ( I + Θ 1 ′ W 1 ∗ ) − 1 f ) , J W 2 ∗ k λ Θ 2 ′ x ⟩ = ⟨ P Θ 2 ( Φ D W 1 ∗ ( I + Θ 1 ′ W 1 ∗ ) − 1 f ) , k λ Θ 2 ( I − W 2 Θ 2 ∗ ) − 1 D W 2 ∗ x ⟩ = ⟨ Φ D W 1 ∗ ( I + Θ 1 ′ W 1 ∗ ) − 1 f , P Θ 2 ( k λ Θ 2 ( I − W 2 Θ 2 ∗ ) − 1 D W 2 ∗ x ) ⟩ = ⟨ Φ D W 1 ∗ ( I + Θ 1 ′ W 1 ∗ ) − 1 f , k λ Θ 2 ( I − W 2 Θ 2 ∗ ) − 1 D W 2 ∗ x ⟩ = ⟨ J W 2 ( Φ D W 1 ∗ ( I + Θ 1 ′ W 1 ∗ ) − 1 f ) , J W 2 ( k λ Θ 2 ( I − W 2 Θ 2 ∗ ) − 1 D W 2 ∗ x ) ⟩ = ⟨ D W 2 ∗ ( I − Θ 2 W 2 ∗ ) − 1 Φ D W 1 ∗ ( I + Θ 1 ′ W 1 ∗ ) − 1 f , k λ Θ 2 ′ x ⟩ = ⟨ P Θ 2 ′ ( Ψ f ) , k λ Θ 2 ′ x ⟩ = ⟨ A Ψ Θ 1 ′ , Θ 2 ′ f , k λ Θ 2 ′ x ⟩ .

Therefore,

⟨ C ( A Φ Θ 1 , Θ 2 ) f , g ⟩ = ⟨ A Ψ Θ 1 ′ , Θ 2 ′ f , g ⟩

for any g in the linear span of k λ Θ 2 ′ x , λ ∈ D , and x ∈ E . The required result follows by the density of the last set in K Θ 2 ′ .

Hence, C ( A Φ Θ 1 , Θ 2 ) = A Ψ Θ 1 ′ , Θ 2 ′ , where the symbol Ψ is given by

Ψ ( z ) = D W 2 ∗ ( I − Θ 2 ( z ) W 2 ∗ ) − 1 Φ ( z ) D W 1 ∗ ( I + Θ 1 ′ ( z ) W 1 ∗ ) − 1 .

It follows that C maps T ( Θ 1 , Θ 2 ) into T ( Θ 1 ′ , Θ 2 ′ ) .

Let A Ψ Θ 1 ′ , Θ 2 ′ ∈ T ( Θ 1 ′ , Θ 2 ′ ) , f ∈ K Θ 1 , and x = ( I − W 2 Θ 2 ( λ ) ∗ ) − 1 D W 2 ∗ y , y ∈ E . Then

⟨ J W 2 ∗ A Ψ Θ 1 ′ , Θ 2 ′ J W 1 f , k λ Θ 2 x ⟩ = ⟨ A Ψ Θ 1 ′ , Θ 2 ′ J W 1 f , J W 2 k λ Θ 2 x ⟩ = ⟨ A Ψ Θ 1 ′ , Θ 2 ′ J W 1 f , k λ Θ 2 ′ y ⟩ = ⟨ P Θ 2 ′ ( Ψ D W 1 ∗ ( I − Θ 1 ( λ ) W 1 ∗ ) − 1 f ) , k λ Θ 2 ′ y ⟩ = ⟨ Ψ D W 1 ∗ ( I − Θ 1 W 1 ∗ ) − 1 f , k λ Θ 2 ′ y ⟩ = ⟨ J W 2 ∗ ( Ψ D W 1 ∗ ( I − Θ 1 W 1 ∗ ) − 1 f ) , J W 2 ∗ k λ Θ 2 ′ y ⟩ = ⟨ D W 2 ∗ ( I + Θ 2 ′ W 2 ∗ ) − 1 Ψ D W 1 ∗ ( I − Θ 1 W 1 ∗ ) − 1 f , k λ Θ 2 x ⟩ = ⟨ P Θ 2 ( Φ f ) , k λ Θ 2 x ⟩ = ⟨ A Φ Θ 1 , Θ 2 f , k λ Θ 2 x ⟩ ,

where

(3.5) Φ ( z ) = D W 2 ∗ ( I + Θ 2 ′ ( z ) W 2 ∗ ) − 1 Ψ ( z ) D W 1 ∗ ( I − Θ 1 ( z ) W 1 ∗ ) − 1 .

Therefore, C maps T ( Θ 1 , Θ 2 ) onto T ( Θ 1 ′ , Θ 2 ′ ) isometrically, with inverse C − 1 given by

□ C − 1 ( A ′ ) = J W 2 ∗ A ′ J W 1 , A ′ ∈ T ( Θ 1 ′ , Θ 2 ′ ) .

4 Condition for A Φ Θ 1 , Θ 2 = 0

Denote by ℳ Θ 1 (respectively by ℳ Θ 2 ) the orthogonal complement of Θ 1 H 2 ( ℒ ( E ) ) (respectively of Θ 2 H 2 ( ℒ ( E ) ) ) in H 2 ( ℒ ( E ) ) endowed with the Hilbert-Schmidt norm.

Lemma 4.1

(Compared with Lemma 6.1 from [9]) If Φ ∈ [ Θ 1 H 2 ( ℒ ( E ) ) ] ∗ + Θ 2 H 2 ( ℒ ( E ) ) , then A Φ Θ 1 , Θ 2 = 0 .

Proof

Let Φ ∈ [ Θ 1 H 2 ( ℒ ( E ) ) ] ∗ + Θ 2 H 2 ( ℒ ( E ) ) . Then Φ = Θ 2 Φ 2 + Φ 1 ∗ Θ 1 ∗ with Φ 1 , Φ 2 ∈ H 2 ( ℒ ( E ) ) . For f ∈ K Θ 1 ∞ , we have

A Φ Θ 1 , Θ 2 f = P Θ 2 ( Θ 2 Φ 2 f ) + P Θ 2 ( Φ 1 ∗ Θ 1 ∗ f ) .

It is clear that P Θ 2 ( Θ 2 Φ 2 f ) = 0 . On the other hand, if g ∈ K Θ 2 , then

⟨ P Θ 2 ( Φ 1 ∗ Θ 1 ∗ f ) , g ⟩ = ⟨ Φ 1 ∗ Θ 1 ∗ f , g ⟩ = ⟨ f , Θ 1 Φ 1 g ⟩ = 0 .

Therefore, P Θ 2 ( Φ 1 ∗ Θ 1 ∗ f ) = 0 and the required result follows.□

The following theorem characterizes the symbol of the zero matrix-valued asymmetric truncated Toeplitz operators.

Theorem 4.2

Let ϕ ∈ L 2 ( ℒ ( E ) ) and Θ 1 , Θ 2 be nonconstant inner functions. Then A ϕ Θ 1 , Θ 2 = 0 if and only if ϕ ∈ [ Θ 1 H 2 ( ℒ ( E ) ) ] ∗ + Θ 2 H 2 ( ℒ ( E ) ) .

Proof

One implication is given by Lemma 4.1. For the other implication, let Φ ∈ L 2 ( ℒ ( E ⊕ E ) ) such that

Φ = 0 0 ϕ 0 ,

where ϕ ∈ L 2 ( ℒ ( E ) ) .

Let A ϕ Θ 1 , Θ 2 : K Θ 1 → K Θ 2 be a matrix-valued asymmetric truncated Toeplitz operator with symbol ϕ . Then the operator

A Φ Θ 1 ⊕ Θ 2 = 0 0 A ϕ Θ 1 , Θ 2 0 : K Θ 1 ⊕ K Θ 2 → K Θ 1 ⊕ K Θ 2

is a matrix-valued truncated Toeplitz operator with symbol Φ . It can also be written as

A Φ Θ 1 ⊕ Θ 2 : K Θ 1 ⊕ K Θ 2 → K Θ 1 ⊕ K Θ 2 .

Suppose that A ϕ Θ 1 , Θ 2 = 0 , then A Φ Θ 1 ⊕ Θ 2 = 0 . Using [9, Theorem 6.3], we have

(4.1) Φ ∈ [ ( Θ 1 ⊕ Θ 2 ) H 2 ( ℒ ( E ⊕ E ) ) ] ∗ + ( Θ 1 ⊕ Θ 2 ) H 2 ( ℒ ( E ⊕ E ) ) .

Now, a function in H 2 ( ℒ ( E ⊕ E ) ) has the form

Ψ 11 Ψ 12 Ψ 21 Ψ 22 .

So (4.1) means that there exist Ψ i j , Ψ i j ′ ∈ H 2 ( ℒ ( E ) ) , such that

Φ = Ψ 11 Ψ 12 Ψ 21 Ψ 22 ∗ Θ 1 0 0 Θ 2 ∗ + Θ 1 0 0 Θ 2 Ψ 11 ′ Ψ 12 ′ Ψ 21 ′ Ψ 22 ′ = Ψ 11 ∗ Θ 1 ∗ + Θ 1 Ψ 11 ′ Ψ 21 ∗ Θ 2 ∗ + Θ 1 Ψ 12 ′ Ψ 12 ∗ Θ 1 ∗ + Θ 2 Ψ 21 ′ Ψ 22 ∗ Θ 2 ∗ + Θ 2 Ψ 22 ′ .

The lower left corner of this equality yields

ϕ = Ψ 12 ∗ Θ 1 ∗ + Θ 2 Ψ 21 ′ ,

which is what we need.□

As a lemma, we show that every asymmetric matrix-valued truncated Toeplitz operator has a symbol in a certain class.

Lemma 4.3

For any A ∈ T ( Θ 1 , Θ 2 ) , there exist Ψ 1 ∈ ℳ Θ 1 and Ψ 2 ∈ ℳ Θ 2 such that A = A Ψ 1 + Ψ 2 ∗ Θ 1 , Θ 2 . If Ψ 1 , Ψ 2 is one such pair, then the other such pairs are Ψ 1 ′ = Ψ 1 + k 0 Θ 2 X and Ψ 2 ′ = Ψ 2 − X ∗ k 0 Θ 1 with X ∈ ℒ ( E ) , such that A = A Ψ 1 ′ + Ψ 2 ′ ∗ Θ 1 , Θ 2 .

Proof

The first assertion follows from Theorem 4.2. For the second part, consider

A Ψ 1 ′ + Ψ 2 ′ ∗ Θ 1 , Θ 2 = A Ψ 1 + k 0 Θ 2 X + ( Ψ 2 − X ∗ k 0 Θ 1 ) ∗ Θ 1 , Θ 2 = A Ψ 1 + Ψ 2 ∗ + k 0 Θ 2 X − ( k 0 Θ 1 ) ∗ X Θ 1 , Θ 2 = A Ψ 1 + Ψ 2 ∗ Θ 1 , Θ 2 + A k 0 Θ 2 X Θ 1 , Θ 2 − A ( k 0 Θ 1 ) ∗ X Θ 1 , Θ 2 .

Now consider

A k 0 Θ 2 X Θ 1 , Θ 2 f = P Θ 2 ( k 0 Θ 2 X f ) = P Θ 2 ( X f ) .

Since Θ 1 ∗ f ⊥ K Θ 1 for all f ∈ K Θ 1 , we obtain

A ( k 0 Θ 1 ) ∗ X Θ 1 , Θ 2 f = P Θ 2 ( ( k 0 Θ 1 ) ∗ X f ) = P Θ 2 ( ( I − Θ 1 ( 0 ) Θ 1 ( z ) ∗ ) X f ) = P Θ 2 ( X f ) − P Θ 2 ( Θ 1 ( 0 ) Θ 1 ( z ) ∗ X f ) = P Θ 2 ( X f ) .

It follows that

A k 0 Θ 2 X Θ 1 , Θ 2 − A ( k 0 Θ 1 ) ∗ X Θ 1 , Θ 2 = 0

and

□ A Ψ 1 ′ + Ψ 2 ′ ∗ Θ 1 , Θ 2 = A = A Ψ 1 + Ψ 2 ∗ Θ 1 , Θ 2 .

It is known that the model space K Θ 1 (respectively K Θ 2 ) is finite dimensional if and only if Θ 1 (respectively Θ 2 ) is finite Blaschke-Potapov product (see, for instance [13], Chapter 2). In this case, we use the previous corollary to obtain the dimension of the space of T ( Θ 1 , Θ 2 ) . The following result is a generalization of [9, Corollary 6.5] and [8, Proposition 2.1].

Corollary 4.4

If dim K Θ 1 = m and dim K Θ 2 = n , then dim T ( Θ 1 , Θ 2 ) = m d + n d − d 2 .

Proof

The proof is similar to Corollary 6.5 given in [9]. It is immediate that dim ℳ Θ 1 = ( dim K Θ 1 ) d = m d and dim ℳ Θ 2 = ( dim K Θ 2 ) d = n d . Consider the map T : ℳ Θ 1 × ℳ Θ 2 → T ( Θ 1 , Θ 2 ) defined by

T ( Ψ 1 , Ψ 2 ) = A Ψ 1 + Ψ 2 ∗ Θ 1 , Θ 2 .

The kernel is obtained from Lemma 4.3 by taking Ψ 1 = Ψ 2 = 0 , that is,

ker T = { ( k 0 Θ 2 X , − ( X ∗ k 0 Θ 1 ) ∗ ) : X ∈ ℒ ( E ) } .

The proof is completed by noting that dim ( ℳ Θ 1 × ℳ Θ 2 ) = m d + n d and dim ker T = dim ℒ ( E ) = d 2 .□

Funding information: Funding sources are not available for the publication of this article.
Author contributions: Both authors contributed equally. The authors are indebted to the referee for many helpful suggestions.
Conflict of interest: The authors declare no conflict of interest in connection with the publication of this article.

References

[1] M. C. Câmara, K. Kliś-Garlicka, B. Lanucha, and M. Ptak, Conjugations in L2(ℋ), Integr. Equ. Oper. Theory 92 (2020), 48. 10.1007/s00020-020-02601-9Search in Google Scholar

[2] C. Câmara, J. Jurasik, K. Kliś-Garlicka, and M. Ptak, Characterizations of asymmetric truncated Toeplitz operators, Banach J. Math. Anal. 11 (2017), no. 4, 899–922. 10.1215/17358787-2017-0029Search in Google Scholar

[3] C. Câmara and J. R. Partington, Spectral properties of truncated Toeplitz operators by equivalence after extension, J. Math. Anal. Appl. 433 (2016), no. 2, 762–784. 10.1016/j.jmaa.2015.08.019Search in Google Scholar

[4] M. C. Câmara and J. R. Partington, Asymmetric truncated Toeplitz operators and Toeplitz operators with matrix symbol, J. Operator Theory 77 (2017), no. 2, 455–479. 10.7900/jot.2016apr27.2108Search in Google Scholar

[5] C. Câmara, K. Kliś-Garlicka, and M. Ptak, Asymmetric truncated Toeplitz operators and conjugations, Filomat 33 (2019), 3697–3710. 10.2298/FIL1912697CSearch in Google Scholar

[6] C. Gu, B. Lanucha, and M. Michalska, Characterizations of asymmetric truncated Toeplitz and Hankel operators, Complex Anal. Oper. Theory 13 (2019), 673–684. 10.1007/s11785-018-0783-8Search in Google Scholar

[7] J. Jurasik and B. Łanucha, Asymmetric truncated Toeplitz operators equal to the zero operator, Ann. Univ. Mariae Curie Sklodowska Sect. A 70 (2016), no. 2, 51–62. 10.17951/a.2016.70.2.51Search in Google Scholar

[8] J. Jurasik and B. Łanucha, Asymmetric truncated Toeplitz operators on finite-dimensional spaces, Oper. Matrices 11 (2017), no. 1, 245–262. 10.7153/oam-11-17Search in Google Scholar

[9] R. Khan and D. Timotin, Matrix valued truncated Toeplitz operators, basic properties, Comple. Analy. Oper. Theory 12 (2017), 997–1014. 10.1007/s11785-017-0675-3Search in Google Scholar

[10] R. Khan, The generalized Crofoot transform, Oper. Matrices 15 (2021), no. 1, 225–237. 10.7153/oam-2021-15-16Search in Google Scholar

[11] B. Łanucha, Asymmetric truncated Toeplitz operators of rank one, Comput. Mathods Funct. Theory 18 (2018), no. 2, 259–267. 10.1007/s40315-017-0219-xSearch in Google Scholar

[12] B. Łanucha, On rank-one asymmetric truncated Toeplitz operators on finite-dimensional model spaces, J. Math. Anal. Appl. 454 (2017), 961–980. 10.1016/j.jmaa.2017.05.033Search in Google Scholar

[13] V. V. Peller, Hankel Operators and their Applications, Springer Verlag, New York, 2003. 10.1007/978-0-387-21681-2Search in Google Scholar

[14] D. Sarason, Algebraic properties of truncated Toeplitz operators, Oper. Matrices 1 (2007), no. 4, 491–526. 10.7153/oam-01-29Search in Google Scholar

Received: 2022-08-02

Revised: 2022-11-12

Accepted: 2022-11-17

Published Online: 2023-02-18

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/conop-2022-0138

Keywords for this article

generalized Crofoot transform; matrix-valued truncated Toeplitz operators; matrix-valued asymmetric truncated Toeplitz operators

Creative Commons

BY 4.0