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Corollary 6.6. If \( \varphi \in \mathrm{{BMO}} \) is real valued, then\n\n\[ \operatorname{dist}\left( {\varphi ,{L}^{\infty }}\right) = \left( {\pi /2}\right) \varepsilon \left( \varphi \right) . \] | Proof. Condition (6.6) implies that\n\n(6.7)\n\n\[ M = \mathop{\sup }\limits_{I}\frac{1}{\left| I\right| }{\int }_{I}\exp \left| {{A\varphi } - A{\varphi }_{I}}\right| {dx} < \infty \]\n\nwhenever \( A < 1/\varepsilon \left( \varphi \right) \), and Chebychev’s inequality shows that \( A \leq 1/\varepsilon \left( \varph... | Yes |
Corollary 6.7. If \( f \in {L}^{\infty } \) is real valued, then\n\n\[ \n\operatorname{dist}\left( {f,\operatorname{Re}{H}^{\infty }}\right) = \mathop{\inf }\limits_{{F \in {H}^{\infty }}}\parallel f - \operatorname{Re}F{\parallel }_{\infty }\n\]\n\nsatisfies\n\n\[ \n\operatorname{dist}\left( {f,\operatorname{Re}{H}^{\... | Proof. This is immediate from Corollary 6.6 and the proof of Corollary 4.7. | No |
Lemma 6.8. If \( 1 < p < \infty \) and if \( w{\left( x\right) }^{\prime } \) satisfies \( \left( {A}_{p}\right) \), then\n\n(a) \( w\left( x\right) \) satisfies \( \left( {A}_{r}\right) \) for all \( r > p \), and\n\n(b) the weight \( {\left( 1/w\right) }^{1/\left( {p - 1}\right) } \) satisfies \( \left( {A}_{q}\right... | Proof. For (a) note that \( 1/\left( {r - 1}\right) < 1/\left( {p - 1}\right) \), so that by Hölder’s inequality\n\n\[ \n\frac{1}{\left| I\right| }{\int }_{I}{\left( \frac{1}{w}\right) }^{1/\left( {r - 1}\right) }{dx} \leq {\left( \frac{1}{\left| I\right| }{\int }_{I}{\left( \frac{1}{w}\right) }^{1/\left( {p - 1}\right... | Yes |
Corollary 6.10. If \( 1 < p < \infty \) and if the weight \( w\left( x\right) \) satisfies the \( \left( {A}_{p}\right) \) condition, then\n\n(a) there are \( \delta > 0 \) and \( C > 0 \) such that, for any interval \( I \), \n\n\[ \n{\left( \frac{1}{\left| I\right| }{\int }_{I}w{\left( x\right) }^{1 + \delta }dx\righ... | Proof. To prove part (a) we can assume that \( p > 2 \), because of Lemma 6.8(a). The Cauchy-Schwarz inequality shows that \n\n\[ \n1 \leq \left( {\frac{1}{\left| I\right| }{\int }_{I}{\left( \frac{1}{w}\right) }^{1/\left( {p - 1}\right) }{dx}}\right) \left( {\frac{1}{\left| I\right| }{\int }_{I}{w}^{1/\left( {p - 1}\r... | Yes |
Lemma 6.11. If \( w\left( x\right) \) satisfies \( \left( {A}_{p}\right) \) for some \( p < \infty \), then \( {d\mu } = w\left( x\right) {dx} \) satisfies \( \left( {\mathrm{A}}_{\infty }\right) \) . | Proof. For \( E \subset I \), Hölder’s inequality and Corollary 6.10(a) give\n\n\[ \n\frac{\mu \left( E\right) }{\left| I\right| } = \frac{1}{\left| I\right| }{\int }_{E}{wdx} \leq {\left( \frac{1}{\left| I\right| }{\int }_{I}{w}^{1 + \delta }dx\right) }^{1/\left( {1 + \delta }\right) }{\left( \frac{\left| E\right| }{\... | Yes |
Theorem 1.1. If \( \left\{ {z}_{j}\right\} \) is a sequence in the upper half plane, then the following conditions are equivalent:\n\n(a) The sequence is an interpolating sequence: Every interpolation problem\n\n\[ f\left( {z}_{j}\right) = {a}_{j},\;j = 1,2,\ldots ,\]\n\nwith \( \left\{ {a}_{j}\right\} \in {l}^{\infty ... | Proof of Theorem 1.1. We have already seen that (a) implies (b), along with the estimate \( M \geq 1/\delta \) .\n\nThere are two remaining steps in the proof. First we show that (b) and (c) are equivalent. This is really only a matter of comparing infinite products to infinite sums. Second, we must show that (b) and (... | No |
Lemma 1.2. Let \( B\left( z\right) \) be the Blaschke product in the upper half plane with zeros \( \left\{ {z}_{j}\right\} \) . Then\n\n\[ - \log {\left| B\left( z\right) \right| }^{2} \geq \mathop{\sum }\limits_{j}\frac{{4y}{y}_{j}}{{\left| z - {\bar{z}}_{j}\right| }^{2}},\;z = x + {iy}. \] | Proof. The inequality \( - \log t \geq 1 - t, t > 0 \), gives\n\n\[ - \log {\left| \frac{z - {z}_{j}}{z - {\bar{z}}_{j}}\right| }^{2} \geq 1 - {\left| \frac{z - {z}_{j}}{z - {\bar{z}}_{j}}\right| }^{2} = \frac{{4y}{y}_{j}}{{\left| z - {\bar{z}}_{j}\right| }^{2}}. \]\n\nSumming now gives (1.9). The reverse inequality,\n... | Yes |
Theorem 2.2. Let \( A \) be a uniform algebra on a compact space \( X \) . Let \( \left\{ {{p}_{1},{p}_{2},\ldots ,{p}_{n}}\right\} \) be a finite set of points in \( X \) and let\n\n\[ \nM = \mathop{\sup }\limits_{{{\begin{Vmatrix}{a}_{j}\end{Vmatrix}}_{\infty } \leq 1}}\inf \left\{ {\parallel g\parallel : g \in A, g\... | Proof. Let \( \omega = {e}^{{2\pi i}/n} \) be a primitive \( n \) th root of unity. Let \( {g}_{j}\left( x\right) \in A,\begin{Vmatrix}{g}_{j}\end{Vmatrix} \leq \) \( M + \delta \), where \( \delta > 0 \), interpolate\n\n\[ \n{g}_{j}\left( {p}_{k}\right) = {\omega }^{jk},\;k = 1,2,\ldots, n.\n\]\n\nSet\n\n\[ \n{f}_{j}\... | Yes |
Theorem 3.1. Assume \( \\left\\{ {z}_{j}\\right\\} \) is a sequence of points in the upper half plane satisfying the separation condition (3.1). Then \( \\left\\{ {z}_{j}\\right\\} \) is an interpolating sequence if and only if, for any \( \\varepsilon > 0 \), there is \( q \) such that for any square \( Q \)\n\n(3.3)\... | Proof. For \( {z}_{j} \\in T\\left( {Q}_{n}\\right) \), we have\n\n\[ \n{y}_{j} \\leq \\ell \\left( {Q}_{n}\\right) \\leq 2{y}_{j} \n\]\n\nEach \( T\\left( {Q}_{n}\\right) \) or \( T\\left( Q\\right) \) contains at most \( C\\left( b\\right) \) points \( {z}_{j} \) and if \( {Q}^{k} \\in {G}_{p}\\left( Q\\right) \), th... | Yes |
Lemma 4.1. If \( 0 < p < \infty \), then\n\n\[{\left( \mathrm{E}\left( {\left| \mathop{\sum }\limits_{{j = 1}}^{n} \pm {\alpha }_{j}\right| }^{p}\right) \right) }^{1/p} \leq {C}_{p}{\left( \sum {\left| {\alpha }_{j}\right| }^{2}\right) }^{1/2},\]\n\nwhere \( {C}_{p} \) is a constant that does not depend on \( n \) . | Proof for \( p \leq 2 \) . The case \( p \leq 2 \) is easier because it is only Hölder’s inequality in disguise. Let \( {X}_{j}\left( \omega \right) = {\omega }_{j}, j = 1,2,\ldots, n \) . Then \( \left| {{X}_{j}^{2}\left( \omega \right) }\right| = 1 \), and for \( j \neq \) \( k,\mathrm{E}\left( {{X}_{j}{X}_{k}}\right... | No |
Lemma 5.2. If \( 0 < \alpha < {\beta }_{n} < 1 \), then\n\n(5.5)\n\n\[ \mathop{\prod }\limits_{{n = 1}}^{\infty }\frac{{\beta }_{n} - \alpha }{1 - \alpha {\beta }_{n}} \geq \frac{\left( {\Pi {\beta }_{n}}\right) - \alpha }{1 - {\alpha \Pi }{\beta }_{n}} \] | Proof. If \( B\left( z\right) \) is the Blaschke product with zeros \( {\beta }_{n} \), then the left side of (5.5) is \( B\left( \alpha \right) \) . By Schwarz’s lemma\n\n\[ \rho \left( {B\left( \alpha \right), B\left( 0\right) }\right) , \leq \alpha, \]\n\nand the euclidean description of the disc \( \{ \rho \left( {... | Yes |
Theorem 1.1. Assume that \( G\left( z\right) \) is bounded and \( {C}^{1} \) on the disc \( D \) and that \( \left| G\right| {dxdy} \) is a Carleson measure on \( D \) , (1.4) \[ {\iint }_{S}\left| G\right| {dxdy} \leq A\ell \left( S\right) \] for every sector \[ S = \left\{ {r{e}^{i\theta } : 1 - \ell \left( S\right) ... | Proof. Let \( F\left( z\right) \) be the solution of (1.3) defined by (1.2). Then every solution of (1.3) has the form \[ b\left( z\right) = F\left( z\right) + h\left( z\right) ,\;h \in {A}_{o}. \] The minimal norm of such solutions is \[ \mathop{\inf }\limits_{{h \in {A}_{o}}}\parallel F + h{\parallel }_{\infty } \] t... | Yes |
Theorem 1.2 (Wolff). Assume that \( G\left( z\right) \) is bounded and \( {C}^{1} \) on the disc \( D \) and assume that the two measures\n\n\[ \n{\left| G\right| }^{2}\log \left( {1/\left| z\right| }\right) {dxdy}\text{ and }\left| {\partial G/\partial z}\right| \log \left( {1/\left| z\right| }\right) {dxdy} \n\]\n\na... | Proof. As before, we have\n\n\[ \n\inf \left\{ {\parallel b{\parallel }_{\infty } : \frac{\partial b}{\partial \bar{z}} = G}\right\} = \sup \left\{ {\left| {\frac{1}{2\pi }{\int }_{0}^{2\pi }{Fkd\theta }}\right| : k \in {H}_{0}^{1},\parallel k{\parallel }_{1} \leq 1}\right\} , \n\]\n\nwhere \( F\left( z\right) \) is de... | Yes |
Theorem 2.3. Suppose \( {f}_{1},{f}_{2},\ldots ,{f}_{n} \) and \( g \) are \( {H}^{\infty } \) functions for which (2.12) holds. Then there are \( {g}_{1},{g}_{2},\ldots ,{g}_{n} \) in \( {H}^{\infty } \) such that\n\n\[ \n{g}^{3} = {g}_{1}{f}_{1} + \cdots + {g}_{n}{f}_{n} \n\] | Proof. As in the proof of Theorem 2.1, we convert smooth solutions of (2.13) into \( {H}^{\infty } \) solutions, using Theorem 1.2 to control the norms of the correcting functions. We assume \( \begin{Vmatrix}{f}_{j}\end{Vmatrix} \leq 1,\parallel g\parallel \leq 1 \), and, by normal families, we can suppose \( g \) and... | Yes |
Theorem 3.1. Let \( u\left( z\right) \) be harmonic in the upper half plane. Assume \( \parallel u{\parallel }_{\infty } \leq \) 1. If \( 0 < \alpha < 1 \) and if \( 0 < \varepsilon < 1 \), then there is \( \beta = \beta \left( {\alpha ,\varepsilon }\right) ,0 < \beta < 1 \), such that if \( Q \) is any square with bas... | Proof. The appropriate tool for this proof is the vertical maximal function\n\n\[ {f}^{ + }\left( x\right) = \mathop{\sup }\limits_{{y > 0}}\left| {f\left( {x + {iy}}\right) }\right| \]\n\nwhere \( f\left( z\right) \) is a harmonic function on the upper half plane.\n\nWe may assume \( Q = \{ 0 \leq x \leq 1,0 < y \leq ... | Yes |
Theorem 4.2. Let \( m \) be a complex homomorphism of \( {H}^{\infty } \) . Then \( m \) is in the Silov boundary of \( {H}^{\infty } \) if and only if \( \left| {m\left( B\right) }\right| = 1 \) for every interpolating Blaschke product \( B\left( z\right) \) . | Proof. If \( m \) is in the Silov boundary, then by Newman’s theorem \( \left| {m\left( B\right) }\right| = 1 \) for every Blaschke product \( B\left( z\right) \) . What requires proof is the reverse implication.\n\nIf \( m \) is not in the Silov boundary, then by Newman’s theorem there is a Blaschke product \( {B}_{0}... | Yes |
Theorem 5.1. Let \( \delta > 0 \) . If \( f\left( z\right) \) is analytic on \( D \) and if \( \left| {f\left( z\right) }\right| \leq 1 \), then there is \( \psi \left( z\right) \in {C}^{\infty }\left( D\right) \) such that\n\n(a) \( 0 \leq \psi \left( z\right) \leq 1 \) ,\n\n(b) \( \psi \left( z\right) = 1 \) if \( \l... | Proof of Theorem 5.1. We do the construction in the upper half plane. In fact, we only construct \( \psi \) in the unit square \( {Q}^{0} = \{ 0 \leq x \leq 1,0 < y \leq 1\} \) . Simple conformal mappings and a partition of unity on \( D \) can then be used to produce \( \psi \) on the disc.\n\nFor each dyadic square \... | Yes |
Corollary 6.2. If \( \varepsilon > 0 \), if \( u\left( z\right) \) is harmonic on \( \{ y > 0\} \), and if \( \parallel u{\parallel }_{\infty } \leq 1 \) , then\n\n\[ \n{\int }_{I}{N}_{\varepsilon }\left( x\right) {dx} \leq C{\varepsilon }^{-7} \n\]\n\nwhenever \( I \) is an interval of unit length. | Proof. Let \( \varphi \in {C}^{\infty } \) satisfy (6.1) and (6.2) with \( \varepsilon /2 \) in place of \( \varepsilon \) . If \( {N}_{\varepsilon }\left( x\right) \geq n \) , then there are \( {y}_{0} < {y}_{1} < {y}_{2} < \cdots < {y}_{n} \leq 1 \) such that \( \mid \varphi \left( {x + i{y}_{j}}\right) - \varphi (x ... | Yes |
Lemma 6.3. Let \( u\left( z\right) = {P}_{y} * u\left( x\right) \) be a bounded harmonic function on the upper half plane and let I be an interval on \( \mathbb{R} \) . Then for \( 0 < \delta < \frac{1}{2} \) , \[ \left| {\frac{1}{\left| I\right| }{\int }_{I}u\left( t\right) {dt} - \frac{1}{\left| I\right| }{\int }_{I}... | Proof. By a change of scale we can take \( I = \left\lbrack {0,1}\right\rbrack \) . We can also assume \( \parallel u{\parallel }_{\infty } = 1 \) . Then the left side of (6.4) has supremum \( \parallel F{\parallel }_{1} \), where \[ F\left( t\right) = {\chi }_{I}\left( t\right) - {\int }_{I}{P}_{\delta }\left( {x - t}... | Yes |
Lemma 6.4. For every \( \varepsilon > 0 \) and for every positive integer \( N \) , \[ \mathop{\sum }\limits_{{p = 1}}^{\infty }\mathop{\sum }\limits_{{{I}_{j} \in {G}_{p}}}\left| {I}_{j}\right| \leq \frac{\parallel u{\parallel }_{\infty }^{2}}{{\varepsilon }^{2}}\left| I\right| \] | Proof. Set \( {G}_{0} = \{ I\} \) and set \( {E}_{p} = I \smallsetminus \mathop{\bigcup }\limits_{{G}_{p}}{I}_{j}, p = 0,1,2,\ldots \) . Define \[ {Y}_{p}\left( t\right) = u\left( t\right) {\chi }_{{E}_{p}}\left( t\right) + \mathop{\sum }\limits_{{G}_{p}}{u}_{{I}_{j}}{\chi }_{{I}_{j}}\left( t\right) . \] Then \( \left|... | Yes |
Theorem 1.1. If \( A \) is a closed subalgebra of \( {L}^{\infty } \) containing \( {H}^{\infty } \), then \( A \) is generated by \( {H}^{\infty } \) and \( {\mathrm{U}}_{A} \) . That is, \( A = \left\lbrack {{H}^{\infty },{\mathrm{U}}_{A}}\right\rbrack \) . | Proof. Let \( f \in {A}^{-1} \) . Then \( \log \left| f\right| \in {L}^{\infty } \) and there is \( g \in {\left( {H}^{\infty }\right) }^{-1} \) such that \( \left| g\right| = \left| f\right| \) almost everywhere. Then \( u = {g}^{-1}f \) is a unimodular function invertible in \( A \) such that \( f = {gu} \in \left\lb... | Yes |
Theorem 1.2. If \( A \) is a closed subalgebra of \( {L}^{\infty } \) containing \( {H}^{\infty } \), then its maximal ideal space \( {\mathfrak{M}}_{A} \) can be identified with a closed subset of \( \mathfrak{M} \) which contains \( X \), and \( X \) is the Silov boundary of \( A \) . | Proof. We can identify \( X \) with a closed subset of \( {\mathfrak{M}}_{A} \) because \( A \) is a closed subalgebra of \( {L}^{\infty } = C\left( X\right) \) and, since \( A \supset {H}^{\infty }, A \) separates the points of \( X \) . This means \( X \) is a closed boundary for \( A \) . But since \( A \) is a logm... | Yes |
Theorem 1.3. Let \( A \) be a closed subalgebra of \( {L}^{\infty } \) containing \( {H}^{\infty } \) and let \( \mathrm{U} \subset {\mathrm{U}}_{A} \) be a set of functions in \( {A}^{-1} \), unimodular on \( X \), such that \( A = \left\lbrack {{H}^{\infty },\mathrm{U}}\right\rbrack \) . Then\n\n\[{\mathfrak{M}}_{A} ... | Proof. Because \( A \subset {L}^{\infty } \), we have \( {u}^{-1} = \bar{u} \) when \( u \in {\mathrm{U}}_{A} \) . We also have \( \bar{u}\left( m\right) = \overline{u\left( m\right) } \) whenever \( u \in {L}^{\infty } \) and \( m \in \mathfrak{M} \), because \( {\mu }_{m} \) is real. Therefore, if \( m \in \mathfrak{... | Yes |
Theorem 1.4. If \( A \) is a closed subalgebra of \( {L}^{\infty } \) containing \( {H}^{\infty } \), then either \( A = {H}^{\infty } \) or \( A \supset \left\lbrack {{H}^{\infty },\bar{z}}\right\rbrack \) . | Proof. If \( z \in {A}^{-1} \), then \( \bar{z} \in A \) and \( \left\lbrack {{H}^{\infty },\bar{z}}\right\rbrack \subset A \) . If \( z \notin {A}^{-1} \) then \( z \) lies in some maximal ideal of \( A \), and there is \( m \in {\mathfrak{M}}_{A} \) with \( z\left( m\right) = 0 \) . Since \( {\mathfrak{M}}_{A} \subse... | Yes |
Lemma 2.1. \( {H}^{\infty } + C \) is uniformly closed. | Proof. Recall from Theorem IV.1.6 that when \( g \in C \)\n\n\[ \operatorname{dist}\left( {g,{H}^{\infty }}\right) = \operatorname{dist}\left( {g,{A}_{o}}\right) ,\]\n\nwhere \( {A}_{o} \) is the disc algebra. If \( h \in {L}^{\infty } \) lies in the closure of \( {H}^{\infty } + C \), there are \( {f}_{n} \in {H}^{\in... | Yes |
Theorem 2.2. \( {H}^{\infty } + C \) is a closed subalgebra of \( {L}^{\infty } \) . In fact,\n\n\[ {H}^{\infty } + C = \left\lbrack {{H}^{\infty },\bar{z}}\right\rbrack \] | Proof. The set of functions\n\n\[ f\left( z\right) + \mathop{\sum }\limits_{1}^{N}{a}_{k}{\bar{z}}^{k},\;f \in {H}^{\infty }, \]\n\nis, by definition, dense in \( \left\lbrack {{H}^{\infty },\bar{z}}\right\rbrack \) . By the Weierstrass theorem, this set is also dense in \( {H}^{\infty } + C \) . Because \( {H}^{\infty... | Yes |
Theorem 2.3. \( {QC} = {L}^{\infty } \cap \mathrm{{VMO}} \) . | Proof. If \( f \in {L}^{\infty } \cap \mathrm{{VMO}} \), then by Chapter VI, Theorem 5.2 there are \( \phi ,\psi \in \) \( C \) such that \( f = \phi + {H\psi } \) . But then \( {H\psi } \in {L}^{\infty } \) and \( \psi + {iH\psi } \in {H}^{\infty } \) . Thus\n\n\[ f = - i\left( {\psi + {iH\psi }}\right) + \left( {\var... | Yes |
Corollary 2.4. Let \( f \in {L}^{\infty } \) and let \( f\left( z\right) \) be its Poisson integral. If \( \left| {f\left( z\right) }\right| \) extends continuously to \( \bar{D} \) then \( f \in {QC} \) . | Proof. If \( 1 - \left| z\right| \) is small enough, and if \( {e}^{i\theta } \) is close to \( z/\left| z\right| \), then by hypothesis \( \left| {f\left( {e}^{i\theta }\right) }\right| \) is close to \( \left| {f\left( z\right) }\right| \) . Thus\n\n\[ \n\frac{1}{2\pi }\int {\left| f\left( {e}^{i\theta }\right) - f\l... | Yes |
Corollary 2.5. Let \( A \) be a closed subalgebra of \( {L}^{\infty } \) containing \( {H}^{\infty } \) . If \( {\mathfrak{M}}_{A} = \mathfrak{M} \smallsetminus D \), then \( A = {H}^{\infty } + C \) . | Proof. It is clear that \( {H}^{\infty } + C \subset A \) because \( z \in {A}^{-1} \) . Now if \( f, g \in A \), then\n\n\[ d\left( m\right) = \left( {fg}\right) \left( m\right) - f\left( m\right) g\left( m\right) \]\n\nis continuous on \( \mathfrak{M} \) and \( d\left( m\right) = 0 \) on \( {\mathfrak{M}}_{A} = \math... | Yes |
Lemma 3.3. If \( b\left( z\right) \) is an interpolating Blaschke product having zeros \( \left\{ {z}_{n}\right\} \) in \( D \) and if \( m \in \mathfrak{M} \) is such that \( b\left( m\right) = 0 \), then \( m \) lies in the closure of \( \left\{ {z}_{n}\right\} \) with respect to the topology of \( \mathfrak{M} \) . | Proof. Assuming the contrary, we have \( {f}_{1},{f}_{2},\ldots ,{f}_{k} \in {H}^{\infty } \), with \( {f}_{k}\left( m\right) = \) 0, such that \( \left\{ {z}_{n}\right\} \) is disjoint from\n\n\[ \mathop{\bigcap }\limits_{{j = 1}}^{k}\left\{ {z : \left| {{f}_{j}\left( z\right) }\right| < 1}\right\} \]\n\nThen \( \math... | Yes |
Theorem 3.4. Let \( A \) and \( B \) be two subalgebras of \( {L}^{\infty } \) containing \( {H}^{\infty } \) . Assume that \( {\mathfrak{M}}_{A} = {\mathfrak{M}}_{B} \) and that \( A \) is a Douglas algebra. Then \( A = B \) . | The proof of Theorem 3.4 depends on a characterization of a Douglas algebra \( A = \left\lbrack {{H}^{\infty },\mathrm{B}}\right\rbrack \) in terms of Poisson integrals. For \( b\left( z\right) \) an inner function and for \( 0 < \delta < 1 \), we define the region\n\n\[ \n{G}_{\delta }\left( b\right) = \{ z : \left| {... | Yes |
Theorem 3.5. Let \( A = \left\lbrack {{H}^{\infty },{\mathrm{B}}_{A}}\right\rbrack \) be a Douglas algebra. When \( f \in {L}^{\infty } \) the following conditions are equivalent.\n\n(i) \( f \in A \) .\n\n(ii) For any \( \varepsilon > 0 \), there are \( b \in {\mathrm{B}}_{A} \) and \( \delta ,0 < \delta < 1 \), such ... | Proof of Theorem 3.5. We show (i) \( \Rightarrow \) (ii) \( \Rightarrow \) (iii) \( \Rightarrow \) (i).\n\nAssume (i) holds. Then there are \( b \in {\mathrm{B}}_{A} \) and \( h \in {H}^{\infty } \) such that \( \parallel f - \) \( \bar{b}h{\parallel }_{\infty } < \varepsilon \) . Fix \( {z}_{0} \in {G}_{\delta }\left(... | Yes |
Theorem 4.1. If \( A \) is a closed subalgebra of \( {L}^{\infty } \) containing \( {H}^{\infty } \), then\n\n\[ \n{Q}_{A} = {L}^{\infty } \cap {\mathrm{{VMO}}}_{A} \n\] | Proof. The theorem follows directly from Theorem 3.5. Since\n\n\[ \n{\lambda }_{f} = 2\left( {{v}_{f} + {v}_{\bar{f}}}\right) \n\]\n\nand \( {v}_{f} \) and \( {v}_{\bar{f}} \) are positive,(4.1) means that both \( f \) and \( \bar{f} \) satisfy condition (3.4), and so \( f \in A \cap \bar{A} = {Q}_{A} \) . Conversely, ... | Yes |
Theorem 4.4. Let \( A \) be a closed subalgebra of \( {L}^{\infty } \) containing \( {H}^{\infty } \) and let \( f \in {BMO} \) . Then the following conditions are equivalent:\n\n(i) \( f \in {\mathrm{{VMO}}}_{A} \) ,\n\n(ii) \( f \in {\mathrm{{VMO}}}_{A} \) ,\n\n(iii) \( f = u + \widetilde{v}, u, v \in {C}_{A} \) ,\n\... | Proof. Clearly (iii) \( \Rightarrow \) (ii), and because \( {\mathrm{{VMO}}}_{A} \) is self-conjugate, Theorem 4.1 shows that (ii) \( \Rightarrow \) (i). To complete the proof we show (i) \( \Rightarrow \) (iii). | No |
Lemma 4.5. If \( f \in \mathrm{{BMO}} \) and iff satisfies (4.1) with respect to some \( b \in {\mathrm{B}}_{A} \) , then when \( n \) is large\n\n(4.9)\n\n\[ \mathop{\sup }\limits_{\substack{{F \in {H}_{0}^{1}} \\ {\parallel F{\parallel }_{1} \leq 1} }}\left| {\frac{1}{2\pi }\int f\left( \theta \right) {b}^{n}\left( \... | The proof of this lemma is exactly the same as the proof of Theorem 3.5. The details are left as an exercise. | No |
Theorem 5.2. Let \( A \) be a closed subalgebra of \( {L}^{\infty } \) containing \( {H}^{\infty } \) and let \( f \in {L}^{\infty } \) . Then \( f \in A \) if and only if for each \( \varepsilon > 0 \), there are \( b \in {\mathrm{B}}_{A} \) and \( \delta ,0 < \delta < 1 \), and \( F \in {H}^{\infty }\left( {{G}_{\del... | Proof. Let us do the easy half first. If \( f \in A \) then there are \( b \in {\mathrm{B}}_{A} \) and \( g \in {H}^{\infty } \) such that \( \parallel f - \bar{b}g{\parallel }_{\infty } < \varepsilon \) . Let \( F\left( z\right) = g\left( z\right) /b\left( z\right) \) . For any \( \delta ,0 < \delta < \) \( 1, F \in {... | Yes |
Lemma 1.1. If \( F : \mathrm{D} \rightarrow \mathfrak{M} \) is an analytic mapping, then \( F\left( \mathrm{D}\right) \) is contained in a single Gleason part. | Proof. This is just Schwarz’s lemma. If \( F : \mathrm{D} \rightarrow \mathfrak{M} \) is analytic and if \( {m}_{j} = \) \( F\left( {\zeta }_{j}\right) ,{\zeta }_{j} \in \mathrm{D}, j = 1,2 \), then\n\n\[ \rho \left( {{m}_{1},{m}_{2}}\right) = \sup \left\{ {\left| {f \circ F\left( {\zeta }_{2}\right) }\right| : f \in {... | Yes |
Lemma 1.2. Let \( m \in \mathfrak{M} \) and let \( g \in {H}^{\infty } \) satisfy \( \parallel g{\parallel }_{\infty } \leq 1 \) and \( g\left( m\right) = 0 \) . Suppose that for \( n = 2,3,\ldots \) there is a factorization\n\n\[ g = {g}_{1}^{\left( n\right) }{g}_{2}^{\left( n\right) }\cdots {g}_{n}^{\left( n\right) }... | Proof. For \( {m}^{\prime } \in \mathfrak{M} \), we have\n\n\[ \left| {g\left( {m}^{\prime }\right) }\right| \leq \mathop{\lim }\limits_{{n \rightarrow \infty }}\mathop{\prod }\limits_{{j = 1}}^{n}\left| {{g}_{j}^{\left( n\right) }\left( {m}^{\prime }\right) }\right| \leq \mathop{\lim }\limits_{{n \rightarrow \infty }}... | Yes |
Lemma 1.3. If \( S \) is an interpolating sequence and if \( m \in \bar{S} \), then there is a unique nonconstant analytic map \( {L}_{m} \in {\mathfrak{M}}^{\mathrm{D}} \) such that whenever \( \left( {z}_{j}\right) \) is a net in \( S \) coverging to \( m \) , | \[ \mathop{\lim }\limits_{j}{L}_{{z}_{j}} = {L}_{m} \] | No |
Corollary 1.6. If \( B\left( z\right) \) is an interpolating Blaschke product with zeros \( \left\{ {z}_{n}\right\} \) and if\n\n\[ \n\delta \left( B\right) = \mathop{\inf }\limits_{n}\left( {1 - {\left| {z}_{n}\right| }^{2}}\right) \left| {{B}^{\prime }\left( {z}_{n}\right) }\right| ,\n\]\n\nthen \( B \) has a factori... | Proof. The corollary is immediate from the lemma. If \( {B}_{1}\left( {z}_{n}\right) = 0 \), then by (1.8),\n\n\[ \n\left( {1 - {\left| {z}_{n}\right| }^{2}}\right) \left| {{B}^{\prime }\left( {z}_{n}\right) }\right| = \left( {1 - {\left| {z}_{n}\right| }^{2}}\right) \left| {{B}_{1}^{\prime }\left( {z}_{n}\right) }\rig... | Yes |
For \( 0 < \delta < 1 \) there are constants \( a = a\left( \delta \right) \) and \( b = b\left( \delta \right) \) such that the Blaschke product \( B\left( z\right) \) has a nontrivial factorization \( B = {B}_{1}{B}_{2} \) such that\n\n\[ a{\left| {B}_{1}\left( z\right) \right| }^{1/b} \leq \left| {{B}_{2}\left( z\ri... | We work in the upper half plane \( \mathrm{H} \) . For \( z \in {K}_{\delta }\left( B\right) \), we have\n\n\[ c\left( \delta \right) \log \left| \frac{z - {\bar{z}}_{n}}{z - {z}_{n}}\right| \leq \frac{{2y}{y}_{n}}{{\left| {z}_{n} - \bar{z}\right| }^{2}} \leq \log \left| \frac{z - {\bar{z}}_{n}}{z - {z}_{n}}\right| ,\]... | Yes |
Theorem 2.2. Suppose \( m \in \mathfrak{M} \smallsetminus G \) . Let \( f \in {H}^{\infty },\parallel f{\parallel }_{\infty } \leq 1 \) . If \( f\left( m\right) = 0 \) then \( f = {f}_{1}{f}_{2},{f}_{j} \in {H}^{\infty } \), with \( {\begin{Vmatrix}{f}_{j}\end{Vmatrix}}_{\infty } \leq 1 \) and \( {f}_{j}\left( m\right)... | Proof. First let us reduce the problem to the critical case where \( f\left( z\right) \) is a Blaschke product with simple zeros. Write \( f = {Bg} \) where \( g \in {H}^{\infty } \) has no zeros and \( B\left( z\right) \) is a Blaschke product. If \( g\left( m\right) = 0 \), then \( f = \left( {B{g}^{1/2}}\right) \lef... | Yes |
Corollary 2.3. If \( m \in \mathfrak{M} \smallsetminus G \), then the Gleason part \( P\left( m\right) \) reduces to the singleton \( \{ m\} \) . | Proof. Suppose \( {m}^{\prime } \neq m \) . We show \( {m}^{\prime } \notin P\left( m\right) \) . There is \( g \in {H}^{\infty },\parallel g{\parallel }_{\infty } = 1 \) , with \( g\left( m\right) = 0 \) but \( g\left( {m}^{\prime }\right) \neq 0 \) . For each \( n = 2,3,\ldots \), Theorem 2.2 gives a factorization\n\... | Yes |
Theorem 2.4. Let \( m \in \mathfrak{M} \) . The following are equivalent:\n\n(i) The Gleason part \( P\left( m\right) \) is trivial.\n\n(ii) If \( \left( {z}_{j}\right) \) is a net in \( D \) converging to \( m \), then \( \lim {L}_{{z}_{j}} \) is a constant map \( L\left( \zeta \right) = m \) .\n\n(iii) \( m \notin G ... | Proof. Clearly (i) implies (ii), because any limit point of \( {L}_{{z}_{j}} \) in \( {\mathfrak{M}}^{\mathrm{D}} \) is an analytic map whose range must be contained in \( P\left( m\right) \) by Lemma 1.1. Lemma 1.3 shows that (ii) imples (iii). By Theorem 2.2, (iv) follows from (iii), and by the proof of Corollary 2.3... | Yes |
Theorem 2.5. Suppose \( F : \mathrm{D} \rightarrow \mathfrak{M} \) is a nonconstant analytic map. Let \( m = \) \( F\left( 0\right) \) . Then \( m \in G \) and there is an analytic function \( \tau : \mathrm{D} \rightarrow \mathrm{D},\tau \left( 0\right) = 0 \) , such that\n\n(2.3)\n\n\[ F\left( \zeta \right) = {L}_{m}... | Proof. By Lemma 1.1, \( P\left( m\right) \) is nontrivial, because \( F \) is analytic but not constant. Hence \( m \in G \) by Theorem 2.4, and we have the one-to-one analytic mapping \( {L}_{m} \) . Now use (2.3) to define the function \( \tau : \mathrm{D} \rightarrow \mathrm{D},\tau \left( 0\right) = 0 \) . Our task... | Yes |
In the topology of \( {\mathfrak{M}}^{\mathrm{D}} \), the set of analytic maps from \( \mathrm{D} \) into \( \mathrm{D} \) is dense in the set of analytic maps from \( \mathrm{D} \) into \( \mathfrak{M} \). The set of maps\n\n\[ \left\{ {\alpha {L}_{z}\left( \zeta \right) = \alpha \frac{\zeta + z}{1 + \bar{z}\zeta } : ... | Proof. Let \( F \in {\mathfrak{M}}^{\mathrm{D}} \) be an analytic map. If \( F \) is constant, the corona theorem yields a net of constants in \( D \) converging to \( F \). If \( F \) is not constant, then \( F \) has the form (2.3) with \( {L}_{m} = \lim {L}_{{z}_{i}} \). Then \( {F}_{i}\left( \zeta \right) = {L}_{{z... | Yes |
Lemma 3.1. Let \( N \) be a positive integer; \( N \equiv 1\left( {\;\operatorname{mod}\;3}\right) \) . Let \( \left\{ {z}_{j}\right\} \) be a separated sequence\n\n(3.2)\n\n\[ \left| {{z}_{j} - {z}_{k}}\right| > \alpha {y}_{j},\;k \neq j \]\n\nwith \( \alpha > 0 \) . Then \( \left\{ {z}_{j}\right\} \) can be partition... | Proof. Consider all intervals of the form\n\n\[ {J}_{k, m} = \left( {k{N}^{m},\left( {k + 3}\right) {N}^{m}}\right) ,\]\n\nwith \( k \) and \( m \) integers. For fixed \( m \) the middle thirds of the \( {J}_{k, m} \) form a paving of \( \mathbb{R} \) . To each \( {z}_{j} \) we let \( {I}_{j} \) be that unique \( {J}_{... | Yes |
Lemma 3.2. Suppose \( \\left\\{ {z}_{j}\\right\\} \) is a sequence of points satisfying (3.2). Let \( N \) be given and let \( {Y}_{1},{Y}_{2},\\ldots ,{Y}_{{n}_{0}},{n}_{0} = {n}_{0}\\left( {\\alpha, N}\\right) \), be the subsequences given by Lemma 3.1. Let \( 0 < \\gamma < 1 \) and let \( r \) be a positive integer.... | Proof. If the contrary holds, then for every \( {Y}_{i} \) and for every \( {z}_{j} \\in {Y}_{i} \), induction gives\n\n\[ \n\\mathop{\\sum }\\limits_{{{G}_{q}\\left( {I}_{j}\\right) }}\\frac{\\left| {I}_{k}\\right| }{\\left| {I}_{j}\\right| } \\leq {\\gamma }^{n} \n\]\n\nwhen \( {nr} \\leq q < \\left( {n + 1}\\right) ... | Yes |
Lemma 3.3. Suppose \( \\left\\{ {z}_{j}\\right\\} \) is a sequence of points satisfying (3.2). For \( \\varepsilon > \) 0 and for \( N = N\\left( \\varepsilon \\right) \\), let \( {Y}_{1},{Y}_{2}\\ldots ,{Y}_{{n}_{0}},{n}_{0} = {n}_{0}\\left( {\\alpha, N}\\right) \) be the subsequences given by Lemma 3.1. There exist \... | Proof. We are going to find weights \( {\\lambda }_{k} \) satisfying (3.10) and (3.11) such that \[ 0 \\leq \\sum {\\lambda }_{k}{K}_{k} \\leq {K}_{j} \] and such that (3.12) \[ {\\begin{Vmatrix}{K}_{j}-\\sum {\\lambda }_{k}{K}_{k}\\end{Vmatrix}}_{1} < \\varepsilon /8. \] Replacing \( {\\lambda }_{k} \) by \( {\\lambda... | Yes |
Theorem 3.4. Let \( \\left\\{ {z}_{j}\\right\\} \) be a sequence in the upper half plane. Assume there are real valued harmonic functions \( \\left\\{ {{u}_{j}\\left( z\\right) }\\right\\} \) such that\n\n(i) \( {\\begin{Vmatrix}{u}_{j}\\end{Vmatrix}}_{\\infty } \\leq 1 \) ,\n\n(ii) \( {u}_{j}\\left( {z}_{j}\\right) \\... | Proof. Clearly, the points are separated. The bounded function \( \\exp \\left( {{u}_{j} + i{\\widetilde{u}}_{j}}\\right) \) separates \( {z}_{j} \) from each other \( {z}_{k} \) . We claim (3.1) holds. That will prove the theorem. If (3.1) fails, then by Lemmas 3.2 and 3.3 there is a point \( {z}_{j} \) and there are ... | Yes |
Lemma 4.2. If disjoint subsets of \( \left\{ {z}_{j}\right\} \) have disjoint closures in \( \mathfrak{M} \), then (4.1) holds with \( K, M \), and \( B = \mathop{\sup }\limits_{{n, m}}{\begin{Vmatrix}{u}_{m}^{\left( n\right) }\end{Vmatrix}}_{\infty } \) not depending on the subsets \( S \) and \( T \) of \( \left\{ {z... | Proof. We can suppose \( T = \left\{ {z}_{j}\right\} \smallsetminus S \) . The set \( \mathrm{L} \) of all subsets \( S \) of \( \left\{ {z}_{j}\right\} \) is a compact space in the product topology, in which a neighborhood \( V \) of a subset \( {S}_{0} \) is determined by finitely many indices \( {j}_{1},{j}_{2},\ldo... | Yes |
Lemma 5.5. Let \( \varepsilon > 0,\delta > 0 \), and \( \eta > 0 \) . Suppose \( {I}_{1},{I}_{2},\ldots ,{I}_{K} \) are pairwise disjoint closed bounded intervals on \( \mathbb{R} \), and suppose \( {\alpha }_{1},{\alpha }_{2},\ldots ,{\alpha }_{K} \) are real numbers, \( 0 < {\alpha }_{j} < {2\pi } \) . Then there exi... | Proof. Because the intervals are pairwise disjoint, it is enough to prove this lemma for one interval \( I \) (with \( \eta \) replaced by \( \eta /K \), with \( \varepsilon \) replaced by \( \varepsilon /2 \), and with \( \delta \) replaced by \( \min \left( {\delta /K,\varepsilon /{2K}}\right) ) \) . Fix a closed int... | Yes |
Theorem 1.1.2. Every function in \( {\mathbf{H}}^{2} \) is analytic on the open unit disk. | Proof. Let \( f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{n} \) and \( \left| {z}_{0}\right| < 1 \) ; it must be shown that \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}_{0}^{n} \) converges. Since \( \left| {z}_{0}\right| < 1 \), the geometric series \( \mathop{\sum }\limits_{{n = ... | Yes |
For each point \( {e}^{i{\theta }_{0}} \in {S}^{1} \), there is a function in \( {\mathbf{H}}^{2} \) that is not analytic at \( {e}^{i{\theta }_{0}} \) . | Proof. Define \( {f}_{{\theta }_{0}} \) by\n\n\[ \n{f}_{{\theta }_{0}}\left( z\right) = \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{{e}^{-{in}{\theta }_{0}}}{n}{z}^{n}\;\text{ for }z \in \mathbb{D}.\n\]\n\nSince \( \left\{ \frac{{e}^{-{in}{\theta }_{0}}}{n}\right\} \in {\ell }^{2},{f}_{{\theta }_{0}} \in {\mathbf{H}... | Yes |
The function \( f\left( z\right) = \frac{1}{1 - z} \) is analytic on \( \mathbb{D} \) but is not in \( {\mathbf{H}}^{2} \). | Proof. Since \( \frac{1}{1 - z} = \mathop{\sum }\limits_{{n = 0}}^{\infty }{z}^{n} \), the coefficients of \( f \) are not square-summable. | Yes |
Theorem 1.1.6. For every \( {z}_{0} \in \mathbb{D} \), the mapping \( f \mapsto f\left( {z}_{0}\right) \) is a bounded linear functional on \( {\mathbf{H}}^{2} \) . | Proof. Fix \( {z}_{0} \in \mathbb{D} \) . Note that the Cauchy-Schwarz inequality yields\n\n\[ \left| {f\left( {z}_{0}\right) }\right| = \left| {\mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}_{0}^{n}}\right| \]\n\n\[ \leq {\left( \mathop{\sum }\limits_{{n = 0}}^{\infty }{\left| {a}_{n}\right| }^{2}\right) }^{1/2}{... | Yes |
Theorem 1.1.8. For \( {z}_{0} \in \mathbb{D} \) and \( f \in {\mathbf{H}}^{2}, f\left( {z}_{0}\right) = \left( {f,{k}_{{z}_{0}}}\right) \) and \( \begin{Vmatrix}{k}_{{z}_{0}}\end{Vmatrix} = \) \( {\left( 1 - {\left| {z}_{0}\right| }^{2}\right) }^{-1/2} \) | Proof. Writing \( {k}_{{z}_{0}} \) as \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{\overline{{z}_{0}}}^{n}{z}^{n} \) yields\n\n\[ \left( {f,{k}_{{z}_{0}}}\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}_{0}^{n} = f\left( {z}_{0}\right) \]\n\nand\n\n\[ {\begin{Vmatrix}{k}_{{z}_{0}}\end{Vmatrix}}^{2} = \math... | Yes |
Theorem 1.1.9. If \( \left\{ {f}_{n}\right\} \rightarrow f \) in \( {\mathbf{H}}^{2} \), then \( \left\{ {f}_{n}\right\} \rightarrow f \) uniformly on compact subsets of \( \mathbb{D} \) . | Proof. For a fixed \( {z}_{0} \in \mathbb{D} \), we have\n\n\[ \left| {{f}_{n}\left( {z}_{0}\right) - f\left( {z}_{0}\right) }\right| = \left| \left( {{f}_{n} - f,{k}_{{z}_{0}}}\right) \right| \leq \begin{Vmatrix}{{f}_{n} - f}\end{Vmatrix}\begin{Vmatrix}{k}_{{z}_{0}}\end{Vmatrix}. \]\n\nIf \( K \) is a compact subset o... | Yes |
Theorem 1.1.10. Let \( \widetilde{f} \) and \( {f}_{r} \) be defined as above. Then\n\n\[ \mathop{\lim }\limits_{{r \rightarrow {1}^{ - }}}\begin{Vmatrix}{\widetilde{f} - {f}_{r}}\end{Vmatrix} = 0\;\text{ in }{\widetilde{\mathbf{H}}}^{2}. \] | Proof. Let \( \varepsilon > 0 \) be given. Since \( \mathop{\sum }\limits_{{n = 0}}^{\infty }{\left| {a}_{n}\right| }^{2} < \infty \), we can choose a natural number \( {n}_{0} \) such that\n\n\[ \mathop{\sum }\limits_{{n = {n}_{0}}}^{\infty }{\left| {a}_{n}\right| }^{2} < \frac{\varepsilon }{2} \]\n\nNow choose \( s \... | Yes |
Corollary 1.1.11. For each \( f \) in \( {\mathbf{H}}^{2} \), there exists an increasing sequence \( \left\{ {r}_{n}\right\} \) of positive numbers converging to 1 such that\n\n\[ \mathop{\lim }\limits_{{n \rightarrow \infty }}f\left( {{r}_{n}{e}^{i\theta }}\right) = \widetilde{f}\left( {e}^{i\theta }\right) \]\n\nfor ... | Proof. It is well known that convergence in \( {\mathbf{L}}^{2} \) implies that a subsequence converges pointwise almost everywhere \( \left\lbrack {{47}\text{, p. 68}}\right\rbrack \), so this follows from the previous theorem. | Yes |
Theorem 1.1.12. Let \( f \) be analytic on \( \mathbb{D} \) . Then \( f \in {\mathbf{H}}^{2} \) if and only if\n\n\[ \mathop{\sup }\limits_{{0 < r < 1}}\frac{1}{2\pi }{\int }_{0}^{2\pi }{\left| f\left( r{e}^{i\theta }\right) \right| }^{2}{d\theta } < \infty . \]\n\nMoreover, for \( f \in {\mathbf{H}}^{2} \) ,\n\n\[ \pa... | Proof. Let \( f \) be an analytic function on \( \mathbb{D} \) with power series\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{n} \]\n\nThen, for \( 0 < r < 1 \) ,\n\n\[ {\left| f\left( r{e}^{i\theta }\right) \right| }^{2} = \mathop{\sum }\limits_{{n = 0}}^{\infty }\mathop{\sum }\limits_... | Yes |
For any function \( f \) analytic on the disk, the function\n\n\[ M\left( r\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }{\left| f\left( r{e}^{i\theta }\right) \right| }^{2}{d\theta } \]\n\nis increasing. Therefore \( \mathop{\lim }\limits_{{r \rightarrow {1}^{ - }}}M\left( r\right) = \mathop{\sup }\limits_{{0 < r < 1}}M... | Proof. This follows immediately from the formula\n\n\[ \frac{1}{2\pi }{\int }_{0}^{2\pi }{\left| f\left( r{e}^{i\theta }\right) \right| }^{2}{d\theta } = \mathop{\sum }\limits_{{n = 0}}^{\infty }{\left| {a}_{n}\right| }^{2}{r}^{2n} \]\n\nestablished in the course of the proof of the preceding theorem. | Yes |
Corollary 1.1.16. Every function in \( {\mathbf{H}}^{\infty } \) is in \( {\mathbf{H}}^{2} \) . | Proof. This follows immediately from the characterization of \( {\mathbf{H}}^{2} \) given in Theorem 1.1.12. | No |
Theorem 1.1.17. If \( f \in {\mathbf{H}}^{\infty } \) and \( f \) is not a constant, then \( \left| {f\left( z\right) }\right| < \parallel f{\parallel }_{\infty } \) for all \( z \in \mathbb{D} \) . | Proof. This is an immediate consequence of the maximum modulus theorem ([9, pp. 79, 128], [47, p. 212]). | Yes |
For each real number \( t \), the function \[ {\left( \frac{1 + z}{1 - z}\right) }^{it} \] is in \( {\mathbf{H}}^{\infty } \) . (Recall that \( {w}^{it} = \exp \left( {{it}\log w}\right) \), where \( \log \) is the principal branch of the logarithm.) | Proof. Note that, for every \( z \in \mathbb{D} \), the number \[ w = \frac{1 + z}{1 - z} \] is in the open right half-plane. For each such \( w \) , \[ {w}^{it} = \exp \left( {{it}\log w}\right) = \exp \left( {{it}\left( {\log r + {i\theta }}\right) }\right) , \] where \( w = r{e}^{i\theta } \) and \( \theta \) is in ... | Yes |
Theorem 1.1.19 (Cauchy Integral Formula). If \( f \) is analytic on an open set containing \( \overline{\mathbb{D}} \) and \( {z}_{0} \in \mathbb{D} \), then\n\n\[ f\left( {z}_{0}\right) = \frac{1}{2\pi i}{\int }_{{S}^{1}}\frac{f\left( z\right) }{z - {z}_{0}}{dz}. \] | Proof. Since \( f \) is analytic on \( \overline{\mathbb{D}} \), Corollary 1.1.11 implies that \( \widetilde{f}\left( {e}^{i\theta }\right) = f\left( {e}^{i\theta }\right) \) for all \( \theta \) . Note that \( {k}_{{z}_{0}} \) is continuous on \( \overline{\mathbb{D}} \), and therefore\n\n\[ \widetilde{{k}_{{z}_{0}}}\... | Yes |
Theorem 1.1.21 (Poisson Integral Formula). If \( f \) is in \( {\mathbf{H}}^{2} \) and \( r{e}^{it} \) is in \( \mathbb{D} \), then\n\n\[ f\left( {r{e}^{it}}\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }\widetilde{f}\left( {e}^{i\theta }\right) {P}_{r}\left( {\theta - t}\right) {d\theta }.\] | Proof. Let \( {z}_{0} \in \mathbb{D} \) . Since\n\n\[ \widetilde{{k}_{{z}_{0}}}\left( {e}^{i\theta }\right) = \frac{1}{1 - \overline{{z}_{0}}{e}^{i\theta }} \]\n\nwe have\n\n\[ f\left( {z}_{0}\right) = \left( {f,{k}_{{z}_{0}}}\right) = \left( {\widetilde{f},\widetilde{{k}_{{z}_{0}}}}\right) = \frac{1}{2\pi }{\int }_{0}... | Yes |
Corollary 1.1.22. For \( r \in \lbrack 0,1) \) and \( t \) any real number,\n\n\[ \n\frac{1}{2\pi }{\int }_{0}^{2\pi }{P}_{r}\left( {\theta - t}\right) {d\theta } = 1 \n\] | Proof. This is an immediate application of Theorem 1.1.21 to the case where \( f \) is the constant function 1 . | Yes |
Corollary 1.1.24. Let \( f \in {\mathbf{H}}^{2} \) and suppose that \( \left| {\widetilde{f}\left( {e}^{i\theta }\right) }\right| \leq K \) a.e. Then \( \left| {f\left( z\right) }\right| \leq K \) for all \( z \in \mathbb{D} \) . In particular, a function in \( {\mathbf{H}}^{2} \) whose boundary function is in \( {\mat... | Proof. Recall that \( {P}_{r}\left( \theta \right) > 0 \) for all \( \theta \) and \( 0 \leq r < 1 \) . For \( r{e}^{it} \in \mathbb{D} \), applying the Poisson integral formula (Theorem 1.1.21) to \( f \) yields\n\n\[ \left| {f\left( {r{e}^{it}}\right) }\right| = \left| {\frac{1}{2\pi }{\int }_{0}^{2\pi }\widetilde{f}... | Yes |
Corollary 1.1.27. Let \( \phi \) be a function in \( {\mathbf{L}}^{1}\left( {{S}^{1},{d\theta }}\right) \) . Define \( u \) by\n\n\[ u\left( {r{e}^{it}}\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }{P}_{r}\left( {\theta - t}\right) \phi \left( {e}^{i\theta }\right) {d\theta }.\]\n\nThen\n\n\[ \mathop{\lim }\limits_{{r \r... | Proof. Define \( \alpha \) by\n\n\[ \alpha \left( \theta \right) = {\int }_{0}^{\theta }\phi \left( {e}^{ix}\right) {dx} \]\n\nThen \( \alpha \) has bounded variation (it is, in fact, absolutely continuous) and \( {\alpha }^{\prime }\left( \theta \right) = \phi \left( {e}^{i\theta }\right) \) a.e. Thus Fatou’s theorem ... | Yes |
Corollary 1.1.28. If \( f \in {\mathbf{H}}^{2} \), then \( \mathop{\lim }\limits_{{r \rightarrow {1}^{ - }}}f\left( {r{e}^{i\theta }}\right) = \widetilde{f}\left( {e}^{i\theta }\right) \) for almost all \( \theta \) . | Proof. Recall that if \( f \in {\mathbf{H}}^{2} \), then\n\n\[ f\left( {r{e}^{i\theta }}\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }{P}_{r}\left( {\theta - t}\right) \widetilde{f}\left( {e}^{i\theta }\right) {d\theta } \]\n\n(by the Poisson integral formula; see Theorem 1.1.21). Thus the previous corollary yields \( \m... | Yes |
Corollary 1.1.29. If \( f \in {\mathbf{H}}^{\infty } \), then \( \widetilde{f} \in {\mathbf{L}}^{\infty } \). | Proof. It follows from the above corollary (Corollary 1.1.28) that the essential supremum of \( \widetilde{f} \) is at most \( \parallel f{\parallel }_{\infty } \). | Yes |
Theorem 1.2.4. Let \( A \) be a bounded linear operator.\n\n(i) If \( \parallel 1 - A\parallel < 1 \), then \( A \) is invertible.\n\n(ii) The spectrum of \( A \) is a nonempty compact subset of \( \mathbb{C} \) .\n\n(iii) If \( A \) is an invertible operator, then\n\n\[ \sigma \left( {A}^{-1}\right) = \left\{ {\frac{1... | Proof. Proofs of the above assertions can be found in most introductory functional analysis textbooks. In particular, see [12, pp. 195-198], [42, pp. 188- 194], [48, pp. 252-255], and [55, Chapter V]. | No |
Theorem 1.2.6. For every bounded linear operator \( A,\sigma \left( A\right) = \Pi \left( A\right) \cup \Gamma \left( A\right) \) . | Proof. Clearly both \( \Pi \left( A\right) \) and \( \Gamma \left( A\right) \) are contained in \( \sigma \left( A\right) \) . If \( \lambda \) is not in \( \Pi \left( A\right) \), it follows that \( A - \lambda \) is bounded below, and hence that \( A - \lambda \) is one-to-one and has closed range. If, in addition, \... | Yes |
Theorem 1.2.9 (Toeplitz-Hausdorff Theorem). The numerical range of a bounded linear operator is a convex subset of the complex plane. | Proof. There are several well-known elementary proofs of this theorem (cf. [27, p. 113], [80]). | No |
Example 1.2.10. If \( A \) is a finite diagonal matrix\n\n\[ A = \left( \begin{matrix} {d}_{1} & 0 & 0 & \cdots & 0 \\ 0 & {d}_{2} & 0 & \cdots & 0 \\ 0 & 0 & {d}_{3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \cdots & {d}_{n} \end{matrix}\right) \]\n\nthen \( W\left( A\right) \) is ... | Proof. If \( f = \left( {{f}_{1},{f}_{2},{f}_{3},\ldots ,{f}_{n}}\right) \), then \( \left( {{Af}, f}\right) = \mathop{\sum }\limits_{{i = 1}}^{n}{d}_{i}{\left| {f}_{i}\right| }^{2} \).\n\nThus, in the case of finite diagonal matrices, the numerical range is the convex hull of the spectrum of \( A \) . | Yes |
For every operator \( A,\sigma \left( A\right) \subset \overline{W\left( A\right) } \) (i.e., the closure of the numerical range). | As was mentioned above, \( \sigma \left( A\right) = \Pi \left( A\right) \cup \Gamma \left( A\right) \) (by Theorem 1.2.6). We first prove that \( \Pi \left( A\right) \subset \overline{W\left( A\right) } \) . Let \( \lambda \in \Pi \left( A\right) \) . Then there exists a sequence \( \left\{ {f}_{n}\right\} \) in \( \ma... | Yes |
Theorem 1.2.12. If \( A \) is normal, then \( \overline{W\left( A\right) } \) (the closure of the numerical range of \( A \) ) is the convex hull of \( \sigma \left( A\right) \) . | Proof. By one form of the spectral theorem ([12, p. 272], [41, p. 13], [42, p. 246]), we may assume that \( A \) is multiplication by an \( {\mathbf{L}}^{\infty }\left( {X,{d\mu }}\right) \) function \( \phi \) acting on a space \( {\mathbf{L}}^{2}\left( {X,{d\mu }}\right) \) for some measurable subset \( X \) of the c... | Yes |
Theorem 1.2.20. Let \( A \) be a bounded linear operator. Then \( \mathcal{M} \in \operatorname{Lat}A \) if and only if \( {\mathcal{M}}^{ \bot } \in \operatorname{Lat}{A}^{ * } \) . | Proof. This follows immediately from the fact that, for \( f \in \mathcal{M} \) and \( g \in {\mathcal{M}}^{ \bot } \) , \( \left( {{Af}, g}\right) = \left( {f,{A}^{ * }g}\right) . | No |
Theorem 1.2.23. If \( \mathcal{M} \in \operatorname{Lat}A \) and \( P \) is the projection onto \( \mathcal{M} \), then \( {AP} = \) PAP. Conversely, if \( P \) is a projection and \( {AP} = {PAP} \), then \( P\mathcal{H} \in \operatorname{Lat}A \) . | Proof. Let \( \mathcal{M} \in \operatorname{Lat}A \) and \( P \) be the projection onto \( \mathcal{M} \) . If \( f \in \mathcal{H} \) then \( {Pf} \in \mathcal{M} \) and therefore \( {APf} \) is contained in \( A\mathcal{M} \) . Since \( A\mathcal{M} \subset \mathcal{M} \) it follows that \( P\left( {APf}\right) = {AP... | Yes |
Theorem 1.2.25. Let \( P \) be the projection onto the subspace \( \mathcal{M} \). Then \( \mathcal{M} \) is a reducing subspace for \( A \) if and only if \( {PA} = {AP} \). Also, \( \mathcal{M} \) reduces \( A \) if and only if \( \mathcal{M} \) is invariant under both \( A \) and \( {A}^{ * } \). | Proof. If \( \mathcal{M} \) is a reducing subspace, then \( \mathcal{M} \) and \( {\mathcal{M}}^{ \bot } \) are invariant under \( A \). If \( P \) is the projection onto \( \mathcal{M} \), it is easily seen that \( I - P \) is the projection onto \( {\mathcal{M}}^{ \bot } \). The previous theorem then implies \( A\lef... | Yes |
Theorem 1.2.28. (i) If \( A \) is an operator of rank 1, then there exist \( f \) and \( g \) in \( \mathcal{H} \) with \( A = f \otimes g \) . | Proof of (i): Let \( f \) be any nonzero vector in the range of \( A \) . Since the range of \( A \) is one-dimensional, there is a bounded linear functional \( \lambda \) such that \( {Ah} = \lambda \left( h\right) f \) for all vectors \( h \) . By the Riesz representation theorem ([12, p. 13],[28, pp. 31-32],[55, p. ... | Yes |
Theorem 2.1.2. (i) The unilateral shift is an isometry (i.e., \( \parallel {Uf}\parallel = \parallel f\parallel \) for all \( f \in {\ell }^{2} \) ). | Proof. To prove (i), we must show that \( \begin{Vmatrix}\left( {{a}_{0},{a}_{1},{a}_{2},\ldots }\right) \end{Vmatrix} = \begin{Vmatrix}\left( {0,{a}_{0},{a}_{1},{a}_{2},\ldots }\right) \end{Vmatrix} \) . But this is trivial since \( \mathop{\sum }\limits_{{k = 0}}^{\infty }{\left| {a}_{k}\right| }^{2} = {\left| 0\righ... | Yes |
Theorem 2.1.5. (i) The bilateral shift is a unitary operator. | Proof. It is clear that \( \parallel {Wx}\parallel = \parallel x\parallel \) for all \( x \in {\ell }^{2}\left( \mathbb{Z}\right) \), and thus \( W \) is an isometry. Define the bounded linear operator \( A \) by\n\n\[ A\left( {\ldots ,{a}_{-2},{a}_{-1},{\mathbf{a}}_{\mathbf{0}},{a}_{1},{a}_{2},\ldots }\right) = \left(... | Yes |
Theorem 2.1.6. Let \( U \) be the unilateral shift on \( {\ell }^{2} \) and let \( {U}^{ * } \) be its adjoint. Then \( {\Pi }_{0}\left( {U}^{ * }\right) = \mathbb{D} \) . Furthermore, for \( \lambda \) in \( \mathbb{D},\left( {{U}^{ * } - \lambda }\right) f = 0 \) for a vector \( f \) in \( {\ell }^{2} \) if and only ... | Proof. Observe first that, since \( \begin{Vmatrix}{U}^{ * }\end{Vmatrix} = \parallel U\parallel = 1 \), the spectral radius formula (Theorem 1.2.4) implies that \( {\Pi }_{0}\left( {U}^{ * }\right) \subset \sigma \left( {U}^{ * }\right) \subset \overline{\mathbb{D}} \) .\n\nIf \( \left| \lambda \right| < 1 \), then th... | Yes |
Theorem 2.1.9. The operator \( {M}_{z} \) on \( {\mathbf{H}}^{2} \) is unitarily equivalent to the unilateral shift. | Proof. If \( V \) is the unitary operator mapping \( {\ell }^{2} \) onto \( {\mathbf{H}}^{2} \) given by\n\n\[ V\left( {{a}_{0},{a}_{1},{a}_{2},\ldots }\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{n} \]\n\nit is trivial to verify that \( {VU} = {M}_{z}V \) .\n\nThus \( {M}_{z} \) is a representation o... | Yes |
Theorem 2.1.11. The operator \( {M}_{{e}^{i\theta }} \) on \( {\mathbf{L}}^{2} \) is unitarily equivalent to the bilateral shift \( W \) on \( {\ell }^{2}\left( \mathbb{Z}\right) \), and the operator \( {M}_{{e}^{-{i\theta }}} \) is unitarily equivalent to \( {W}^{ * } \) | Proof. If \( V \) is the unitary operator mapping \( {\ell }^{2}\left( \mathbb{Z}\right) \) onto \( {\mathbf{L}}^{2} \) given by\n\n\[ \nV\left( {\ldots ,{a}_{-2},{a}_{-1},{\mathbf{a}}_{\mathbf{0}},{a}_{1},{a}_{2},\ldots }\right) = \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{a}_{n}{e}^{in\theta }, \n\]\n\nit is e... | Yes |
Theorem 2.1.12. The operator \( {M}_{{e}^{i\theta }} \) leaves the subspace \( {\widetilde{\mathbf{H}}}^{2} \) of \( {\mathbf{L}}^{2} \) invariant and the restriction of \( {M}_{{e}^{i\theta }} \) to \( {\widetilde{\mathbf{H}}}^{2} \) is the unilateral shift on \( {\widetilde{\mathbf{H}}}^{2} \) . On \( {\ell }^{2}\lef... | Proof. This is immediate. | No |
Theorem 2.2.1. The only reducing subspaces of the unilateral shift are \( \{ 0\} \) and the entire space. | Proof. This is easily proven using any representation of \( U \) . Suppose \( \mathcal{M} \) is a subspace of \( {\ell }^{2} \) that reduces \( U \) and is different from \( \{ 0\} \) . We must show that \( \mathcal{M} = {\ell }^{2} \n\nSince \( \mathcal{M} \neq \{ 0\} \), it follows that there exists a nonzero vector\... | Yes |
Theorem 2.2.4. If \( \phi \) is a function in \( {\mathbf{L}}^{\infty } \), then \( \begin{Vmatrix}{M}_{\phi }\end{Vmatrix} = \parallel \phi {\parallel }_{\infty } \) . | Proof. Let \( f \in {\mathbf{L}}^{2} \) with \( \parallel f\parallel = 1 \) . Since \( \left| {\phi \left( {e}^{i\theta }\right) }\right| \leq \parallel \phi {\parallel }_{\infty } \) a.e., it follows that\n\n\[ \n{\begin{Vmatrix}{M}_{\phi }f\end{Vmatrix}}^{2} = \frac{1}{2\pi }{\int }_{0}^{2\pi }{\left| \phi \left( {e}... | Yes |
Corollary 2.2.6. The reducing subspaces of the bilateral shift on \( {\mathbf{L}}^{2} \) are the subspaces \[ {\mathcal{M}}_{E} = \left\{ {f \in {\mathbf{H}}^{2} : f\left( {e}^{i\theta }\right) = 0\text{ a.e. on }E}\right\} \] for measurable subsets \( E \subset {S}^{1} \) . | Proof. Fix any measurable subset \( E \) of \( {S}^{1} \) and let \[ {\mathcal{M}}_{E} = \left\{ {f \in {\mathbf{H}}^{2} : f\left( {e}^{i\theta }\right) = 0\text{ a.e. on }E}\right\} . \] If \( f\left( {e}^{i{\theta }_{0}}\right) = 0 \), then \( {e}^{i{\theta }_{0}}f\left( {e}^{i{\theta }_{0}}\right) = 0 \), so \( {\ma... | Yes |
Theorem 2.2.8. If \( \left| {{\phi }_{1}\left( {e}^{i\theta }\right) }\right| = \left| {{\phi }_{2}\left( {e}^{i\theta }\right) }\right| = 1 \), a.e., then \( {\phi }_{1}{\widetilde{\mathbf{H}}}^{2} = {\phi }_{2}{\widetilde{\mathbf{H}}}^{2} \) if and only if there is a constant \( c \) of modulus 1 such that \( {\phi }... | Proof. Clearly \( {\phi }_{1}{\widetilde{\mathbf{H}}}^{2} = c{\phi }_{1}{\widetilde{\mathbf{H}}}^{2} \) when \( \left| c\right| = 1 \) . Conversely, suppose that \( {\phi }_{1}{\widetilde{\mathbf{H}}}^{2} = {\phi }_{2}{\widetilde{\mathbf{H}}}^{2} \) with \( \left| {{\phi }_{1}\left( {e}^{i\theta }\right) }\right| = \le... | Yes |
Theorem 2.2.10. If \( \phi \) is a nonconstant inner function, then \( \left| {\phi \left( z\right) }\right| < 1 \) for all \( z \in \mathbb{D} \) . | Proof. This follows immediately from Corollary 1.1.24 and Theorem 1.1.17. | No |
Theorem 2.2.11. Let \( \phi \in {\mathbf{H}}^{2} \) . If \( \left| {\widetilde{\phi }\left( {e}^{i\theta }\right) }\right| = 1 \) a.e., then \( \phi \) is an inner function. | Proof. It only needs to be shown that \( \phi \in {\mathbf{H}}^{\infty } \) ; this follows from Corollary 1.1.24. | No |
Corollary 2.2.12 (Beurling's Theorem). Every invariant subspace of the unilateral shift other than \( \{ 0\} \) has the form \( \phi {\mathbf{H}}^{2} \), where \( \phi \) is an inner function. | Proof. The unilateral shift is the restriction of multiplication by \( {e}^{i\theta } \) to \( {\widetilde{\mathbf{H}}}^{2} \), so if \( \mathcal{M} \) is an invariant subspace of the unilateral shift, it is an invariant subspace of the bilateral shift contained in \( {\widetilde{\mathbf{H}}}^{2} \) . Thus, by Theorem ... | Yes |
Corollary 2.2.13. Every invariant subspace of the unilateral shift is cyclic. (See Definition 1.2.17.) | Proof. If \( \mathcal{M} \) is an invariant subspace of the unilateral shift, it has the form \( \phi {\mathbf{H}}^{2} \) by Beurling’s theorem (Corollary 2.2.12). For each \( n,{U}^{n}\phi = {z}^{n}\phi \), so \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}\phi }\right\} \) contains all functions of th... | Yes |
Theorem 2.3.2. If \( F \) is an outer function, then \( F \) has no zeros in \( \mathbb{D} \) . | Proof. If \( F\left( {z}_{0}\right) = 0 \), then \( \left( {{U}^{n}F}\right) \left( {z}_{0}\right) = {z}_{0}^{n}F\left( {z}_{0}\right) = 0 \) for all \( n \) . Since the limit of a sequence of functions in \( {\mathbf{H}}^{2} \) that all vanish at \( {z}_{0} \) must also vanish at \( {z}_{0} \) (Theorem 1.1.9), \[ \mat... | Yes |
Theorem 2.3.3 (The F. and M. Riesz Theorem). If \( f \in {\mathbf{H}}^{2} \) and the set\n\n\[ \left\{ {{e}^{i\theta } : \widetilde{f}\left( {e}^{i\theta }\right) = 0}\right\} \]\n\nhas positive measure, then \( f \) is identically 0 on \( \mathbb{D} \) . | Proof. Let \( E = \left\{ {{e}^{i\theta } : \widetilde{f}\left( {e}^{i\theta }\right) = 0}\right\} \) and let\n\n\[ \mathcal{M} = \mathop{\bigvee }\limits_{{k = 0}}^{\infty }\left\{ {{U}^{k}\widetilde{f}}\right\} = \mathop{\bigvee }\limits_{{k = 0}}^{\infty }\left\{ {{e}^{ik\theta }\widetilde{f}}\right\} \]\n\nThen eve... | Yes |
Theorem 2.3.4. If \( f \) is a function in \( {\mathbf{H}}^{2} \) that is not identically zero, then \( f = {\phi F} \), where \( \phi \) is an inner function and \( F \) is an outer function. This factorization is unique up to constant factors. | Proof. Let \( f \in {\mathbf{H}}^{2} \) and consider \( \mathop{\bigvee }\limits_{{n = 0}}^{\infty }\left\{ {{U}^{n}f}\right\} \) . If this span is \( {\mathbf{H}}^{2} \), then \( f \) is outer by definition, and we can take \( \phi \) to be the constant function 1 and \( F = f \) to obtain the desired conclusion.\n\nI... | Yes |
Theorem 2.3.6. The zeros of an \( {\mathbf{H}}^{2} \) function are precisely the zeros of its inner part. | Proof. This follows immediately from Theorem 2.3.2 and Theorem 2.3.4. | No |
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