Intertwining relations for composition operators and integral-type operators between the Bloch-type spaces

1.   Introduction

For two Banach spaces $X$ and $Y$, $\mathcal{B}(X, Y)$ denotes the collection of all bounded linear operators from $X$ to $Y$ and $\mathcal{K}(X, Y)$ denotes the collection of all compact operators in $\mathcal{B}(X, Y)$. The Calkin algebra $\mathcal{Z}(X, Y)$ is the quotient Banach algebra $\mathcal{B}(X, Y)/\mathcal{K}(X, Y)$.

For bounded linear operators $A\in\mathcal{B}(X, X)$, $B\in\mathcal{B}(Y, Y)$ and $T\in\mathcal{B}(X, Y)$, we say "$T$ intertwines $A$ and $B$" if

When it is convenient to deemphasize the intertwining operator $T\in\mathcal{B}(X, Y)$, we write $A\varpropto B$ (sometimes we also use $A\varpropto B (T)$) as the intertwining relation above for simplicity. In ^[2] Bourdon and Shapiro showed that the intertwining relation is neither symmetric nor transitive. Furthermore, we say "$T$ intertwines $A$ and $B$ in $\mathcal{Z}(X, Y)$" (or "$T$ intertwines $A$ and $B$ compactly") if

For simplicity, the notation $A\varpropto_K B(T)$ represents the compact intertwining relations above. The relation $\varpropto_K$ turns to be symmetric when $T\in\mathcal{B}(X, Y)$ is invertible.

As usual, $S(\mathbb{D})$ denotes the collection of all analytic self-maps of the unit disk $\mathbb{D}$ of the complex plane $\mathbb{C}$. The composition operator $C_\varphi$ induced by $\varphi\in S(\mathbb{D})$ is defined as $C_\varphi f = f\circ\varphi$ for each $f\in H(\mathbb{D})$, where $H(\mathbb{D})$ is the collection of all holomorphic functions on the unit disk.

We next recall the spaces to work on, one of which is a classical Banach space of analytic functions, the Bloch space, which is defined as

The Bloch space $\mathcal{B}$ is maximal among all M$\ddot{o}$bius-invariant Banach spaces of analytic functions on $\mathbb{D}$, which implies that $\|f\circ\varphi\|_1 = \|f\|_1$ holds for all $f\in\mathcal{B}$ and $\varphi\in Aut(\mathbb{D})$ with the seminorm $\|\cdot\|_1$. It is well-known that $\mathcal{B}$ is a Banach space endowed with the norm $\|f\|_{\mathcal{B}} = |f(0)|+\|f\|_1$.

For $0 < \alpha < \infty$, the $\alpha$-Bloch space (or Bloch-type space) is defined as:

The little $\alpha$-Bloch space defined as:

$\mathcal{B}^{\alpha}$ is a Banach space endowed with the norm $\|f\|_{\mathcal{B}^\alpha} = |f(0)|+\|f\|_\alpha$.

For $0 < \alpha, \beta < \infty$, the weighted logarithmic Bloch space and the little weighted logarithmic Bloch space were introduced in ^[13,14]. It is defined as:

$\mathcal{B}_{\log^\beta}^\alpha$ is a Banach space endowed with the norm $\|f\|_{\mathcal{B}_{\log^\beta}^\alpha} = |f(0)|+\|f\|_{\log^\beta}^\alpha$, which reduces to $\mathcal{B}^\alpha$ if $\beta = 0$.

For $0 < \alpha < \infty$, the classical weighted space is defined as:

The little weighted space is defined as:

$H_{\alpha}^\infty$ is a Banach space endowed with the norm $\|f\|_{H_\alpha^\infty} = |f(0)|+\|f\|_\alpha$.

For $g\in H(\mathbb{D})$, two integral-type operators are defined by

and

where $f\in H(\mathbb{D})$ and $z\in\mathbb{D}$. Obviously, integration by parts gives

which shows the close relation among the integral-type operators $J_g$, $I_g$ and the multiplication operator $M_g$. Here, the miltiplication operator $M_g$ is defined by

Conveniently, the symbol $V_g$ is used to represent $J_g$ or $I_g$. Composition, integral type operators and their products from or to the weighted logarithmic Bloch space and the little weighted logarithmic Bloch space have been investigated a lot recently (see, for example, ^[9,15,16]). For more information on the logarithmic Bloch spaces, interested readers can refer to ^{[3,8,11,12,17,18,19,20]}.

Suppose that $\alpha, \beta > 0$, $\varphi\in S(\mathbb{D})$ and $g, h\in H(\mathbb{D})$. For two composition operators $C_\varphi\in\mathcal{B}(\mathcal{B}^\alpha, \mathcal{B}^\alpha)$, $C_\varphi\in\mathcal{B}(\mathcal{B}^\beta, \mathcal{B}^\beta)$, we concentrate on the compact intertwining relations of $C_\varphi$ whose intertwining operator is the integral-type operators $V_g\in\mathcal{B}(\mathcal{B}^\alpha, \mathcal{B}^\beta)$. In other word, we will study the properties of the difference operator

By $V[\varphi, \psi; g, h]$ we denote the following expression

We also say that $C_\varphi$ and $V_g$ are essentially commutative if

Moreover, the notation $\Omega_{co}^{\alpha, \beta}(V_g)$ is denotes the collection of $g\in H(\mathbb{D})$ such that

● $V_g\in\mathcal{B}(\mathcal{B}^\alpha, \mathcal{B}^\beta)$.

● $V_g$ are essentially commutative with $C_\varphi$ for all $\varphi$ such that $C_\varphi$ is bounded on both $\mathcal{B}^\alpha$ and $\mathcal{B}^\beta$.

Here, the lower symbol "co" represents "composition operator".

Some authors in their papers such as ^{[21,22,23,25]} investigate the compact intertwining relations of the integral-type operators and the composition operators on various spaces of analytic functions on the unit disk.

In this paper, we investigate the compact intertwining relations of integral-type operators $V_g$ from $\mathcal{B}^\alpha$ to $\mathcal{B}^\beta$ and the relevant composition operators $C_\varphi$. In Section 2, we present some lemmata to be used later in this paper. In Section 3, we investigate the intertwining relations of integral-type operators and composition operators, in which the equivalent conditions of $V[\varphi, \psi; g, h] = 0$ is given. In Section 4, boundedness and compactness of $V[\varphi; g, h]$ are investigated. In Sections 5 and 6, two questions of the compact intertwining relations of $V_g$ and $C_\varphi$ are investigated respectively.

For simplification, the hypotheses $0 < \alpha, \beta < \infty, \varphi\in S(\mathbb{D}), g, h\in H(\mathbb{D})$ are available throughout this paper which will not be specified later.

Specially, for two real numbers $A$ and $B$, we say $A\lesssim B$ if there exists a constant $C\neq0$ such that $A\leq CB$.

2.   Auxiliary results

In this section, we introduce some basic properties of the Bloch-type spaces and the integral-type operators to be used in this paper.

The following folklore lemma is proved in a standard way (see, e.g., ^[10]), which also implies that the point evaluation functional is continuous on the Bloch-type space.

Proposition 2.1. For each $f\in\mathcal{B}^\alpha$ and $z\in\mathbb{D}$, we have

The following result can be also found in ^[10].

Lemma 2.2. The composition operator $C_\varphi$ is bounded from $\mathcal{B}^\alpha$ to $\mathcal{B}^\beta$ if and only if

The following lemma was proved, e.g., in ^[6,7] even in much more general settings.

Lemma 2.3. The integral-type operators $J_g$ is bounded from $\mathcal{B}^\alpha$ to $\mathcal{B}^\beta$ if and only if

Lemma 2.4. ^[5] The integral-type operators $I_g$ is bounded from $\mathcal{B}^\alpha$ to $\mathcal{B}^\beta$ if and only if

The proposition below is a crucial criterion for the compactness of $V[\varphi; g, h]$, which can be proved by a little modification of Proposition 3.11 in ^[4].

Proposition 2.5. $V[\varphi; g, h]$ is compact from $\mathcal{B}^\alpha$ to $\mathcal{B}^\beta$ if and only if whenever $\{f_n\}$ is bounded in $\mathcal{B}^\alpha$ and $f_n\rightarrow0$ uniformly on any compact subset of the unit disk, then

3.   Intertwining relations of $C_\varphi$ and $V_g$

Theorem 3.1. Assume that $J[\varphi, \psi; g, h]$ is defined as (1.2), then

$J[\varphi, \psi; g, h] = 0$ if and only if

(a) either $\varphi(0) = 0$ or $g$ is a constant;

(b) $\varphi = \psi$;

(c) $h = g\circ\varphi+C,$ where $C$ is an arbitrary constant.

Proof. The sufficiency is easily checked by calculation. To prove the necessity, we only show some different details from what Tong and Zhou did in ^[22] for the study of the intertwining relations for Volterra operators and composition operators on the Bergman space.

$J[\varphi, \psi; g, h] = 0$ implies that

which further implies that, for each $f\in\mathcal{B}^\alpha$,

Hence, for each $f\in\mathcal{B}^\alpha$, $|(C_\varphi J_g-J_h C_\psi)f(0)| = 0$ and

hold. And the latter one shows that for each $z\in\mathbb{D}$,

To this end, the remaining part of the theorem is parallel with Proposition 3.1 in ^[22]. This completes the proof.

Theorem 3.2. Assume that $I[\varphi, \psi; g, h]$ is defined as (1.2), then

$I[\varphi, \psi; g, h] = 0$ if and only if

(a) either $\varphi(0) = 0$ or $g\equiv0$;

(b) $\varphi = \psi$;

(c) $h = g\circ\varphi.$

Proof. The sufficiency is easily verified by calculation. To prove the necessity, we only show some essential details. $I[\varphi, \psi; g, h] = 0$ implies that for each $f\in\mathcal{B}^\alpha$,

Hence, for each $f\in\mathcal{B}^\alpha$, $|(C_\varphi I_g-I_h C_\psi)f(0)| = 0$ and

hold. If $\varphi(0) = 0$, then the first of the conditions is automatically satisfied. If $\varphi(0)\neq0$, then we can obtain that $g\equiv0$ by the same method used in Proposition 3.1 in ^[22]. Moreover, the second equality

implies that for each $z\in\mathbb{D}$,

By choosing $f(z) = z\in\mathcal{B}^\alpha$ and $f(z) = z^2\in\mathcal{B}^\alpha$, we have that

Combining these two quantities, we have that

Hence, $\varphi = \psi$ and $h = g\circ\varphi$. This completes the proof.

Specially, we consider the operators mapping a Bloch-type space into and onto itself, that is $C_\varphi:\mathcal{B}^\alpha\rightarrow \mathcal{B}^\alpha, J_g, J_h, I_g, I_h:\mathcal{B}^{\alpha}\rightarrow \mathcal{B}^\alpha$. Combining Theorems 3.1 and 3.2 and the equivalent conditions of the boundedness of $J_g$ and $I_g$ from Lemmas 2.3 and 2.4, we conclude the statements in the following.

Corollary 3.3. (a) If $\alpha > 1, g, h\in\mathcal{B}^{\beta-\alpha+1}$, then $J_g\propto J_h(C_\varphi)$ if and only if $g$ is a constant or $\varphi(0) = 0$, $h = g\circ\varphi+C,$ where $C$ is a constant.

(b) If $\alpha = 1, g, h\in\mathcal{B}_{\log^1}^\beta$, then $J_g\propto J_h(C_\varphi)$ if and only if $g$ is a constant or $\varphi(0) = 0$, $h = g\circ\varphi+C,$ where $C$ is a constant.

(c) If $0\leq\alpha < 1, g, h\in\mathcal{B}^\beta$, then $J_g\propto J_h(C_\varphi)$ if and only if $g$ is a constant or $\varphi(0) = 0$, $h = g\circ\varphi+C,$ where $C$ is a constant.

Corollary 3.4. If $g, h\in H_{\beta-\alpha}^\infty$, then $I_g\propto I_h(C_\varphi)$ if and only if $g\equiv0$ or $\varphi(0) = 0$, $h = g\circ\varphi.$

4.   Intertwining relations between the composition operators and the integral-type operators between the Bloch-type spaces

In this section, we characterize the boundedness and the compactness of $V[\varphi; g, h]$ defined as (1.1), in which the compactness is essential for our study in this paper. The method of the proof is basic, which is also parallel with Proposition 3.1 in ^[22] and Corollary 4.3 in ^[22].

Theorem 4.1. $J[\varphi; g, h]$ is bounded from $\mathcal{B}^\alpha$ to $\mathcal{B}^\beta$ if and only if

(a)

when $\alpha > 1$;

(b)

when $\alpha = 1$;

(c) $g\circ\varphi-h\in\mathcal{B}^\beta$ when $0 < \alpha < 1$.

Proof. The proof of the theorem is essentially given in ^[7], but since there are some technical differences, we will give some details. We firstly prove the sufficiency. By Proposition 2.1, we estimate the semi-norm $\|J[\varphi; g, h]\|_\beta$ respectively.

(a) when $\alpha > 1$,

(b) when $\alpha = 1$,

(c) when $0 < \alpha < 1$,

Therefore, we conclude that the difference operator $J[\varphi; g, h]$ is bounded by the estimations above and the boundedness of the point evaluation functional at 0.

To prove the necessity, we aim to find a contradiction if we suppose that the hypotheses do not hold. When $\alpha > 1,$ there exists a sequence $\{z_n\}\subset\mathbb{D}$ satisfying

For $n\in\mathbb{N}$ and $z\in\mathbb{D}$, assume that the test function is

An easy estimation shows that $\|f_{n, 1}\|_\alpha\leq4^\alpha(\alpha-1)$ and thus $f_{n, 1}\in\mathcal{B}^\alpha$. By the boundedness of $J[\varphi; g, h]$, we have

as $n\rightarrow \infty$, which contradicts to our hypothesis. Assume that the test function is $f_{n, 2}(z) = \log\frac{2(1-|\varphi(z_n)|^2)}{(1-\overline{\varphi(z_n)}z)^2}$ when $\alpha = 1$ and is $f_{n, 3}(z) = 1$ when $0 < \alpha < 1$ respectively. Further observe the fact that $\|f_{n, 2}\|_\alpha\leq4$ and $\|f_{n, 3}\|_\alpha\leq2^{\alpha+2}$. To this end, we conclude the results in a similar way shown above. This completes the proof.

Theorem 4.2. $J[\varphi; g, h]$ is compact from $\mathcal{B}^\alpha$ to $\mathcal{B}^\beta$ if and only if $J[\varphi; g, h]$ is bounded and

(a)

when $\alpha > 1$;

(b)

when $\alpha = 1$;

(c)

when $0 < \alpha < 1$.

Proof. The proof of the theorem is also essentially given in ^[7], but since there are some technical differences, we will give some details. To the sufficiency of the theorem, we present the proof of the hypothesis $\alpha > 1$. Assume that $\{f_n\}$ is bounded in $\mathcal{B}^\alpha$ and $f_n\rightarrow0$ uniformly on any compact subset of $\mathbb{D}$, by Proposition 2.5 we are only supposed to check that

For convenience, suppose that there exists a positive number $M_1$ such that $\sup_{n\in\mathbb{Z}}\|f_n\|_{\mathcal{B}^\alpha}\leq M_1.$ For any $\epsilon > 0$, there exists a $\delta > 0$ such that

and

Furthermore, there exists another positive number $M_2$ such that

Hence,

Thus $J[\varphi; g, h]$ is compact from $\mathcal{B}^\alpha$ to $\mathcal{B}^\beta$ when $\alpha > 1$. Similarly, the compactness of $J[\varphi; g, h]$ can be obtained when $\alpha = 1$ and $0 < \alpha < 1$ respectively if the hypotheses hold.

To the necessity of the theorem, we firstly consider the situation when $\alpha > 1$. Like what we do in Theorem 4.1, we aim to find a contradiction if we suppose that the hypotheses do not hold. Thus there exists a sequence $\{z_n\}\subset\mathbb{D}$ such that for any $\epsilon > 0$,

whenever $|\varphi(z_n)|\rightarrow1.$ For $n\in\mathbb{Z}$ and $z\in\mathbb{D}$, assume that the same test function $f_{n, 1}$ defined as (4.1) and it is easy to check that $\{f_{n, 1}\}\rightarrow0$ as $n\rightarrow \infty$ on any compact subset of $\mathbb{D}$. Then for the given $\epsilon > 0$ above, we have

as $|\varphi(z_n)|\rightarrow1$, which contradicts to our hypothesis.

When $0 < \alpha < 1$, we still aim to find a contradiction if we suppose that the hypotheses do not hold. Thus there exists a sequence $\{u_n\}\subset\mathbb{D}$ such that for any $\epsilon > 0$,

whenever $|\varphi(u_n)|\rightarrow1.$ For $n\in\mathbb{Z}$, let $\varphi(u_n) = r_ne^{i\theta_n}$. Assume that the test function

Observe that $\|\tilde{f}_{n, 3}\|_{\mathcal{B}^\alpha}\leq4(1-\alpha)$ and

as $|\varphi(z_n)|\rightarrow1$, which contradicts to our hypothesis.

Moreover, The result of $\alpha = 1$ can be obtained in a similar way by choosing the testing function which is different from $f_{n, 2}$ in Theorem 4.1

where $\{w_n\}\subset\mathbb{D}$ is the sequence such that for any $\epsilon > 0$,

whenever $|\varphi(w_n)|\rightarrow1.$ This completes the proof.

Theorem 4.3. Suppose that $\beta-\alpha\geq0$, then $I[\varphi; g, h]$ is bounded from $\mathcal{B}^\alpha$ to $\mathcal{B}^\beta$ if and only if

Proof. This theorem essentially follows from the proofs of some known results such as ^[5,7], but since there are some technical differences, we will give some details. The sufficiency is obvious. To prove the necessity, we aim to find a contradiction if we suppose that the hypotheses do not hold. Thus there exists a sequence $\{z_n\}\subset\mathbb{D}$ satisfying

For $n\in\mathbb{Z}$ and $z\in\mathbb{D}$, assume that the test function is

An easy estimation shows that $\|g_n\|_\alpha\leq2^{2\alpha}$ and thus $g_n\in\mathcal{B}^\alpha$. By the boundedness of $I[\varphi; g, h]$, we have

as $n\rightarrow \infty$, which contradicts to our hypothesis. This completes the proof.

Remark 4.4. We can observe that Theorem 4.3 holds for each $\alpha > 0$ and $\beta > 0$. However, if $\beta-\alpha < 0$, then $g\equiv0$ by the maximal modulus principal, which can be simplified in the following.

Corollary 4.5. Suppose that $\beta-\alpha < 0$, then $I[\varphi; g, h]$ is bounded from $\mathcal{B}^\alpha$ to $\mathcal{B}^\beta$ if and only if

We just present the compactness of $I[\varphi; g, h]$ and omit the proof, which can be similarly proved by the method used in Theorem 4.2 with the same test function used in Theorem 4.3.

Theorem 4.6. Suppose that $\beta-\alpha\geq0$, then $I[\varphi; g, h]$ is compact from $\mathcal{B}^\alpha$ to $\mathcal{B}^\beta$ if and only if $I[\varphi; g, h]$ is bounded and

Corollary 4.7. Suppose that $\beta-\alpha < 0$, then $I[\varphi; g, h]$ is compact from $\mathcal{B}^\alpha$ to $\mathcal{B}^\beta$ if and only if $I[\varphi; g, h]$ is bounded and

5.   Essential commutation of $C_\varphi$ and $J_g$ between the Bloch-type spaces

In this section, we concentrate on two questions of the compactly intertwining relations of $J_g$ and $C_\varphi$.

Problem 5.1. What properties does a non-constant $g\in H(\mathbb{D})$ have if $V_g$ essentially commutes with $C_\varphi$ for all $C_\varphi$ that are bounded on both $\mathcal{B}^\alpha$ and $\mathcal{B}^\beta$?

Problem 5.2. What properties does $\varphi\in S(\mathbb{D})$ have if the bounded $V_g$ essentially commutes with $C_\varphi$ for all $C_\varphi$ that are bounded on both $\mathcal{B}^\alpha$ and $\mathcal{B}^\beta$?

We firstly answer the first problem. Recall that the notation $\Omega_{co}^{\alpha, \beta}(V_g)$ is denotes the collection of $g\in H(\mathbb{D})$ such that

● $V_g\in\mathcal{B}(\mathcal{B}^\alpha, \mathcal{B}^\beta)$.

● $V_g$ are essentially commutative with $C_\varphi$ for all $\varphi$ such that $C_\varphi$ is bounded on both $\mathcal{B}^\alpha$ and $\mathcal{B}^\beta$.

Theorem 5.3. $\Omega_{co}^{\alpha, \beta}(J_g) = \mathcal{B}^0_{\beta-\alpha+1}$ if $\alpha > 1$ and $\beta-\alpha+1\geq0$.

Proof. In the proof we use the ideas in Theorem 5.1 in ^[22]. We firstly prove that $\mathcal{B}^0_{\beta-\alpha+1}\subset\Omega_{co}^{\alpha, \beta}(J_g).$ For any $g\in\mathcal{B}^0_{\beta-\alpha+1}$, obviously, $J_g$ is bounded by Lemma 2.3. Furthermore, by the boundedness of $C_\varphi$ on $\mathcal{B}^\beta$ (see Lemma 2.2),

as $|\varphi(z)|\rightarrow1$. Here, we use the following inequality (see, for example, Corollary 2.40 in ^[4])

This implies that $\mathcal{B}^0_{\beta-\alpha+1}\subset\Omega_{co}^{\alpha, \beta}(J_g)$ by Theorem 4.2.

Next we prove that $\Omega_{co}^{\alpha, \beta}(J_g)\subset\mathcal{B}^0_{\beta-\alpha+1}$. For any $g\in\Omega_{co}^{\alpha, \beta}(J_g)$, by Theorem 4.2, we have that

Choose specifically $\varphi(z) = e^{i\theta}z\in S(\mathbb{D})$, it follows that,

Note that

If we assume that $g(z) = \sum_{n = 0}^\infty a_nz^n$, then by integrating the left side of (5.2) with respect to $\theta$ from $0$ to $2\pi$, we obtain that

where the Dominant Convergent Theorem is applied in the second line. This implies that $g\in\mathcal{B}^{\beta-\alpha+1}_0$. This completes the proof.

Corollary 5.4. $\Omega_{co}^{\alpha, \beta}(J_g) = \mathbb{C}$ if $\alpha > 1$ and $\beta-\alpha+1 < 0$.

Proof. Obviously by the maximal modulus principle.

Theorem 5.5. $\Omega_{co}^{\alpha, \beta}(J_g) = \mathcal{B}^\beta_{\log^1, 0}$ if $\alpha = 1$.

Proof. For any $g\in\mathcal{B}^\beta_{\log^1, 0}$, obviously, $J_g$ is bounded. Furthermore, by the boundedness of $C_\varphi$ on $\mathcal{B}^\beta$ and observing that $g\in\mathcal{B}^0_\beta$,

as $|\varphi(z)|\rightarrow1$. This implies that $\mathcal{B}^\beta_{\log^1, 0}\subset\Omega_{co}^{\alpha, \beta}(J_g)$ by Theorem 4.2.

Next we prove that $\Omega_{co}^{\alpha, \beta}(J_g)\subset\mathcal{B}^\beta_{\log^1, 0}$. For any $g\in\Omega_{co}^{\alpha, \beta}(J_g)$, by Theorem 4.2, we have that

Choose specifically $\varphi(z) = e^{i\theta}z\in S(\mathbb{D})$, it follows that,

Note that

If we assume that $g(z) = \sum_{n = 0}^\infty a_nz^n$, then by integrating the left side of (5.2) with respect to $\theta$ from $0$ to $2\pi$, we obtain that

where the Dominant Convergent Theorem is applied in the second line. This implies that $\mathcal{B}^\beta_{\log^1, 0}$. This completes the proof.

Theorem 5.6. $\Omega_{co}^{\alpha, \beta}(J_g) = \mathcal{B}^0_{\beta+\alpha-1}$ when $0 < \alpha < 1$.

Proof. We firstly prove that $\mathcal{B}^0_{\beta+\alpha-1}\subset\Omega_{co}^{\alpha, \beta}(J_g).$ For any $g\in\mathcal{B}^0_{\beta+\alpha-1}$, obviously, $g\in\mathcal{B}^0_{\beta}$ and hence $J_g$ is bounded by Lemma 2.3. Furthermore, by the boundedness of $C_\varphi$ on $\mathcal{B}^\beta$ (see Lemma 2.2), observe that

as $|\varphi(z)|\rightarrow1$. The left part remains to be proved in a similar way from Theorem 5.3. This completes the proof.

Proof. The proof is similar with Theorem 5.3.

In the following, we partly answer the second problem. We only prove the first result and the other two results can be proved similarly.

Proposition 5.7. If $\alpha > 1$, $g\in\mathcal{B}^{\beta-\alpha+1}$,

and $\varphi$ has finite angular derivative at any point of the unit circle, then $C_\varphi\propto_K C_\varphi(J_g).$

Proof. By Theorem 4.2, we only ought to check that

Since $\varphi$ has finite angular derivative at any point of the unit circle, it follows that

as $|\varphi(z)|\rightarrow1$.

Proposition 5.8. If $\alpha = 1$, $g\in\mathcal{B}^\beta_{\log^1}$,

and $\varphi$ has finite angular derivative at any point of the unit circle, then $C_\varphi\propto_K C_\varphi(J_g).$

Proposition 5.9. If $0 < \alpha < 1$, $g\in\mathcal{B}^0_\beta$,

and $\varphi$ has finite angular derivative at any point of the unit circle, then $C_\varphi\propto_K C_\varphi(J_g).$

6.   Essential commutation of $C_\varphi$ and $I_g$ between the Bloch-type spaces

In this section, we answer the two questions of the compactly intertwining relations of $I_g$ and $C_\varphi$ respectively.

Theorem 6.1. $\Omega_{co}^{\alpha, \beta}(I_g) = H^\infty_{\beta-\alpha, 0}$ if $\beta-\alpha\geq0$.

Proof. We only prove that $H^\infty_{\beta-\alpha, 0}\subset\Omega_{co}^{\alpha, \beta}(I_g).$ For any $f\in H^\infty_{\beta-\alpha, 0}$, obviously, $I_g$ is bounded. Furthermore, by the boundedness of $C_\varphi$,

as $|\varphi(z)|\rightarrow1$, which implies that $H^\infty_{\beta-\alpha, 0}\subset\Omega_{co}^{\alpha, \beta}(J_g).$ The left part remains to be proved in a similar way from Theorem 5.3. This completes the proof.

Corollary 6.2. $\Omega_{co}^{\alpha, \beta}(I_g) = \{0\}$ if $\beta-\alpha < 0$.

Proposition 6.3. If $\beta-\alpha\geq0$, $g\in H^\infty_{\beta-\alpha}$ and

then $C_\varphi\propto_K C_\varphi(I_g)$.

Proof. The proof can be completed in a similar way from Proposition 5.7.

Remark 6.4. Obviously, under the hypothesis of Proposition 6.3, we can further conclude that $C_\varphi\in\mathcal{B}(\mathcal{B}^\alpha, \mathcal{B}^\alpha)$ and $C_\varphi\in\mathcal{B}(\mathcal{B}^\beta, \mathcal{B}^\beta)$ are both compact linear operators.

Theorem 6.5. If $\beta-\alpha\geq0$, $g\in A(\mathbb{D})$, then $C_\varphi\propto_K C_\varphi(I_g)$ if and only if

where $A(\mathbb{D})$ denotes the disk algebra.

Proof. The necessity is obvious by setting $g = Id$ in Theorem 4.6, where $Id$ denoted the identity function. Next we prove the sufficiency. Suppose that $h_n(z) = z^n$, it follows that

as $|\varphi(z)|\rightarrow1$. For each $g\in A(\mathbb{D})$, there exists a subsequence of $\{h_n\}$, denoted by $\{h_n^{[g]}\}$ such that $\lim_{n\rightarrow \infty}h_n^{[g]} = g.$ Thus,

as $n\rightarrow \infty.$ This completes the proof.

7.   Conclusions

Main conclusions are given in the following.

Theorem $\quad \Omega_{co}^{\alpha, \beta}(J_g) = \mathcal{B}^0_{\beta-\alpha+1}$ if $\alpha > 1$ and $\beta-\alpha+1\geq0$.

Theorem $\quad \Omega_{co}^{\alpha, \beta}(J_g) = \mathcal{B}^\beta_{\log^1, 0}$ if $\alpha = 1$.

Theorem $\quad \Omega_{co}^{\alpha, \beta}(J_g) = \mathcal{B}^0_{\beta+\alpha-1}$ if $0 < \alpha < 1$.

Theorem $\quad \Omega_{co}^{\alpha, \beta}(I_g) = H^\infty_{\beta-\alpha, 0}$ if $\beta-\alpha\geq0$.

Conflict of interest

The author declares no conflicts of interest in this paper.