0$, we now use the fact that $\mu _\lambda ^s\perp \nu _\lambda ^s$ to choose some open set $F\subset \partial \mathbf{D}$ such that $\mu _\lambda ^s(F)\geq \left\| \mu _\lambda ^s\right\| -\varepsilon $ while at the same time $\nu _\lambda ^s\left( F\right) <\varepsilon $. $F$ is a union of open arcs, so we can pick a finite collection of these arcs, whose union we will call $E$, with $\mu _\lambda ^s\left( F\right) -\mu _\lambda ^s\left( E\right) <\varepsilon $, and, by shrinking them a small amount if necessary, we can be sure that none of these arcs has an endpoint at which $\mu _\lambda $ has an atom. $E^c$ is now a finite union of closed arcs (we can ignore any possible single point sets), and has the property that $\mu _\lambda ^s\left( E^c\right) <2\varepsilon .$ With this $E$, we can see that by the extended version of the lemma, as $a\rightarrow \lambda $ nontangentially, the right hand side of the inequality \ref{split} is at most \begin{equation*} \left( \mu _\lambda ^s\left( E\right) \nu _\lambda ^s\left( E\right) \right) ^{1/2}+\left( \mu _\lambda ^s\left( E^c\right) \nu _\lambda ^s\left( E^c\right) \right) ^{1/2}<\left\| \mu _\lambda ^s\right\| ^{1/2}\varepsilon ^{1/2}+\left\| \nu _\lambda ^s\right\| ^{1/2}\left( 2\varepsilon \right) ^{1/2}. \end{equation*} Since this holds for any $\varepsilon >0$, we conclude that as $a\rightarrow \lambda $ nontangentially, $\left\langle C_\varphi f_a,C_\psi f_a\right\rangle \rightarrow 0$, and the theorem is proved.\hfill$\square $% \vspace{4mm} If for some $\lambda \in \partial \mathbf{D},$ $\mu _\lambda ^s$ and $\nu _\lambda ^s$ are not mutually singular, but are different, then we can still get a lower estimate for $\left\| C_\varphi -C_\psi \right\| _e^2$. If $E$ is any arc (or finite set of arcs) in $\partial \mathbf{D}$, with $\mu _\lambda ^s\left( E\right) \neq \nu _\lambda ^s\left( E\right) $, then we can use equations \ref{essnormest}, \ref{integral2}, and the Cauchy-Schwarz inequality, with the integrals all taken over the set $E$ to get \begin{eqnarray*} \left\| C_\varphi -C_\psi \right\| _e^2 &\geq &\left\| \left( C_\varphi -C_\psi \right) f_a\right\| _2^2 \\ &\geq &\int_E\frac{1-\left| a\right| ^2}{\left| 1-\overline{a}\varphi \left( \xi \right) \right| ^2}dm\left( \xi \right) +\int_E\frac{1-\left| a\right| ^2% }{\left| 1-\overline{a}\psi \left( \xi \right) \right| ^2}dm\left( \xi \right) \\ &&-2\left( \int_E\frac{1-\left| a\right| ^2}{\left| 1-\overline{a}\varphi \left( \xi \right) \right| ^2}dm\left( \xi \right) \int_E\frac{1-\left| a\right| ^2}{\left| 1-\overline{a}\psi \left( \xi \right) \right| ^2}% dm\left( \xi \right) \right) ^{1/2} \\ &\geq &\left( \left( \int_E\frac{1-\left| a\right| ^2}{\left| 1-\overline{a}% \varphi \left( \xi \right) \right| ^2}dm\left( \xi \right) \right) ^{1/2}-\left( \int_E\frac{1-\left| a\right| ^2}{\left| 1-\overline{a}\psi \left( \xi \right) \right| ^2}dm\left( \xi \right) \right) ^{1/2}\right) ^2. \end{eqnarray*} By the lemma, this last difference approaches $\left( \mu _\lambda ^s\left( E\right) ^{1/2}-\nu _\lambda ^s\left( E\right) ^{1/2}\right) ^2$ as $% a\rightarrow \lambda $ nontangentially. This gives us \begin{theorem} \label{diffestthm}If $\varphi $ and $\psi $ are holomorphic self-maps of the disk which have corresponding measures $\mu _\lambda $ and $\nu _\lambda $, $% \lambda \in \partial \mathbf{D}$, then for any $\lambda $ and any set $E$ that is a finite union of arcs in $\partial \mathbf{D}$ whose endpoints do not contain atoms of the measures $\mu _\lambda $ or $\nu _\lambda $, we have \begin{equation*} \left\| C_\varphi -C_\psi \right\| _e\geq \left| \mu _\lambda ^s\left( E\right) ^{1/2}-\nu _\lambda ^s\left( E\right) ^{1/2}\right| . \end{equation*} In particular, unless $\mu _\lambda ^s=\nu _\lambda ^s$ for all $\lambda $, we can find such a set $E$ for which the right side of the above is positive, and we will thus have \begin{equation*} \left\| C_\varphi -C_\psi \right\| _e>0. \end{equation*} \end{theorem} We can generalize the methods above to give a lower bound on the essential norm of a linear combination of composition operators. \begin{theorem} \label{lincomb}Let $\varphi _1,\ldots ,\varphi _n$ be holomorphic self-maps of the disk, with corresponding measures $\mu _{1,\lambda },\ldots ,\mu _{n,\lambda }$. If for some $\lambda \in \partial \mathbf{D}$, the measures $% \mu _{1,\lambda }^s,\ldots ,\mu _{n,\lambda }^s$ are mutually singular, then \begin{equation*} \left\| \sum_{j=1}^na_jC_{\varphi _j}\right\| _e^2\geq \left| a_1\right| ^2\left\| \mu _{1,\lambda }^s\right\| +\cdots +\left| a_n\right| ^2\left\| \mu _{n,\lambda }^s\right\| . \end{equation*} \end{theorem} \noindent \emph{\ Proof. }First we write \begin{equation*} \left\| \sum_{j=1}^na_jC_{\varphi _j}\right\| _e^2\geq \left\| \left( \sum_{j=1}^na_jC_{\varphi _j}\right) f_a\right\| _2^2 \end{equation*} and \begin{eqnarray*} \left\| \left( \sum_{j=1}^na_jC_{\varphi _j}\right) f_a\right\| _2^2 &=&\left\langle \left( \sum_{j=1}^na_jC_{\varphi _j}\right) f_a,\left( \sum_{j=1}^na_jC_{\varphi _j}\right) f_a\right\rangle \\ &=&\left| a_1\right| ^2\left\| C_{\varphi _1}f_a\right\| _2^2+\cdots +\left| a_n\right| ^2\left\| C_{\varphi _n}f_a\right\| _2^2 \\ &&+\sum_{i\neq j}a_i\overline{a_j}\left\langle C_{\varphi _i}f_a,C_{\varphi _j}f_a\right\rangle . \end{eqnarray*} The first terms on the right approach $\left| a_1\right| ^2\left\| \mu _{1,\lambda }^s\right\| +\cdots +\left| a_n\right| ^2\left\| \mu _{n,\lambda }^s\right\| $ as $a\rightarrow \lambda $ nontangentially, and the remaining terms approach zero, as we showed earlier. \hfill$\square $\vspace{4mm} \section{Compactness of the Difference of Composition Operators\label {compdiffsec}} In \cite{BDM}, MacCluer gives several theorems regarding the essential norms of composition operator differences. One of them, \cite[Theorem 2.2]{BDM}, tells us \begin{theorem}[MacCluer] Let $\varphi :\mathbf{D}\rightarrow \mathbf{D}$ and suppose that $\varphi $ has finite angular derivative at a point $e^{i\theta }\in \partial \mathbf{D} $. Let $\psi :\mathbf{D}\rightarrow \mathbf{D}$ be holomorphic and consider $% C_\varphi $ and $C_\psi $ acting on $H^2$. Then, unless both \begin{equation*} \psi (e^{i\theta })=\varphi (e^{i\theta }) \end{equation*} and \begin{equation*} \psi ^{\prime }(e^{i\theta })=\varphi ^{\prime }(e^{i\theta }), \end{equation*} we have $\left\| C_\varphi -C_\psi \right\| _e^2\geq \left| \varphi ^{\prime }(e^{i\theta })\right| ^{-1}$. \end{theorem} From this theorem, we can get the immediate corollary \begin{corollary} If $C_\varphi -C_\psi $ is a compact operator (on $H^2$), then $\varphi $ and $\psi $ must have angular derivatives at the same places on the unit circle, and at those places the values of the angular derivatives must be the same. \end{corollary} From our point of view, we can use Theorem \ref{diffestthm} to get the following: \begin{theorem} \label{compdiffthm}If $C_\varphi -C_\psi $ is a compact operator, then we must have $\mu _\lambda ^s=\nu _\lambda ^s$ for all $\lambda \in \partial \mathbf{D}$. \end{theorem} This theorem is a generalization of the previous one, since if $\mu _\lambda ^s=\nu _\lambda ^s$ for all $\lambda \in \partial \mathbf{D}$, then $\varphi $ and $\psi $ have the same angular derivatives - wherever any $\mu _\lambda ^s$ or $\nu _\lambda ^s$ has an atom, with magnitude the inverse of the magnitude of the atom. The angular derivative condition mentioned above, though necessary for compactness of the difference of two composition operators, is, by this last theorem, not sufficient. The situation is seemingly quite similar to the compactness question for a single composition operator, in which the Aleksandrov measure condition was both necessary and sufficient for compactness. In this case, however, the converse to Theorem \ref{compdiffthm} is unknown. We need to have an upper bound to $\left\| C_\varphi -C_\psi \right\| _e^2$ of a form similar to the lower bound to prove the converse, which would then give us an answer to a question raised in \cite{JHSCS2} \begin{conjecture} Given holomorphic self maps of the disk $\varphi $ and $\psi $, with associated measures $\mu _\lambda $ and $\nu _\lambda $, $C_\varphi -C_\psi $ is a compact operator on $H^2$ if and only if $\mu _\lambda ^s=\nu _\lambda ^s$ for all $\lambda \in \partial \mathbf{D}$. \end{conjecture} \section{Other Consequences of the Main Theorems\label{conseqsec}} The Aleksandrov measure approach to the essential norm of a composition operator leads to several interesting inequalities and equalities, both by generalizing some theorems, as we have shown above, and by providing new proofs to some theorems already known. We will begin by presenting new proofs for some theorems by Shapiro and Sundberg in \cite[Theorem 2.3, Corollary 2.4]{JHSCS2}. \begin{theorem} \label{avg}For $\varphi $ a holomorphic self-map of the disk, let\\ $% E=\left\{ \zeta \in \partial \mathbf{D}:\left| \varphi \left( \zeta \right) \right| =1\right\} $. Then $\left\| C_\varphi \right\| _e^2\geq m\left( E\right) $. \end{theorem} \noindent \emph{Proof.} We can relate the size of $E$ to properties of the corresponding family $\left\{ \mu _\lambda \right\} $. The average (over all $\lambda \in \partial \mathbf{D}$) of the norms of the measures $\mu _\lambda ^{a.c.}$ is given by $\iint \frac{1-\left| \varphi \left( \zeta \right) \right| ^2}{\left| \lambda -\varphi \left( \zeta \right) \right| ^2}% dm\left( \zeta \right) dm\left( \lambda \right) $. When we switch the order of integration, we can carry out the inside integral for each $\lambda $. For any $\zeta $ in $E$, the integrand (and hence the integral) is zero, whereas for any $\zeta $ not in $E$, the integral is the that of the Poisson kernel for the point $\varphi \left( \zeta \right) $ (with $\left| \varphi \left( \zeta \right) \right| $ $<1$), which has value $1$. Therefore we have $\iint \frac{1-\left| \varphi \left( \zeta \right) \right| ^2}{\left| \lambda -\varphi \left( \zeta \right) \right| ^2}dm\left( \lambda \right) dm\left( \zeta \right) =1-m\left( E\right) $. The average value of the norm of $\mu _\lambda $ is given by $\int \frac{1-\left| \varphi \left( 0\right) \right| ^2}{\left| \lambda -\varphi \left( 0\right) \right| ^2}dm\left( \lambda \right) $, which is equal to $1$. So we find that the average value of $\left\| \mu _\lambda ^s\right\| $ is just $1-(1-m\left( E\right) )=m\left( E\right) $. Since $\left\| C_\varphi \right\| _e^2=\sup\limits_{\lambda \in \partial \mathbf{D}}\left\| \mu _\lambda ^s\right\| \geq $ average$\left\| \mu _\lambda ^s\right\| $, the theorem is proved.\hfill$\square $\vspace{4mm} Using Theorem \ref{diffthm}, we can get a new proof for the similar lower bound for the essential norms of the difference of two composition operators. The previous theorem is an immediate corollary of the following: \begin{theorem}[Shapiro and Sundberg] Let $\varphi $ $\neq $ $\psi $ be holomorphic self-maps of the disk. Then \begin{equation*} \left\| C_\varphi -C_\psi \right\| _e^2\geq m\left( E_\varphi \right) +m\left( E_\psi \right) , \end{equation*} where $E_\varphi =\left\{ \zeta \in \partial \mathbf{D}:\left| \varphi \left( \zeta \right) \right| =1\right\} $ and $E_\psi =\left\{ \zeta \in \partial \mathbf{D}:\left| \psi \left( \zeta \right) \right| =1\right\} $. \end{theorem} \noindent \emph{New Proof.} For $m$-a.e. $\lambda \in \partial \mathbf{D}$, $% \mu _\lambda ^s$ is singular to $\nu _\lambda ^s$ , since if they are not mutually singular for a.e. $\lambda $, then we use the fact that $\varphi (\zeta )=\lambda $ for $\mu _\lambda ^s$ a.e.-$\zeta $ and $\psi (\zeta )=\lambda $ for $\nu _\lambda ^s$ a.e.-$\zeta $ to deduce that $\left\{ \lambda \in \partial \mathbf{D:\;}\varphi ^{-1}(\lambda )\cap \psi ^{-1}(\lambda )\text{ is nonempty}\right\} $ has positive measure, thus $% \left\{ \zeta \in \partial \mathbf{D:\;}\varphi (\zeta )=\psi (\zeta )\right\} $ has positive measure, which is impossible for two unequal holomorphic functions. We can thus use Theorem \ref{diffthm} to tell us that \begin{eqnarray*} \left\| C_\varphi -C_\psi \right\| _e^2 &\geq &\sup\limits_{\lambda \in \partial \mathbf{D,\,}\mu _\lambda ^s\perp \nu _\lambda ^s}\left( \left\| \mu _\lambda ^s\right\| +\left\| \nu _\lambda ^s\right\| \right) \\ &\geq &\limfunc{avg}\limits_{\lambda \in \partial \mathbf{D}}\left( \left\| \mu _\lambda ^s\right\| +\left\| \nu _\lambda ^s\right\| \right) \\ &=&\limfunc{avg}\limits_{\lambda \in \partial \mathbf{D}}\left\| \mu _\lambda ^s\right\| +\limfunc{avg}\limits_{\lambda \in \partial \mathbf{D}% }\left\| \nu _\lambda ^s\right\| =m\left( E_\varphi \right) +m\left( E_\psi \right) . \end{eqnarray*} The last step follows by the argument in the proof of the previous theorem, Theorem \ref{avg}.\hfill$\square $\vspace{4mm} Indeed, with the use of Theorem \ref{lincomb}, we can even get a new proof of the theorem from \cite[Theorem 2.3]{JHSCS2} which gives a similar lower bound for the essential norm of a linear combination of composition operators.\vspace{4mm} These theorems, as pointed out in \cite{JHSCS2}, provide a generalization of a theorem of Berkson \cite{EB}, and can be used to show that if $m\left( E_\varphi \right) >0$, then $C_\varphi $ is isolated in the space of composition operators acting on $H^2$. This is clear, since its distance in operator norm, and, in fact, its essential distance, from any other composition operator is, by the theorem, bounded below by $m\left( E_\varphi \right) ^{1/2}$. A different sort of lower bound for the essential norm of a composition operator is given by Cowen in \cite[Theorem 2.4]{Cowen} along with a similar upper bound, under the added condition of the continuity of $\varphi ^{\prime }$ on $\overline{\mathbf{D}}$. It also appears in \cite[Theorem 3.3] {JHS1}. It is proved by Cima and Matheson in \cite{CM} as a corollary of the formula for the essential norm of a composition operator. It is presented here to show further uses of the Aleksandrov measure approach to essential norm inequalities. \begin{theorem}[Cowen] \label{atomthm}Let $\delta \left( \omega \right) =\sum \left\{ \left| \varphi ^{\prime }\left( \zeta \right) \right| ^{-1}:\zeta \in \partial \mathbf{D}\text{ and }\varphi \left( \zeta \right) =\omega \right\} $. Then \begin{equation*} \left\| C_\varphi \right\| _e^2\geq \sup \left\{ \delta \left( \omega \right) :\omega \in \partial \mathbf{D}\right\} . \end{equation*} \end{theorem} \noindent \emph{Proof }(Cima and Matheson). This theorem is an immediate consequence of Theorem \ref{mainthm}, since the measure $\mu _\omega $ has atoms precisely at those points $\zeta \in \partial \mathbf{D}$ with $% \varphi \left( \zeta \right) =\omega $, for which $\varphi $ has an angular derivative, and the magnitude of each atom is just the reciprocal of the absolute value of the angular derivative. Thus the sum of the magnitudes of the atoms of $\mu _\omega $ is exactly $\delta \left( \omega \right) $, so this is certainly $\leq \left\| \mu _\omega ^s\right\| $. The theorem follows.\hfill$\square $\vspace{4mm} It should be noted here that we also get information about when we have equality in the theorem. If $\varphi $ is univalent, or even of bounded valence, then each measure $\mu _\omega $ has a singular part which consists of either at most a single atom (in the univalent case), or some number of atoms (bounded by the valence). The measure $\mu _\omega $ can have a nonatomic singular part only if $\varphi $ has infinite valence near some $% \omega $. If there is no nonatomic singular part of $\mu _\omega $, then we have equality in Theorem \ref{atomthm}. Thus we see that though bounded valence for $\varphi $ is sufficient for equality, it is not necessary. Finally, we can obtain very easily an exact expression for the essential norm of the composition operator generated by an inner function. We get the same answer, of course, as found by Shapiro in \cite[Theorem 2.5]{JHS1}, but with a different simple proof. \begin{theorem} If $\varphi $ is an inner function, then \begin{equation*} \left\| C_\varphi \right\| _e=\left[ \frac{1+\left| \varphi \left( 0\right) \right| }{1-\left| \varphi \left( 0\right) \right| }\right] ^{1/2}. \end{equation*} \end{theorem} \noindent \emph{New Proof.} This follows from our formula for the essential norm of a composition operator, since for an inner function $\varphi $, all of the measures $\mu _{\lambda \text{ }}$are singular. We already know that $% \left\| \mu _\lambda \right\| =\limfunc{Re}\left( \frac{\lambda +\varphi \left( 0\right) }{\lambda -\varphi \left( 0\right) }\right) =\frac{1-\left| \varphi \left( 0\right) \right| ^2}{\left| \lambda -\varphi \left( 0\right) \right| ^2}$ (this was proven in our list of properties of the $\mu _\lambda $). The largest value of $\frac{1-\left| \varphi \left( 0\right) \right| ^2}{% \left| \lambda -\varphi \left( 0\right) \right| ^2}$ is taken when the denominator is as small as possible, i.e., $\lambda $ is the boundary point closest to $\varphi \left( 0\right) $ . We thus have \begin{eqnarray*} \left\| C_\varphi \right\| _e^2 &=&\sup\limits_{\lambda \in \partial \mathbf{% D}}\left\| \mu _\lambda ^s\right\| \\ &=&\sup\limits_{\lambda \in \partial \mathbf{D}}\left\| \mu _\lambda \right\| \\ &=&\sup\limits_{\lambda \in \partial \mathbf{D}}\frac{1-\left| \varphi \left( 0\right) \right| ^2}{\left| \lambda -\varphi \left( 0\right) \right| ^2} \\ &=&\frac{1-\left| \varphi \left( 0\right) \right| ^2}{\left( 1-\left| \varphi \left( 0\right) \right| \right) ^2}=\frac{1+\left| \varphi \left( 0\right) \right| }{1-\left| \varphi \left( 0\right) \right| }. \end{eqnarray*} \hfill$\square $\vspace{4mm} \begin{thebibliography}{99} \bibitem{EB} E. 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