2. Operator Lipschitz functions

Let $\Phi\;:\;\mathbb{B}(\mathscr{H}\;\,)\to \mathbb{B}(\mathscr{H}\;\,)$ be a linear map. Let

\begin{equation*}\|\Phi\|=\sup\{\|\Phi(T)\|\;:\; \|T\|\leq 1\},\end{equation*}

\begin{equation*}\|\Phi\|_1=\sup\{\|\Phi(T)\|_1\;:\; \|T\|_1\leq 1\},\end{equation*}

It is well known that if $\|\Phi\|=\|\Phi\|_1=d$ , then

(2.1) \begin{equation}\||\Phi(X)\||\leq d \||X\||,\end{equation}

for all $X\in C_{\||.\||}$ . For details, see the first part of proof of [Reference Hiai and Kosaki7, Proposition 2.7.].

Let $A(\mathbb{D})$ be the disk algebra of all continuous complex-valued functions on the unit disk $\mathbb{D}$ , which are holomorphic in the interior of $\mathbb{D}$ . It is well known that any function in $A(\mathbb{D})$ acts on the set of all contraction operators in $\mathbb{B}(\mathscr{H}\;\,)$ .

A continuous function f on the unit disk $\mathbb{D}$ is called operator Lipschitz with constant d, if

(2.2) \begin{equation} || f (S)-f (T)|| \leq d \ ||S-T||,\end{equation}

for all normal contraction operators T and S on any Hilbert space $\mathscr{H}$ .

Kissin and Shulman in [Reference Kissin and Shulman9] proved that if $f\in A(\mathbb{D})$ is an operator Lipschitz function with constant d, then

\begin{equation*} || f (S)-f (T)|| \leq d \ ||S-T||,\end{equation*}

for all arbitrary contraction operators S and T. Moreover, by using the interpolation theory, they proved that if $f\in A(\mathbb{D})$ is an operator Lipschitz function with constant d, then

\begin{equation*} || f (T)-f (S)||_p \leq d \ ||T-S||_p,\end{equation*}

for any $1\leq p< \infty$ and any contraction operators S and T with $S-T\in S_p$ ; see also [Reference Kissin, Potapov, Shulman and Sukochev8, Theorem 6.4].

We can extend the results of [Reference Kissin and Shulman9] for a unitarily invariant norm ideals by using the majorization property that state in the first part of this section. Although, the proof of the following theorem is similar to [Reference Kissin and Shulman9, Theorem 4.2.], for the convenience of the reader we prove the following theorem. For more results on Lipschitz-type estimates for general symmetrically normed ideals, we refer the reader to [Reference Skripka and Tomskova14].

Theorem 2.1. Let $f \in A(\mathbb{D})$ be operator Lipschitz with constant d. Then, for arbitrary contraction operators S and T and an arbitrary operator $X\in C_{\||.\||}$ , we have

\begin{equation*}\||f (S)X-X f (T)\|| \leq d \ \||SX-XT\||.\end{equation*}

Proof. First, assume that $\sigma(S)\cap \sigma(T)=\varnothing$ . As the operator $\Delta = L_S -R_T$ on $\mathbb{B}(\mathscr{H}\;\,)$ is invertible, we can consider the operator $F = (L_{f (S)}-R_{f (T)})\Delta^{-1}$ . The proof of [Reference Kissin and Shulman9, Theorem 4.2.] shows that $\|F\|\leq d$ on $\mathbb{B}(\mathscr{H}\;\,)$ and $\|F|_{S_1}\|_1\leq d$ . Now, by interpolation theory (equation (2.1)), for each unitarily invariant norm $\||.\||$ and for each $X\in C_{\|||.\||}$ , we have

\begin{equation*}\||F(X)\||\leq d\ \||X\||.\end{equation*}

The definition of F implies that for each $S,T\in \mathbb{B}(\mathscr{H}\;\,)$ with $\sigma(S)\cap \sigma(T)=\varnothing$ and for each $X\in C_{\||.\||}$ , we have

(2.3) \begin{equation}\||f (S)X-X f (T)\|| \leq d \ \||SX-XT\||.\end{equation}

Now, if ${\rm{dim}}(\mathscr{H}\;\,)<\infty$ and $S,T\in \mathbb{B}(\mathscr{H}\;\,)$ , we can see that there exist contractions $S_n$ such that $\sigma(S_n)\cap \sigma(T)=\varnothing$ and $\|S_n-S\|\to 0$ . We have

\begin{eqnarray*}\||f (S)X-Xf (T)\|| &\leq & \ \||f(S_n)X-Xf(T)\||+\||f(S_n)X-f(S)X\||\\[5pt]&\leq & \ \||f(S_n)X-Xf(T)\||+ \|f(S_n)-f(S)\| \ \||X\||\\[5pt]&\leq & d \ \||S_nX-XT\||+ \|f(S_n)-f(S)\| \ \||X\||.\end{eqnarray*}

By the previous observation, we can prove (2.3) for finite rank operators S, T.

In the general case, let $P_n$ be an increasing sequence of finite-dimensional projections such that $P_n \rightarrow I$ in the strong operator topology. We have

\begin{eqnarray*} \||f (P_nS)XP_n-P_nXf (TP_n)\||&\leq & d \ \||P_nS XP_n - P_n XT P_n\||\\[5pt] &=& d\ \||P_n(SX-XT)P_n\||\\[5pt] &\leq & d\ ||P_n||\ \||SX-XT\|| \ ||P_n\||\\[5pt] &\leq & d\ \||SX-XT\||. \end{eqnarray*}

Since $C_{\||.\||}$ is an ideal of compact operators, $f(S)X-Xf(T)$ is compact. Now $f (P_nS)XP_n-P_nXf (TP_n) \rightarrow f (S)X-Xf(T)$ in the strong operator topology and $f (S)X-Xf(T)$ is compact, so by the noncommutative Fatou’s lemma [Reference Simon13], we have

\begin{equation*}\||f (S)X-Xf(T)\||\leq \sup_{n \in \mathbb{N}} \ \||f (P_nS)XP_n-P_nXf (TP_n)\|| \leq d\ \||SX-XT\||.\end{equation*}

Let $\mathcal{O}_r(z_0)=\{z\in \mathbb{C}\;:\; |z-z_0|\leq r\}$ be a closed disk in $\mathbb{C}$ . We can see that $f\in A(\mathbb{D})$ is an operator Lipschitz function with constant d, if and only if $g(z)=f\left(\frac{1}{r}(z- z_0)\right)$ is an operator Lipschitz function with constant d on $\mathcal{O}_r(z_0)$ . Hence, we have the following corollary.

Corollary 2.2. Let f be an analytic function on the disk $\mathcal{O}_r(z_0)$ such that

(2.4) \begin{equation} || f (S)-f (T)|| \leq d \ ||S-T||,\end{equation}

for all normal operators T,S on any Hilbert space $\mathscr{H}$ with ${\rm{sp}}(S),{\rm{sp}}(T)\subseteq \mathcal{O}_r(z_0)$ . Then, for arbitrary operators S and T with ${\rm{sp}}(S),{\rm{sp}}(T)\subseteq \mathcal{O}_r(z_0)$ and an arbitrary operator $X\in C_{\||.\||}$ , we have

\begin{equation*}\||f (S)X-X f (T)\|| \leq d \ \||SX-XT\||.\end{equation*}

3. Operator monotone functions

Let $\Pi_+$ be the upper half-plane and $\Pi_-$ be the lower half-plane. Let $\Omega =\Pi_{+} \cup \Pi_{-} \cup [0,\infty)$ . Let f be an operator monotone function on $[0,\infty)$ . The Löwner theorem [Reference Löwner11] states that f is analytic on $(0,\infty)$ and has an analytic continuation to $\Omega$ , which again we denote by f, such that $f(\Pi_+)\subseteq \Pi_+$ . Let $S\in \mathbb{B}(\mathscr{H}\;\,)$ with ${\rm{sp}}(S) \subseteq \Omega \setminus \{0\}$ and f be an operator monotone function on $[0,\infty)$ . Since f is analytic on $\Omega$ , we can define the operator f(S) by the integral representation:

(3.1) \begin{equation}f(S)=\frac{1}{2\pi i}\int_{\gamma}f(z)(z-S)^{-1}\ dz,\end{equation}

where $\gamma$ is a closed rectifiable curve in $\Omega$ such that ${\rm{sp}}(S) \subset \rm{ins}(\gamma)$ .

Let $P[0,\infty)$ denote the set of all positive operator monotone functions defined in the positive half-line and consider the convex set:

\begin{equation*}\mathcal{P}= \{ f \in P[0,\infty) | f(1) = 1 \}.\end{equation*}

Hansen in [Reference Hansen5] showed that $\mathcal{P}$ is compact in the topology of point-wise convergence and extreme points in $\mathcal{P}$ are necessarily of the form:

\begin{eqnarray*}f_{\alpha}(t)=\frac{t}{\alpha +(1-\alpha)t},\end{eqnarray*}

where $0\leq \alpha \leq 1$ . The next theorem shows that the family $\mathcal{P}$ is generated in the uniformly compact topology by the convex hull of its extreme points.

Theorem 3.1. [Reference Najafi12, Theorem 3.1] Let f be a nonnegative operator monotone function on $[0,\infty)$ such that $f(1)=1$ . Then, there exists a sequence $f_{n}$ which is uniformly convergent to f on every compact subset of $\Omega$ . Moreover, for each n the following property hold:

(3.2) \begin{equation} f_{n}=\sum_{i=1}^{k_n}\gamma_i f_{\alpha_i},\end{equation}

where $\alpha_1, \alpha_2,\ldots,\alpha_{k_n}$ and $\gamma_1, \gamma_2,\ldots,\gamma_{k_n}$ are positive scalars such that $\sum_{i=1}^{k_n}\gamma_i=1$ .

In the last theorem, since $f_{n}$ converges uniformly on compact sets to f, we can conclude that $f_n^{'}$ is also uniformly convergent to $f'$ on compact sets. The following lemma will be useful.

Lemma 3.2. Let $0\leq \alpha \leq 1$ , and let S,T be bounded invertible operators such that $\left({\rm{sp}}(S) \cup {\rm{sp}}(T)\right) \cap $ $(-\infty,0)=\emptyset$ . Then,

\begin{equation*} f_{\alpha}(S)- f_{\alpha}(T)=\alpha \ f_{1-\alpha}(S^{-1})(S-T) f_{1-\alpha}(T^{-1}).\end{equation*}

Proof. We can see that $f_{\alpha}(t)=(\alpha t^{-1}+(1-\alpha))^{-1}$ . Since $\alpha S^{-1}+(1-\alpha)$ and $\alpha T^{-1}+(1-\alpha)$ are invertible, so

\begin{eqnarray*} f_{\alpha}(S)- f_{\alpha}(T)&=&(\alpha S^{-1}+(1-\alpha))^{-1}-(\alpha T^{-1}+(1-\alpha))^{-1}\\[5pt]&=& \alpha (\alpha S^{-1}+(1-\alpha))^{-1}( T^{-1}-S^{-1})(\alpha T^{-1}+(1-\alpha))^{-1}\\[5pt]&=& \alpha (\alpha S^{-1}+(1-\alpha))^{-1}S^{-1}(S -T)T^{-1}(\alpha T^{-1}+(1-\alpha))^{-1}\\[5pt]&=& \alpha (\alpha+(1-\alpha)S)^{-1}(S -T)(\alpha+(1-\alpha)T)^{-1}\\[5pt]&=&\alpha \ f_{1-\alpha}(S^{-1})(S-T) f_{1-\alpha}(T^{-1}).\end{eqnarray*}

Proposition 3.3. Let f be an operator monotone function on $[0,\infty)$ . Let S and T be bounded normal operators in $\mathbb{B}(\mathscr{H}\;\,)$ such that ${\rm{sp}}(S) \subseteq \Gamma_a$ and ${\rm{sp}}(T) \subseteq \Gamma_b$ for some $a,b>0$ . Then,

\begin{equation*}\|f(S)-f(T)\| \leq d_{a,b}(f) \ \|S-T\|,\end{equation*}

for each $X\in C_{\||.\||}$ .

Proof. Without loss of generality, we can assume that f is nonconstant. Let $T_{\alpha}=\alpha+(1-\alpha)T$ and $S_{\alpha}=\alpha+(1-\alpha)S$ for each $0\leq \alpha \leq 1$ . As ${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \Gamma_a$ , we can conclude that $T_{\alpha},S_{\alpha}$ are invertible for each $0\leq \alpha \leq 1$ . Moreover,

\begin{equation*}S_{\alpha}^*S_{\alpha}= \alpha^2+(1-\alpha)^2 S^*S+ \alpha(1-\alpha)(S+S^*).\end{equation*}

Since S is normal, $S+S^* \geq 2 a$ and $S^*S \geq a^2$ . Therefore,

\begin{eqnarray*}S_{\alpha}^*S_{\alpha}&\geq& \alpha^2+(1-\alpha)^2 S^*S+2 a \alpha(1-\alpha)\\[5pt]&\geq& \alpha^2+(1-\alpha)^2 a^2+2 a \alpha(1-\alpha)\\[5pt]&=& (\alpha+ (1-\alpha)a)^{2}.\end{eqnarray*}

Hence, $(S_{\alpha}^*S_{\alpha})^{-1} \leq (\alpha+ (1-\alpha)a)^{-2}$ , and so

\begin{equation*}||S_{\alpha}^{-1}||=||S_{\alpha}^{*-1}S_{\alpha}^{-1}||^{\frac{1}{2}}=||(S_{\alpha}^*S_{\alpha})^{-1}||^{\frac{1}{2}}\leq (\alpha+ (1-\alpha)a)^{-1}.\end{equation*}

A similar argument implies that $||T_{\alpha}^{-1}||\leq (\alpha+ (1-\alpha)b)^{-1}.$ By Lemma 3.2, we have

\begin{eqnarray*}|| f_{\alpha}(S)- f_{\alpha}(T)||&=& \alpha ||S_{\alpha}^{-1}(S-T)T_{\alpha}^{-1}||\\[5pt]&\leq & \alpha ||S_{\alpha}^{-1}|| \ ||S-T|| \ ||T_{\alpha}^{-1}||\\[5pt]&\leq & \frac{\alpha}{(\alpha+ (1-\alpha)a)(\alpha+ (1-\alpha)b)} ||S-T||\\[5pt] &=& d_{a,b}( f_{\alpha}) ||S-T||.\end{eqnarray*}

Now, assume that f is an arbitrary operator monotone function on $[0,\infty)$ . By replacing f(t) with $\frac{f(t)-f(0)}{f(1)-f(0)}$ , we can assume that f is nonnegative and $f(1)=1$ (as f is non-constant, Lemma 3.2. in [Reference Bendat and Sherman2], implies that $f(1)\neq f(0)$ ). By Theorem 3.1, there exists a sequence $\{f_{n}\}$ in $\mathcal{P}$ that satisfies (3.2) and is uniformly convergent to f on compact sets. If

\begin{equation*} f_{n}=\sum_{i=1}^{k_n}\gamma_i f_{\alpha_i},\end{equation*}

then $d_{a,b}(f_n)=\sum_{i=1}^{k_n}\gamma_i d_{a,b}(f_{\alpha_i})$ and we have

\begin{eqnarray*}||f_n(S)- f_n(T)||&=& || \sum_{i=1}^{k_n}\gamma_i f_{\alpha_i}(S)-\sum_{i=1}^{k_n}\gamma_i f_{\alpha_i}(T)||\\[5pt]&\leq & \sum_{i=1}^{k_n}\gamma_i \ || f_{\alpha_i}(S)- f_{\alpha_i}(T)||\\[5pt]&\leq & \sum_{i=1}^{k_n}\gamma_i d_{a,b}( f_{\alpha_i}) \ ||S-T||\\[5pt] &=&d_{a,b}(f_n) ||S-T||.\end{eqnarray*}

Letting $n\to \infty$ to get

(3.3) \begin{equation}||f(S)-f(T)||\leq d_{a,b}(f) ||S-T||.\end{equation}

We obtain the following theorem.

Theorem 3.4. Let f be an operator monotone function on $[0,\infty)$ . Let S and T be bounded operators in $\mathbb{B}(\mathscr{H}\;\,)$ such that ${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \Gamma_a$ for some $a>0$ . Then,

\begin{equation*}\||f(S)X-Xf(T)\|| \leq f'(a) \ \||SX-XT\||,\end{equation*}

for each $X\in C_{\||.\||}$ .

Proof. Let S, T be arbitrary and ${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \{z\in \mathbb{C} \ | \ {\rm{re}}(z)> a\}$ . Since ${\rm{sp}}(S)$ and ${\rm{sp}}(T)$ are compact, there exists a closed disk $\mathcal{O} \subset \Gamma_a$ such that ${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \mathcal{O}$ . Proposition 3.3 shows that f is operator Lipschitz with constant $f'(a)$ on the closed disk $\mathcal{O}$ . Hence, Corollary 2.2 implies that

\begin{equation*} \||f(S)X-Xf(T)\||\leq f'(a) \||SX-XT\||,\end{equation*}

for any symmetric norm $\||.\||$ and any $X\in C_{\||.\||}$ .

In the general case, the assumptions ${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \Gamma_a$ imply that ${\rm{sp}}(S+1/n),{\rm{sp}}(T+1/n) \subseteq \{z\in \mathbb{C} \ | \ {\rm{re}}(z)> a\}$ for each $n\in \mathbb{N}$ . We use the noncommutative Fatou’s lemma to get

\begin{eqnarray*}\||f(S)X-Xf(T)\|| &\leq & \sup_{n\in \mathbb{N}} \ \||f(S+1/n)X-Xf(T+1/n)\|| \\[5pt] & \leq & f'(a) \ \sup_{n\in \mathbb{N}} \||(S+1/n)X-X(T+1/n)\||\\[5pt] &=& f'(a) \ \limsup_n \||(S+1/n)X-X(T+1/n)\||\\[5pt] &=& f'(a) \||SX-XT\||.\end{eqnarray*}

Corollary 3.5. Let f be an operator monotone function on $[0,\infty)$ . Let S and T be bounded operators in $\mathbb{B}(\mathscr{H}\;\,)$ such that ${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \Gamma_a$ for some $a>0$ and $T-S \in C_{|||.|||}$ . Then,

\begin{equation*}\||f(S)-f(T)\|| \leq\;f'(a) \ \||S-T\||.\end{equation*}

Proof. Let $P_n$ be an increasing sequence of finite-dimensional projections such that $P_n \rightarrow I$ in the strong operator topology. We have

\begin{eqnarray*} |||f (P_nS)P_n-P_nf (TP_n)|||&\leq & f'(a) |||P_nS P_n - P_n T P_n|||\\[5pt] &=& f'(a) |||P_n(S-T)P_n|||\\[5pt] &\leq & f'(a) ||P_n||\ |||S-T||| \ ||P_n|||\\[5pt] &\leq & f'(a) |||S-T|||. \end{eqnarray*}

Since f is an analytic function and $S-T$ is a compact operator, $f(S)-f(T)$ is compact. Now $f (P_nS)P_n-P_nf (TP_n) \rightarrow f (S)-f(T)$ in the strong operator topology and $f (S)-f(T)$ is compact, so by the noncommutative Fatou’s lemma, we have

\begin{equation*}|||f (S)-f(T)|||\leq \sup_{n \in \mathbb{N}} \ |||f (P_nS)P_n-P_nf (TP_n)||| \leq\;f'(a) |||S-T|||.\end{equation*}

As $t\mapsto t^r$ and $t\mapsto \log(t+1)$ are operator monotone functions on $[0,\infty)$ for each $0\leq r \leq 1$ , we obtain the following corollaries.

Corollary 3.6. Let $0\leq r \leq 1$ , and let S,T be bounded operators such that ${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \Gamma_a$ . Then

\begin{equation*}\||S^r X-XT^r\|| \leq r a^{r-1} \ \||SX-XT\||,\end{equation*}

for each $X\in C_{\||.\||}$ . In particular, if $T-S\in C_{\||.\||}$ , then

\begin{equation*}\||S^r-T^r\|| \leq r a^{r-1} \ \||S-T\||.\end{equation*}

Corollary 3.7. If S and T are bounded operators such that ${\rm{sp}}(S),{\rm{sp}}(T) \subseteq \Gamma_a$ , then

\begin{equation*}\||\log(S+1) X-X\log(T+1)\|| \leq \frac{1}{a+1} \ \||SX-XT\||,\end{equation*}

for each $X\in C_{\||.\||}$ .