Q stringlengths 4 3.96k | A stringlengths 1 3k | Result stringclasses 4
values |
|---|---|---|
For each \( {z}_{0} \in \mathbb{D} \), the function\n\n\[ \psi \left( z\right) = \frac{{z}_{0} - z}{1 - \overline{{z}_{0}}z} \]\n\nis an inner function and \( {\mathcal{M}}_{{z}_{0}} = \left\{ {f \in {\mathbf{H}}^{2} : f\left( {z}_{0}\right) = 0}\right\} = \psi {\mathbf{H}}^{2} \) . | The function \( \psi \) is clearly in \( {\mathbf{H}}^{\infty } \) . Moreover, it is continuous on the closure of \( \mathbb{D} \) . Therefore, to show that \( \psi \) is inner, it suffices to show that \( \left| {\psi \left( z\right) }\right| = 1 \) when \( \left| z\right| = 1 \) . For this, note that \( \left| z\righ... | Yes |
Theorem 2.4.2. If \( {z}_{1},{z}_{2},\ldots ,{z}_{n} \in \mathbb{D} \) , \n\n\[ \n\mathcal{M} = \left\{ {f \in {\mathbf{H}}^{2} : f\left( {z}_{1}\right) = f\left( {z}_{2}\right) = \cdots = f\left( {z}_{n}\right) = 0}\right\} , \n\] \n\nand \n\n\[ \n\psi \left( z\right) = \mathop{\prod }\limits_{{k = 1}}^{n}\frac{{z}_{k... | Proof. It is obvious that a product of a finite number of inner functions is inner. Thus Theorem 2.4.1 above implies that \( \psi \) is inner. \n\nIt is clear that \( \psi {\mathbf{H}}^{2} \) is contained in \( \mathcal{M} \) . The proof of the opposite inclusion is very similar to the proof of the case of a single fac... | No |
Corollary 2.4.3. Suppose that the inner function \( \phi \) has a zero of multiplicity \( s \) at 0 and also vanishes at the nonzero points \( {z}_{1},{z}_{2},\ldots ,{z}_{n} \in \mathbb{D} \) (allowing repetition according to multiplicity). Let\n\n\[ \psi \left( z\right) = {z}^{s}\mathop{\prod }\limits_{{k = 1}}^{n}\f... | Proof. Since \( \psi \) is a product of inner functions, \( \psi \) is inner. The function \( \phi \) is in the subspace \( \mathcal{M} \) of the preceding Theorem 2.4.2, so that theorem implies that \( \phi = {\psi S} \), where \( S \) is in \( {\mathbf{H}}^{2} \) . Moreover, \( \widetilde{\phi } = \widetilde{\psi }\w... | Yes |
Theorem 2.4.4. If \( \phi \) is an inner function and \( \phi \left( 0\right) \neq 0 \), and if \( \left\{ {z}_{j}\right\} \) is a sequence in \( \mathbb{D} \) such that \( \phi \left( {z}_{j}\right) = 0 \) for all \( j \), then \( \left| {\phi \left( 0\right) }\right| < \mathop{\prod }\limits_{{j = 1}}^{n}\left| {z}_{... | Proof. For each natural number \( n \), let\n\n\[ \n{B}_{n}\left( z\right) = \mathop{\prod }\limits_{{j = 1}}^{n}\frac{{z}_{j} - z}{1 - \overline{{z}_{j}}z}.\n\]\n\nAs shown in Corollary 2.4.3, each \( {B}_{n} \) is an inner function and, for each \( n \), there is an inner function \( {S}_{n} \) such that \( \phi = {B... | Yes |
If \( {z}_{k} = \frac{k}{k + 1} \) for natural numbers \( k \), there is no function \( f \) in \( {\mathbf{H}}^{2} \) whose set of zeros is exactly \( \left\{ {z}_{k}\right\} \) . | Proof. Suppose that \( f \) was such a function and let \( \phi \) be its inner part. In particular, \( \phi \left( 0\right) \neq 0 \) . By the previous theorem,\n\n\[ \left| {\phi \left( 0\right) }\right| < \mathop{\prod }\limits_{{j = k}}^{n}\left| {z}_{k}\right| \]\n\nfor every natural number \( n \) . But\n\n\[ \ma... | Yes |
Corollary 2.4.7. If \( {\left\{ {z}_{k}\right\} }_{k = 1}^{\infty } \) are nonzero zeros of a function \( f \) in \( {\mathbf{H}}^{2} \) that is not identically zero, then\n\n\[ \mathop{\prod }\limits_{{k = 1}}^{\infty }\left| {z}_{k}\right| \;\text{ converges. } \] | Proof. If \( {p}_{n} = \mathop{\prod }\limits_{{k = 1}}^{n}\left| {z}_{k}\right| \), then \( \left\{ {p}_{n}\right\} \) is a decreasing sequence (since \( \left| {z}_{k}\right| < 1 \) for all \( k \) ) and hence converges to some \( P \geq 0 \) . It must be shown that \( P > 0 \) .\n\nIf \( f \) has a zero of multiplic... | Yes |
Theorem 2.4.8. If \( 0 < {r}_{k} < 1 \) for all \( k \), then \( \mathop{\prod }\limits_{{k = 1}}^{\infty }{r}_{k} \) converges if and only if \( \mathop{\sum }\limits_{{k = 0}}^{\infty }\left( {1 - {r}_{k}}\right) \) converges. | Proof. Assume \( \mathop{\prod }\limits_{{k = 1}}^{\infty }{r}_{k} \) converges. Since \( \left\{ {\mathop{\prod }\limits_{{k = 1}}^{n}{r}_{k}}\right\} \) converges as \( n \rightarrow \infty \) to a number different from 0 , it follows that\n\n\[ \left\{ \frac{\mathop{\prod }\limits_{{k = 1}}^{n}{r}_{k}}{\mathop{\prod... | Yes |
Theorem 2.4.9. Let \( 0 < {r}_{k} < 1 \) for all \( k \) . If \( \mathop{\prod }\limits_{{k = 1}}^{\infty }{r}_{k} \) converges, then\n\n\[ \left\{ {\mathop{\prod }\limits_{{k = m + 1}}^{n}{r}_{k}}\right\} \]\n\nconverges to 1 as \( n \) and \( m \) approach infinity. | Proof. Observe that the above sequence is just\n\n\[ \frac{\mathop{\prod }\limits_{{k = 1}}^{n}{r}_{k}}{\mathop{\prod }\limits_{{k = 1}}^{m}{r}_{k}} \]\n\nand hence, when \( m \) and \( n \) approach infinity, the sequence approaches 1 . | Yes |
Corollary 2.4.10. If \( {\left\{ {z}_{k}\right\} }_{k = 1}^{\infty } \) are zeros of a function \( f \in {\mathbf{H}}^{2} \) and \( f \) is not identically zero, then\n\n\[ \mathop{\sum }\limits_{{k = 1}}^{\infty }\left( {1 - \left| {z}_{k}\right| }\right) < \infty \] | Proof. It follows from Corollary 2.4.7 and Theorem 2.4.8 that the subseries obtained by including only the nonzero zeros converges. If the multiplicity of the zero at 0 is \( m \), adding in the terms corresponding to the zeros at 0 adds \( m \) to the sum of the subseries.\n\nThus if a function in \( {\mathbf{H}}^{2} ... | Yes |
Theorem 2.4.12 (Hurwitz’s Theorem). Let \( \left\{ {g}_{n}\right\} \) be a sequence of functions that are analytic and have no zeros on a domain \( V \) . If \( \left\{ {g}_{n}\right\} \rightarrow g \) uniformly on compact subsets of \( V \), then either \( g \) has no zero in \( V \) or \( g \) is identically 0 on \( ... | Proof. Suppose that \( g \) is not identically 0 on \( V \) . We will show that assuming that \( g \) has a zero in \( V \) leads to a contradiction. Suppose, then, that \( g\left( {z}_{0}\right) = 0 \) for some \( {z}_{0} \in V \) . Choose \( r > 0 \) such that \( {z}_{0} \) is the only zero of \( g \) in the disk \( ... | Yes |
Corollary 2.4.14. Suppose that the inner function \( \phi \) has a zero of multiplicity \( s \) at 0 and has nonzero zeros at the points \( {z}_{1},{z}_{2},{z}_{3},\ldots \) in \( \mathbb{D} \) (repeated according to multiplicity). Let\n\n\[ B\left( z\right) = {z}^{s}\mathop{\prod }\limits_{{k = 1}}^{\infty }\frac{\ove... | Proof. For each positive integer \( n \), let\n\n\[ {B}_{n}\left( z\right) = \mathop{\prod }\limits_{{k = 1}}^{n}\frac{\overline{{z}_{k}}}{\left| {z}_{k}\right| }\frac{{z}_{k} - z}{1 - \overline{{z}_{k}}z} \]\n\nand\n\n\[ {B}_{0}\left( z\right) = \mathop{\prod }\limits_{{k = 1}}^{\infty }\frac{\overline{{z}_{k}}}{\left... | Yes |
There is a Blaschke product that is not analytic at any point of \( {S}^{1} \) . | Proof. Let \( \left\{ {c}_{n}\right\} \) be any sequence dense in \( {S}^{1} \) and, for each \( n \), let\n\n\[ \n{z}_{n} = \left( {1 - \frac{1}{{n}^{2}}}\right) {c}_{n} \n\]\n\nThen \( 1 - \left| {z}_{n}\right| = \frac{1}{{n}^{2}} \), so the \( \left\{ {z}_{n}\right\} \) are the zeros of a Blaschke product \( B \) .\... | Yes |
Every continuous function on \( \left\lbrack {0,1}\right\rbrack \) is a uniform limit of polynomials whose exponents are prime numbers. | Proof. Euler's theorem (Theorem 2.5.2) and the Müntz-Szasz theorem (Theorem 2.5.1) immediately imply the corollary. | No |
If \[ f\left( z\right) = \exp \left( \frac{z + 1}{z - 1}\right) \] then \( f \) is a singular inner function. | Proof. Recall that \( \left| {e}^{w}\right| = \left| {e}^{\operatorname{Re}w + i\operatorname{Im}w}\right| = \left| {e}^{\operatorname{Re}w}\right| = {e}^{\operatorname{Re}w} \) for every complex number \( w \) . Hence \[ \left| {f\left( z\right) }\right| = \exp \left( {\operatorname{Re}\left( \frac{z + 1}{z - 1}\right... | Yes |
Corollary 2.6.6. If \( \phi \) is an inner function, then \( \phi \) can be written as \( \phi = {BS} \), where \( B \) is the Blaschke product formed from the zeros of \( \phi \) and \( S \) is a singular inner function given by an integral as in Theorem 2.6.5. | Proof. Given an inner function \( \phi \), let \( B \) be the Blaschke product formed from the zeros of \( \phi \). By Theorem 2.4.14, \( \phi /B \) is an inner function with no zeros in \( \mathbb{D} \). Thus \( \phi /B = S \) is a singular inner function and therefore has the form given in Theorem 2.6.5. | Yes |
Theorem 2.6.10. Every collection of inner functions has a greatest common divisor. Every finite collection of inner functions has a least common multiple. | Proof. Let \( \left\{ {\phi }_{\alpha }\right\} \) be a collection of inner functions. Define\n\n\[ \mathcal{M} = \bigvee \left\{ {f : f \in {\phi }_{\alpha }{\mathbf{H}}^{2}\text{ for some }\alpha }\right\} \]\n\nIt is easily seen that \( \mathcal{M} \) is invariant under \( U \) . Thus \( \mathcal{M} = {\phi }_{g}{\m... | Yes |
Corollary 2.6.11. If \( \mathcal{M} \) is an invariant subspace, other than \( \{ 0\} \), of the unilateral shift, then \( \mathcal{M} = \phi {\mathbf{H}}^{2} \), where \( \phi \) is the greatest common divisor of all the inner parts of all the functions in \( \mathcal{M} \) . | Proof. Since \( \mathcal{M} \) is an invariant subspace for the unilateral shift, Beurling’s theorem (Theorem 2.2.12) guarantees that there exists an inner function \( \phi \) such that \( \mathcal{M} = \phi {\mathbf{H}}^{2} \) . We will show that \( \phi \) is the greatest common divisor of all the inner parts of all ... | Yes |
Theorem 2.6.13. Let \( A \) be a bounded operator on an infinite-dimensional separable Hilbert space. Suppose that \( \mathcal{M} \) and \( \mathcal{N} \) are in Lat \( A \) and \( \mathcal{M} \subset \mathcal{N} \) . If \( \mathcal{N} \ominus \mathcal{M} \) is infinite-dimensional, then the lattice\n\n\[ \n\{ \mathcal... | Proof. Let \( P \) be the projection onto the subspace \( \mathcal{N} \ominus \mathcal{M} \) . Define the bounded linear operator \( B \) on \( \mathcal{N} \ominus \mathcal{M} \) as \( B = {\left. PA\right| }_{\mathcal{N} \ominus \mathcal{M}} \) . We will show that Lat \( B \) is order-isomorphic to the lattice of the ... | Yes |
Let\n\n\[ \phi \left( z\right) = \exp \left( \frac{z + 1}{z - 1}\right) \]\n\nand let \( \mathcal{M} = {\left( \phi {\mathbf{H}}^{2}\right) }^{ \bot } \) . Then Lat \( \left( {\left. {U}^{ * }\right| }_{\mathcal{M}}\right) \) is order-isomorphic to the closed unit interval \( \left\lbrack {0,1}\right\rbrack \) with its... | Proof. The function \( \phi \) is inner singular. The measure \( \mu \) defined by \( \mu \left( A\right) = {2\pi } \) for any Borel set \( A \) containing 0 and \( \mu \left( B\right) = 0 \) for Borel sets \( B \) that do not contain zero is the measure provided by Theorem 2.6.5.\n\nIt follows from Beurling’s theorem ... | Yes |
Corollary 2.7.2 (The F. and M. Riesz Theorem). If \( f \in {\mathbf{H}}^{2} \) and the set \[ \left\{ {{e}^{i\theta } : \widetilde{f}\left( {e}^{i\theta }\right) = 0}\right\} \] has positive measure, then \( f \) is identically 0 on \( \mathbb{D} \) . | Proof. If the set \( \left\{ {{e}^{i\theta } : \widetilde{f}\left( {e}^{i\theta }\right) = 0}\right\} \) has positive measure, the function \( \log \left| {\widetilde{f}\left( {e}^{i\theta }\right) }\right| \) is not integrable on \( \left\lbrack {0,{2\pi }}\right\rbrack \) . Thus the previous lemma implies that \( f \... | Yes |
Theorem 2.7.3. (i) For \( x > 0,\log x \leq x - 1 \) . | Proof of (i): If \( g\left( x\right) = x - 1 - \log x \), then \( g\left( 1\right) = 0 \) and \( {g}^{\prime }\left( x\right) = 1 - \frac{1}{x} \) for all \( x > 0 \) . Thus \( g \) is decreasing on \( \left( {0,1}\right) \) and increasing on \( \left( {1,\infty }\right) \), so \( g\left( x\right) \geq g\left( 1\right)... | Yes |
Theorem 2.7.4. If \( f \) is in \( {\mathbf{H}}^{2} \) and \( f \) is not identically 0 on \( \mathbb{D} \), define\n\n\[ F\left( z\right) = \exp \left( {\frac{1}{2\pi }{\int }_{0}^{2\pi }\frac{{e}^{i\theta } + z}{{e}^{i\theta } - z}\log \left| {\widetilde{f}\left( {e}^{i\theta }\right) }\right| {d\theta }}\right) .\n\... | Proof. For each fixed \( z \in \mathbb{D} \) ,\n\n\[ \left| \frac{{e}^{i\theta } + z}{{e}^{i\theta } - z}\right| \]\n\nis a bounded function of \( {e}^{i\theta } \in {S}^{1} \) . Since \( \log \left| {\widetilde{f}\left( {e}^{i\theta }\right) }\right| \) is in \( {\mathbf{L}}^{1}\left( {{S}^{1},{d\theta }}\right) \) by... | Yes |
Corollary 2.7.5. If \( f \) is in \( {\mathbf{H}}^{2}, f \) is not identically 0, and \( F \) is defined by\n\n\[ F\left( z\right) = \exp \left( {\frac{1}{2\pi }{\int }_{0}^{2\pi }\frac{{e}^{i\theta } + z}{{e}^{i\theta } - z}\log \left| {\widetilde{f}\left( {e}^{i\theta }\right) }\right| {d\theta }}\right) ,\]\n\nthen ... | Proof. Since \( F \) is in \( {\mathbf{H}}^{2} \) ,\n\n\[ \left| {\widetilde{F}\left( {e}^{i\theta }\right) }\right| = \mathop{\lim }\limits_{{r \rightarrow {1}^{ - }}}\left| {F\left( {r{e}^{i\theta }}\right) }\right| = \exp \left( {\mathop{\lim }\limits_{{r \rightarrow {1}^{ - }}}\frac{1}{2\pi }{\int }_{0}^{2\pi }{P}_... | Yes |
Theorem 2.7.7. If \( f \) is in \( {\mathbf{H}}^{2} \) and \( F \) is defined by\n\n\[ F\left( z\right) = \exp \left( {\frac{1}{2\pi }{\int }_{0}^{2\pi }\frac{{e}^{i\theta } + z}{{e}^{i\theta } - z}\log \left| {f\left( {e}^{i\theta }\right) }\right| {d\theta }}\right) ,\]\nthen \( F \) is outer. | Proof. It was shown above (Theorem 2.7.4) that \( F \) is in \( {\mathbf{H}}^{2} \) . Thus \( F = {\phi G} \) for \( \phi \) an inner function and \( G \) an outer function (Theorem 2.3.4). It suffices to show that \( \phi \) is a constant function. Since \( \left| {\widetilde{\phi }\left( {e}^{i\theta }\right) }\right... | Yes |
Corollary 2.7.8. The function \( G \) in \( {\mathbf{H}}^{2} \) is outer if and only if there exists a constant \( K \) of modulus 1 such that\n\n\[ G\left( z\right) = K\exp \left( {\frac{1}{2\pi }{\int }_{0}^{2\pi }\frac{{e}^{i\theta } + z}{{e}^{i\theta } - z}\log \left| {\widetilde{G}\left( {e}^{i\theta }\right) }\ri... | Proof. Since a nonzero constant times an outer function is outer, the previous theorem establishes that every function of the given form is outer.\n\nConversely, suppose \( G \) is any outer function and define\n\n\[ F\left( z\right) = \exp \left( {\frac{1}{2\pi }{\int }_{0}^{2\pi }\frac{{e}^{i\theta } + z}{{e}^{i\thet... | Yes |
Corollary 2.7.9. Let \( f \) be a function in \( {\mathbf{H}}^{2} \) that is not identically 0, and let \( B \) denote the Blaschke product formed from its zeros. Then there exists a constant \( K \) of modulus 1 and a singular measure \( \mu \) on \( {S}^{1} \) such that \( f\left( z\right) \) has the form\n\n\[ \n{KB... | Proof. This follows immediately from Theorem 2.3.4, Corollary 2.4.14, Theorem 2.6.5, and Corollary 2.7.8. | No |
Theorem 2.7.10. The function \( F \) in \( {\mathbf{H}}^{2} \) is outer if and only if\n\n\[ \log \left| {F\left( 0\right) }\right| = \frac{1}{2\pi }{\int }_{0}^{2\pi }\log \left| {\widetilde{F}\left( {e}^{i\theta }\right) }\right| {d\theta }. \] | Proof. If \( F \in {\mathbf{H}}^{2} \) is outer, then\n\n\[ F\left( z\right) = K\exp \left( {\frac{1}{2\pi }{\int }_{0}^{2\pi }\frac{{e}^{i\theta } + z}{{e}^{i\theta } - z}\log \left| {\widetilde{F}\left( {e}^{i\theta }\right) }\right| {d\theta }}\right) \]\n\nfor some \( K \) (Corollary 2.7.8).\n\nThen,\n\n\[ F\left( ... | Yes |
Theorem 3.1.2. Let \( \phi \) be a function in \( {\mathbf{L}}^{\infty } \) with Fourier series\n\n\[ \mathop{\sum }\limits_{{n = - \infty }}^{\infty }{\phi }_{n}{e}^{in\theta } \]\n\nThen the matrix of \( {M}_{\phi } \) with respect to the orthonormal basis \( {\left\{ {e}^{in\theta }\right\} }_{n = - \infty }^{\infty... | Proof. We compute, for each pair of integers \( \left( {m, n}\right) \),\n\n\[ \left( {{M}_{\phi }{e}_{n},{e}_{m}}\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }\phi \left( {e}^{i\theta }\right) {e}^{in\theta }\overline{{e}^{im\theta }}{d\theta } = \frac{1}{2\pi }{\int }_{0}^{2\pi }\phi \left( {e}^{i\theta }\right) {e}^{-... | Yes |
Theorem 3.1.4. A bounded linear operator on \( {\mathbf{L}}^{2} \) is multiplication by an \( {\mathbf{L}}^{\infty } \) function if and only if its matrix with respect to the standard basis in \( {\mathbf{L}}^{2} \) is a Toeplitz matrix. | Proof. It was shown in Theorem 3.1.2 that every multiplication on \( {\mathbf{L}}^{2} \) has a Toeplitz matrix. To establish the converse, assume that \( A \) has a Toeplitz matrix.\n\nTo show that \( A = {M}_{\phi } \) for some \( \phi \in {\mathbf{L}}^{\infty } \), it suffices, by Theorem 2.2.5, to show that \( {AW} ... | Yes |
Theorem 3.1.6. If \( \phi \in {\mathbf{L}}^{\infty } \), then \( \sigma \left( {M}_{\phi }\right) = \Pi \left( {M}_{\phi }\right) = \operatorname{ess}\operatorname{ran}\phi \) . | Proof. We prove this in two steps. We first show that ess \( \operatorname{ran}\phi \subset \Pi \left( {M}_{\phi }\right) \) , and then show that \( \sigma \left( {M}_{\phi }\right) \subset \) ess ran \( \phi \) . These two assertions together imply the theorem.\n\nLet \( \lambda \in \operatorname{essran}\phi \) . For ... | Yes |
Theorem 3.2.2. The matrix of the Toeplitz operator with symbol \( \phi \) with respect to the basis \( {\left\{ {e}^{in\theta }\right\} }_{n = 0}^{\infty } \) of \( {\widetilde{\mathbf{H}}}^{2} \) is\n\n\[ \n{T}_{\phi } = \left( \begin{matrix} {\phi }_{0} & {\phi }_{-1} & {\phi }_{-2} & {\phi }_{-3} & \\ {\phi }_{1} & ... | Proof. This can easily be computed in the same way as the corresponding result for multiplication operators (Theorem 3.1.2). Alternatively, since \( P \) is the projection onto \( {\widetilde{\mathbf{H}}}^{2} \) and \( {T}_{\phi } \) is defined on \( {\widetilde{\mathbf{H}}}^{2} \), the matrix of \( {T}_{\phi } \) is t... | Yes |
Theorem 3.2.4. If \( {T}_{\phi } \) is an analytic Toeplitz operator, then the matrix of \( {T}_{\phi } \) with respect to the basis \( {\left\{ {e}^{in\theta }\right\} }_{n = 0}^{\infty } \) is\n\n\[ \n{T}_{\phi } = \left( \begin{matrix} {\phi }_{0} & 0 & 0 & 0 & 0 & \\ {\phi }_{1} & {\phi }_{0} & 0 & 0 & 0 & \ddots \... | Proof. The Fourier coefficients of \( \phi \) with negative indices are 0 since \( \phi \) is in \( {\widetilde{\mathbf{H}}}^{2} \), so this follows immediately from Theorem 3.2.2. | Yes |
Corollary 3.2.7. The operator \( T \) is a Toeplitz operator if and only if \( {U}^{ * }{TU} = \) \( T \), where \( U \) is the unilateral shift. | Proof. Note that, for nonnegative integers \( n \) and \( m \) ,\n\n\[ \left( {{U}^{ * }{TU}{e}_{n},{e}_{m}}\right) = \left( {{TU}{e}_{n}, U{e}_{m}}\right) = \left( {T{e}_{n + 1},{e}_{m + 1}}\right) . \]\n\nThus if \( T \) is a Toeplitz operator, \( \left( {{U}^{ * }{TU}{e}_{n},{e}_{m}}\right) = \left( {T{e}_{n},{e}_{m... | Yes |
Theorem 3.2.8. The mapping \( \phi \mapsto {T}_{\phi } \) is an injective, bounded, linear, adjoint-preserving (i.e., \( {T}_{\phi }^{ * } = {T}_{\bar{\phi }} \) ) mapping from \( {\mathbf{L}}^{\infty } \) onto the space of Toeplitz operators regarded as a subspace of the algebra of bounded linear operators on \( {\wid... | Proof. The map is obviously linear, and\n\n\[ \begin{Vmatrix}{T}_{\phi }\end{Vmatrix} = \begin{Vmatrix}{P{M}_{\phi }}\end{Vmatrix} \leq \begin{Vmatrix}{M}_{\phi }\end{Vmatrix} = \parallel \phi {\parallel }_{\infty }, \]\n\nso the mapping is bounded. If \( {T}_{\phi } \) and \( {T}_{\psi } \) are equal, then comparing t... | Yes |
Lemma 3.2.9. If \( {T}_{\psi } \) and \( {T}_{\phi } \) are Toeplitz operators and \( U \) is the unilateral shift, then\n\n\[ \n{U}^{ * }{T}_{\psi }{T}_{\phi }U - {T}_{\psi }{T}_{\phi } = P\left( {{e}^{-{i\theta }}\psi }\right) \otimes P\left( {{e}^{-{i\theta }}\bar{\phi }}\right) \;\text{ (see Notation 1.2.27),} \n\]... | Proof. Note that \( I = U{U}^{ * } + {e}_{0} \otimes {e}_{0} \), where \( {e}_{0} \otimes {e}_{0} \) is the orthogonal projection from \( {\widetilde{\mathbf{H}}}^{2} \) onto the constants. Therefore\n\n\[ \n{U}^{ * }{T}_{\psi }{T}_{\phi }U = {U}^{ * }{T}_{\psi }\left( {U{U}^{ * } + {e}_{0} \otimes {e}_{0}}\right) {T}_... | Yes |
Theorem 3.2.11. For \( \psi \) and \( \phi \) in \( {\mathbf{L}}^{\infty },{T}_{\psi }{T}_{\phi } \) is a Toeplitz operator if and only if either \( {T}_{\psi } \) is coanalytic or \( {T}_{\phi } \) is analytic. In both of those cases, \( {T}_{\psi }{T}_{\phi } = {T}_{\psi \phi } \) . | Proof. That \( {T}_{\psi }{T}_{\phi } = {T}_{\psi \phi } \) when \( {T}_{\phi } \) is analytic is trivial, as remarked above. If \( {T}_{\psi } \) is coanalytic, then\n\n\[{\left( {T}_{\psi }{T}_{\phi }\right) }^{ * } = {T}_{\phi }^{ * }{T}_{\psi }^{ * } = {T}_{\bar{\phi }}{T}_{\bar{\psi }} = {T}_{\bar{\phi }\bar{\psi ... | Yes |
Corollary 3.2.12. The product of two Toeplitz operators is 0 if and only if one of the factors is 0. | Proof. Assume that \( {T}_{\psi }{T}_{\phi } = 0 \) . Since 0 is a Toeplitz operator, the previous theorem implies that either \( {T}_{\psi } \) is coanalytic or \( {T}_{\phi } \) is analytic, and that \( {T}_{\psi }{T}_{\phi } = \) \( {T}_{\psi \phi } = 0 \) . Hence \( {\psi \phi } = 0 \) .\n\nIf \( {T}_{\phi } \) is ... | Yes |
Corollary 3.2.14. If two Toeplitz operators commute with each other and neither is a linear combination of the identity and the other operator, then their product is a Toeplitz operator. | Proof. It follows from the previous theorem that either the first operator is analytic or the second operator is coanalytic. Theorem 3.2.11 finishes the proof. | No |
Theorem 3.2.15. A Toeplitz operator is self-adjoint if and only if its symbol is real-valued almost everywhere. | Proof. This follows immediately from the fact that \( {T}_{\phi } = {T}_{\phi }^{ * } \) if and only if \( \phi = \bar{\phi } \) | Yes |
Corollary 3.2.16. The Toeplitz operator \( {T}_{\phi } \) is normal if and only if there exist complex numbers \( c \) and \( d \) and a real-valued function \( \psi \) in \( {\mathbf{L}}^{\infty } \) such that \( \phi = {c\psi } + d \) a.e. That is, the only normal Toeplitz operators are affine functions of self-adjoi... | Proof. Since \( {T}_{\psi }^{ * } = {T}_{\psi } \), it is clear that every operator of the given form is normal.\n\nThe converse follows easily from Theorem 3.2.13. To prove it, suppose that \( {T}_{\phi } \) commutes with \( {T}_{\phi }^{ * } = {T}_{\bar{\phi }} \) . At least one of the three cases of Theorem 3.2.13 h... | Yes |
Theorem 3.2.17. The only compact Toeplitz operator is 0. | Proof. Let \( {T}_{\phi } \) be a compact Toeplitz operator. Recall that compact operators map weakly convergent sequences to norm-convergent sequences [12, p. 173]. Therefore we have\n\n\[ \left\{ \left( {{T}_{\phi }{e}_{s + n},{e}_{t + n}}\right) \right\} \rightarrow 0\;\text{ as }n \rightarrow \infty ,\]\n\nsince \(... | Yes |
Theorem 3.3.1 (The Spectral Inclusion Theorem). For all \( \phi \) in \( {\mathbf{L}}^{\infty } \) , the spectrum of \( {M}_{\phi } \) is contained in the spectrum of \( {T}_{\phi } \) . More precisely,\n\n\[ \n\text{ess}\operatorname{ran}\phi = \Pi \left( {M}_{\phi }\right) = \sigma \left( {M}_{\phi }\right) \subset \... | Proof. It has already been shown (Theorem 3.1.6) that ess ran \( \phi = \Pi \left( {M}_{\phi }\right) = \) \( \sigma \left( {M}_{\phi }\right) \) .\n\nAssume \( \lambda \in \Pi \left( {M}_{\phi }\right) \) . Then there exists a sequence \( \left\{ {f}_{n}\right\} \) of functions in \( {\mathbf{L}}^{2} \) with \( \begin... | No |
Corollary 3.3.2. For \( \phi \) in \( {\mathbf{L}}^{\infty },\parallel \phi {\parallel }_{\infty } = \begin{Vmatrix}{M}_{\phi }\end{Vmatrix} = \begin{Vmatrix}{T}_{\phi }\end{Vmatrix} = r\left( {T}_{\phi }\right) \) (where \( r\left( {T}_{\phi }\right) \) is the spectral radius; see Definition 1.2.2). | Proof. We have already shown that \( \parallel \phi {\parallel }_{\infty } = \begin{Vmatrix}{M}_{\phi }\end{Vmatrix} \) (see Theorem 2.2.4).\n\nIt is an easy consequence of the spectral radius formula that the spectral radius of a normal operator is equal to its norm (see, for example, [41, p. 11] or \( \left\lbrack {{... | Yes |
Corollary 3.3.3. The only quasinilpotent Toeplitz operator is the operator 0. | Proof. If \( r\left( {T}_{\phi }\right) = 0 \), the previous corollary gives \( \parallel \phi {\parallel }_{\infty } = 0 \), which implies that \( \phi = 0 \) a.e.; i.e., \( {T}_{\phi } = 0 \) . | Yes |
Corollary 3.3.4. If \( \phi \) is in \( {\mathbf{L}}^{\infty } \) and \( K \) is compact, then \( \begin{Vmatrix}{{T}_{\phi } - K}\end{Vmatrix} \geq \begin{Vmatrix}{T}_{\phi }\end{Vmatrix} \) . | Proof. Since \( \begin{Vmatrix}{T}_{{e}^{-{in\theta }}}\end{Vmatrix} = 1 \) for each natural number \( n \) ,\n\n\[ \begin{Vmatrix}{{T}_{\phi } - K}\end{Vmatrix} \geq \begin{Vmatrix}{{T}_{{e}^{-{in\theta }}}\left( {{T}_{\phi } - K}\right) }\end{Vmatrix} \]\n\n\[ = \begin{Vmatrix}{{T}_{{e}^{-{in\theta }}\phi } - {T}_{{e... | Yes |
There is a rank-one operator \( K \) such that \( \begin{Vmatrix}{{S}^{ * } - K}\end{Vmatrix} = \begin{Vmatrix}{S}^{ * }\end{Vmatrix} \) , where \( {S}^{ * } \) is the backward unilateral shift. | Proof. Let \( K = {e}_{0} \otimes {e}_{1} \) . Clearly, \( \begin{Vmatrix}{{S}^{ * } - K}\end{Vmatrix} = 1 = \begin{Vmatrix}{S}^{ * }\end{Vmatrix} \). | Yes |
Theorem 3.3.6. For \( \phi \) in \( {\mathbf{L}}^{\infty } \), the following sets are identical:\n\n(i) the closed convex hull of \( \sigma \left( {T}_{\phi }\right) \) ;\n\n(ii) the closed convex hull of \( \sigma \left( {M}_{\phi }\right) \) ;\n\n(iii) the closure of the numerical range of \( {T}_{\phi } \) ;\n\n(iv)... | Proof. The closure of the numerical range of \( {M}_{\phi } \) is the convex hull of its spectrum, since \( {M}_{\phi } \) is normal (Theorem 1.2.12). By the spectral inclusion theorem (Theorem 3.3.1), the closed convex hull of \( \sigma \left( {M}_{\phi }\right) \) is contained in the closed convex hull of \( \sigma \... | Yes |
Corollary 3.3.7. For every \( \phi \) in \( {\mathbf{L}}^{\infty } \), ess \( \operatorname{ran}\phi \) is contained in \( \sigma \left( {T}_{\phi }\right) \) and \( \sigma \left( {T}_{\phi }\right) \) is contained in the closed convex hull of ess \( \operatorname{ran}\phi \) . In particular, if ess ran \( \phi \) is c... | Proof. This is an immediate consequence of the previous theorem together with the spectral inclusion theorem (Theorem 3.3.1). | No |
Theorem 3.3.8. If \( \phi \in {\mathbf{H}}^{\infty } \), then \( \sigma \left( {T}_{\phi }\right) \) is the closure of \( \phi \left( \mathbb{D}\right) \) . | Proof. For the proof of this theorem, it is convenient to regard \( {T}_{\phi } \) as acting on \( {\mathbf{H}}^{2} \) rather than on \( {\widetilde{\mathbf{H}}}^{2} \) . To establish one inclusion, suppose that \( \lambda = \phi \left( {z}_{0}\right) \) for some \( {z}_{0} \in \mathbb{D} \) . Then\n\n\[ \left( {\left(... | Yes |
Corollary 3.3.9. If \( {T}_{\phi } \) is a coanalytic Toeplitz operator, and if \( \bar{\phi } \) is the function in \( {\mathbf{H}}^{\infty } \) whose boundary function is the complex conjugate of \( \phi \) a.e., then \( \sigma \left( {T}_{\phi }\right) \) is the closure of the set of complex conjugates of \( \bar{\p... | Proof. Recall that the adjoint of a Toeplitz operator is the Toeplitz operator whose symbol is the complex conjugate of the given one (Theorem 3.2.8). Since the complex conjugate of \( \phi \) is in \( {\widetilde{\mathbf{H}}}^{2} \), the result follows from the previous theorem and the fact that the spectrum of the ad... | Yes |
Theorem 3.3.10 (The Coburn Alternative). If \( \phi \) is a function in \( {\mathbf{L}}^{\infty } \) other than 0, then at least one of \( {T}_{\phi } \) and \( {T}_{\phi }^{ * } \) is injective. | Proof. Suppose that \( {T}_{\phi }f = 0 \) for some \( f \neq 0 \) . Suppose also that\n\n\[ {T}_{\phi }^{ * }g = P\left( {\bar{\phi }g}\right) = 0. \]\n\nIt must be shown that \( g = 0 \) .\n\nWe are given \( P\left( {\phi f}\right) = 0 \), where \( P \) is the projection of \( {\mathbf{L}}^{2} \) onto \( {\widetilde{... | Yes |
Corollary 3.3.11. A Toeplitz operator, other than 0, has dense range if it is not injective. | Proof. If \( {T}_{\phi } \) is not injective, it follows from the Coburn alternative (Theorem 3.3.10) that \( {T}_{\phi }^{ * } \) is injective. If the range of \( {T}_{\phi } \) were not dense, there would exist a \( g \) different from 0 such that \( \left( {{T}_{\phi }f, g}\right) = 0 \) for all \( f \in {\widetilde... | Yes |
Corollary 3.3.12. For \( \phi \) a nonconstant function in \( {\mathbf{L}}^{\infty } \) , \[ {\Pi }_{0}\left( {T}_{\phi }\right) \bigcap \overline{{\Pi }_{0}\left( {T}_{\phi }^{ * }\right) } = \varnothing \] where \( \overline{{\Pi }_{0}\left( {T}_{\phi }^{ * }\right) } \) denotes the set of complex conjugates of the e... | Proof. Suppose that \( \lambda \in {\Pi }_{0}\left( {T}_{\phi }\right) \) . Then there is a function \( f \) other than zero such that \( \left( {{T}_{\phi } - \lambda }\right) f = \left( {T}_{\phi - \lambda }\right) f = 0 \) . Suppose that \( \left( {{T}_{\phi }^{ * } - \bar{\lambda }}\right) g = 0 \) ; it must be sho... | Yes |
Corollary 3.3.13. If \( \phi \) is a real-valued nonconstant function in \( {\mathbf{L}}^{\infty } \), then \( {\Pi }_{0}\left( {T}_{\phi }\right) = \varnothing \) . | Proof. In this case, \( {T}_{\phi } \) is self-adjoint, so its spectrum is real. Thus if there was a \( \lambda \) in \( {\Pi }_{0}\left( {T}_{\phi }\right) ,\lambda = \bar{\lambda } \) would be in \( {\Pi }_{0}\left( {T}_{\phi }^{ * }\right) \), contradicting the previous corollary. | Yes |
Lemma 3.3.17. Let \( \\beta \) and \( \\gamma \) be continuous functions on \( {S}^{1} \) neither of which assumes the value 0 . Then\n\n\[ \n{\\operatorname{Ind}}_{0}\\left( {\\beta \\gamma }\\right) = {\\operatorname{Ind}}_{0}\\beta + {\\operatorname{Ind}}_{0}\\gamma \n\] | Proof. See [9, p. 81]. | No |
Theorem 4.1.4. The Hankel operators \( {H}_{\phi } \) and \( {H}_{\psi } \) are equal if and only if \( \phi - \psi \) is in \( {e}^{i\theta }{\widetilde{\mathbf{H}}}^{2} \) . | Proof. Since the matrix of a Hankel operator depends only on the Fourier coefficients in nonpositive positions (Definition 4.1.3), two \( {\mathbf{L}}^{\infty } \) functions induce the same Hankel operator if and only if their Fourier coefficients agree for nonpositive indices. This is equivalent to the difference betw... | Yes |
Theorem 4.1.7. The operator \( A \) has a Hankel matrix with respect to the standard basis of \( {\widetilde{\mathbf{H}}}^{2} \) if and only if it satisfies the equation \( {U}^{ * }A = {AU} \), where \( U \) is the unilateral shift. | Proof. First note that\n\n\[ \left( {{U}^{ * }A{e}_{n},{e}_{m}}\right) = \left( {A{e}_{n}, U{e}_{m}}\right) = \left( {A{e}_{n},{e}_{m + 1}}\right) .\n\]\n\nAlso,\n\n\[ \left( {{AU}{e}_{n},{e}_{m}}\right) = \left( {A{e}_{n + 1},{e}_{m}}\right) .\n\]\n\nTherefore,\n\n\[ \left( {{U}^{ * }A{e}_{n},{e}_{m}}\right) = \left( ... | Yes |
Corollary 4.1.8. If A has a Hankel matrix with respect to the standard basis of \( {\widetilde{\mathbf{H}}}^{2} \), and \( U \) is the unilateral shift, then \( {U}^{ * }{AU} \) has a Hankel matrix. | Proof. This is easily seen by noticing the effect on the matrix of \( A \) of multiplying on the left by \( {U}^{ * } \) and on the right by \( U \) . Alternatively,\n\n\[ \n{U}^{ * }\left( {{U}^{ * }{AU}}\right) = {U}^{ * }\left( {{U}^{ * }A}\right) U \n\]\n\n\[ \n= {U}^{ * }\left( {AU}\right) U\;\text{ (by Theorem 4.... | Yes |
Theorem 4.1.9 (Douglas’s Theorem). Let \( \mathcal{H},\mathcal{K} \), and \( \mathcal{L} \) be Hilbert spaces and suppose that \( E : \mathcal{H} \rightarrow \mathcal{K} \) and \( F : \mathcal{H} \rightarrow \mathcal{L} \) are bounded operators. If \( {E}^{ * }E \leq {F}^{ * }F \), then there exists an operator \( R : ... | Proof. First of all, observe that the hypothesis \( {E}^{ * }E \leq {F}^{ * }F \) is equivalent to \( \parallel {Ex}\parallel \leq \parallel {Fx}\parallel \) for all \( x \in \mathcal{H} \), since \( \left( {{E}^{ * }{Ex}, x}\right) = \parallel {Ex}{\parallel }^{2} \) and \( \left( {{F}^{ * }{Fx}, x}\right) = \) \( \pa... | Yes |
Lemma 4.1.10. Let \( \mathcal{H} \) and \( \mathcal{K} \) be Hilbert spaces and let \( B : \mathcal{H} \rightarrow \mathcal{K} \) be a bounded operator with \( \parallel B\parallel \leq 1 \) . Then \( {\left( I - {B}^{ * }B\right) }^{1/2}{B}^{ * } = {B}^{ * }{\left( I - B{B}^{ * }\right) }^{1/2} \) . | Proof. First of all, since \( \parallel B\parallel = \begin{Vmatrix}{B}^{ * }\end{Vmatrix} \leq 1 \), it follows that \( I - {B}^{ * }B \geq 0 \) and \( I - B{B}^{ * } \geq 0 \) . Thus \( {\left( I - {B}^{ * }B\right) }^{1/2} \) and \( {\left( I - B{B}^{ * }\right) }^{1/2} \) exist, since every positive operator has a ... | Yes |
Theorem 4.1.11 (The Julia-Halmos Theorem). Let \( \mathcal{H} \) and \( \mathcal{K} \) be Hilbert spaces and let \( A : \mathcal{H} \rightarrow \mathcal{K} \) be a bounded operator with \( \parallel A\parallel \leq 1 \) . If \( U \) is the operator mapping \( \mathcal{K} \oplus \mathcal{H} \) into \( \mathcal{H} \oplus... | Proof. Applying Lemma 4.1.10 (with \( B = {A}^{ * } \) and with \( B = A \) ) gives\n\n\[ {\left( I - A{A}^{ * }\right) }^{1/2}A = A{\left( I - {A}^{ * }A\right) }^{1/2}\;\text{ and }\;{\left( I - {A}^{ * }A\right) }^{1/2}{A}^{ * } = {A}^{ * }{\left( I - A{A}^{ * }\right) }^{1/2}. \]\n\nMultiplying matrices shows that ... | Yes |
Theorem 4.2.2. For a fixed element \( w \) of \( \mathbb{D} \), let \( {k}_{w} \) be defined by\n\n\[ \n{k}_{w}\left( z\right) = \frac{1}{1 - \bar{w}z} \n\]\n\n(see Definition 1.1.7). Then the rank-one operator \( {k}_{\bar{w}} \otimes {k}_{w} \) (see Notation 1.2.27) is the Hankel operator \( {H}_{{\breve{k}}_{\bar{w}... | Proof. To see that \( {k}_{\bar{w}} \otimes {k}_{w} \) has the stated matrix representation, fix any nonnegative integer \( n \) and compute\n\n\[ \n\left( {{k}_{\bar{w}} \otimes {k}_{w}}\right) {e}_{n} = \left( {{e}_{n},{k}_{w}}\right) {k}_{\bar{w}} = {w}^{n}\left( {{e}_{0} + w{e}_{1} + {w}^{2}{e}_{2} + {w}^{3}{e}_{3}... | Yes |
Theorem 4.2.3 (Kronecker’s Theorem). Let\n\n\\[ \nH = \\left( \\begin{matrix} {a}_{0} & {a}_{1} & {a}_{2} & {a}_{3} & & \\cdots \\\\ {a}_{1} & {a}_{2} & {a}_{3} & & \\cdots & \\\\ {a}_{2} & {a}_{3} & & \\cdots & & \\\\ {a}_{3} & & \\cdots & & & \\\\ & \\cdots & & & & \\\\ \\vdots & & & & & \\\\ & & & & & \\\\ & & & & &... | Proof. Suppose that the columns of \\( H \\) are linearly dependent. Then there is a natural number \\( s \\) and complex numbers \\( {c}_{0},{c}_{1},{c}_{2},\\ldots ,{c}_{s} \\), not all zero, such that\n\n\\[ \n{c}_{0}\\left( \\begin{matrix} {a}_{0} \\\\ {a}_{1} \\\\ {a}_{2} \\\\ {a}_{3} \\\\ \\vdots \\end{matrix}\\r... | Yes |
Corollary 4.2.4 (Kronecker’s Theorem). For \( \phi \in {\mathbf{L}}^{\infty } \), the Hankel operator \( {H}_{\phi } \) has finite rank if and only if the function\n\n\[ f\left( z\right) = \mathop{\sum }\limits_{{k = 0}}^{\infty }\frac{{\phi }_{-k}}{{z}^{k}} \]\n\nis a rational function all of whose poles are in \( \ma... | Proof. Suppose \( {H}_{\phi } \) has finite rank. Then \( f \) is a rational function, by the version of Kronecker's theorem we have already established (Theorem 4.2.3). It must be shown that \( f \) has no poles outside \( \mathbb{D} \) .\n\nIf \( f \) had a pole outside \( \mathbb{D} \), say \( \lambda \), consider t... | Yes |
Corollary 4.2.5 (Kronecker's Theorem). Let \( \mathcal{R} \) be the set of rational functions with poles inside \( \mathbb{D} \) . Then \( H \) is a bounded Hankel operator of finite rank if and only if \( H = {H}_{\psi } \) for some \( \psi \in {e}^{i\theta }{\widetilde{\mathbf{H}}}^{\infty } + \mathcal{R} \) . | Proof. The previous form of Kronecker's theorem (Corollary 4.2.4) immediately implies this form since Hankel operators are the same if and only if their symbols have identical coanalytic parts (Theorem 4.1.4). | No |
Corollary 4.2.7. If the matrix of a Hankel operator with respect to the standard basis for \( {\widetilde{\mathbf{H}}}^{2} \) has only a finite number of entries different from 0, then there exists a function \( \psi \), continuous on \( {S}^{1} \), that is a symbol of the Hankel operator and satisfies \( \begin{Vmatri... | Proof. Since any finite Blaschke product is continuous on \( {S}^{1} \), the result follows from the previous theorem. | No |
Corollary 4.3.3 (Hartman’s Theorem). The Hankel operator \( {H}_{\phi } \) is compact if and only if \( \phi \) is in \( {\widetilde{\mathbf{H}}}^{\infty } + \mathbf{C} \) . | Proof. If \( {H}_{\phi } \) is compact, the previous version of Hartman’s theorem (Theorem 4.3.1) implies that there is a continuous function \( \psi \) such that \( {H}_{\phi } = {H}_{\psi } \) . Then \( \phi \) and \( \psi \) have the same coanalytic parts, so \( \phi - \psi \) is a function in \( {\widetilde{\mathbf... | Yes |
Theorem 4.4.2. If \( H \) is a Hankel operator, then \( {H}^{ * }{f}^{ * } = {\left( Hf\right) }^{ * } \) for every \( f \in {\widetilde{\mathbf{H}}}^{2} \) . In particular, \( \begin{Vmatrix}{{H}^{ * }{f}^{ * }}\end{Vmatrix} = \parallel {Hf}\parallel \) for every \( f \) in \( {\widetilde{\mathbf{H}}}^{2} \) . | Proof. Recall that the matrix of \( {A}^{ * } \) with respect to a given orthonormal basis can be obtained from that of \( A \) by taking the \ | No |
For every \( \phi \) in \( {\mathbf{L}}^{\infty },{H}_{\phi }^{ * } = {H}_{{\phi }^{ * }} \) . Moreover, if \( {H}_{\phi } \) is selfadjoint, then there exists a \( \psi \) in \( {\mathbf{L}}^{\infty } \) such that \( \psi = {\psi }^{ * } \) a.e. and \( {H}_{\phi } = {H}_{\psi } \) . | The fact that \( {H}_{\phi }^{ * } = {H}_{{\phi }^{ * }} \) follows from the matrix representation of \( {H}_{\phi } \) as indicated above. Alternatively, it is also a consequence of the following computation. For any \( f \) and \( g \) in \( {\mathbf{H}}^{2} \) ,\n\n\[ \left( {{H}_{\phi }^{ * }f, g}\right) = \left( {... | Yes |
Lemma 4.4.4. If \( {H}_{\phi } \) and \( {H}_{\psi } \) are Hankel operators and \( U \) is the unilateral shift, then\n\n\[ \n{H}_{\phi }{H}_{\psi } - {U}^{ * }{H}_{\phi }{H}_{\psi }U = \left( {P\breve{\phi }}\right) \otimes \left( {P\bar{\psi }}\right) ,\n\]\n\nwhere \( P \) is the projection of \( {\mathbf{L}}^{2} \... | Proof. Note that\n\n\[ \n{H}_{\phi }{H}_{\psi } - {U}^{ * }{H}_{\phi }{H}_{\psi }U = {H}_{\phi }{H}_{\psi } - {H}_{\phi }U{U}^{ * }{H}_{\psi }\;\text{ (by Theorem 4.1.7) }\n\]\n\n\[ \n= {H}_{\phi }\left( {I - U{U}^{ * }}\right) {H}_{\psi }\n\]\n\nRecall that \( I - U{U}^{ * } \) is the projection of \( {\widetilde{\mat... | Yes |
Theorem 4.4.5. The product of two nonzero Hankel operators is a Hankel operator if and only if both of the operators are constant multiples of the same Hankel operator of rank 1. | Proof. We have seen that every Hankel operator of rank 1 has the form \( {k}_{\bar{w}} \otimes {k}_{w} \) for some \( w \) in \( \mathbb{D} \), where \( {k}_{w} \) is the kernel function given by \( {k}_{w}\left( z\right) = \frac{1}{1 - \bar{w}z} \) (Theorem 4.2.2). Thus, to prove the first implication of the theorem, ... | Yes |
Corollary 4.4.6. The product of two Hankel operators is a Toeplitz operator only if at least one of the Hankel operators is 0 . | Proof. By Lemma 4.4.4,\n\n\[ \n{H}_{\phi }{H}_{\psi } - {U}^{ * }{H}_{\phi }{H}_{\psi }U = \left( {P\breve{\phi }}\right) \otimes \left( {P\bar{\psi }}\right) .\n\]\n\nIf \( {H}_{\phi }{H}_{\psi } \) is a Toeplitz operator, then \( {U}^{ * }{H}_{\phi }{H}_{\psi }U = {H}_{\phi }{H}_{\psi } \) (by Corollary 3.2.7), so\n\... | Yes |
Corollary 4.4.7. The product of two Hankel operators is 0 if and only if one of them is 0. | Proof. If the product of two Hankel operators is the Toeplitz operator 0, the previous corollary implies that at least one of the Hankel operators is zero. | No |
Theorem 4.4.8. Let \( \phi \) and \( \psi \) be in \( {\mathbf{L}}^{\infty } \) and suppose that \( {H}_{\psi } \neq 0 \) . If \( {H}_{\phi } \) and \( {H}_{\psi } \) commute, then there exists a complex number \( c \) such that \( {H}_{\phi } = c{H}_{\psi } \) . | Proof. When \( {H}_{\phi } \neq 0 \), by Lemma 4.4.4,\n\n\[ \n{H}_{\phi }{H}_{\psi } - {U}^{ * }{H}_{\phi }{H}_{\psi }U = \left( {P\breve{\phi }}\right) \otimes \left( {P\bar{\psi }}\right) , \n\]\n\nand\n\n\[ \n{H}_{\psi }{H}_{\phi } - {U}^{ * }{H}_{\psi }{H}_{\phi }U = \left( {P\breve{\psi }}\right) \otimes \left( {P... | Yes |
Corollary 4.4.9. Every normal Hankel operator is a multiple of a self-adjoint Hankel operator. | Proof. Let \( H \) be a normal Hankel operator; i.e., \( H{H}^{ * } = {H}^{ * }H \) . If \( H = 0 \), the result is trivial. In the other case, by the previous theorem, there is a constant \( c \) such that \( H = c{H}^{ * } \) . Since \( \parallel H\parallel = \begin{Vmatrix}{H}^{ * }\end{Vmatrix} \), we have \( \left... | Yes |
Theorem 4.4.11. Every hyponormal Hankel operator is normal. | Proof. By Theorem 4.4.2, it follows that \( \begin{Vmatrix}{{H}^{ * }{f}^{ * }}\end{Vmatrix} = \parallel {Hf}\parallel \) for every \( f \in {\widetilde{\mathbf{H}}}^{2} \) , where \( {f}^{ * } \) is the vector whose coefficients are the conjugates of those of \( f \) (Notation 4.4.1). Applying this to \( {f}^{ * } \) ... | Yes |
Theorem 4.5.1. Let \( \phi \) and \( \psi \) be in \( {\mathbf{L}}^{\infty } \) . Then\n\n\[ \n{H}_{{e}^{i\theta }\breve{\phi }}{H}_{{e}^{i\theta }\psi } = {T}_{\phi \psi } - {T}_{\phi }{T}_{\psi }\n\] | Proof. The flip operator, \( J \), and the projection onto \( {\widetilde{\mathbf{H}}}^{2}, P \), satisfy the following equation:\n\n\[ \n{JPJ} = {M}_{{e}^{i\theta }}\left( {I - P}\right) {M}_{{e}^{-{i\theta }}}.\n\]\n\n(This can easily be verified by applying each side to the basis vectors \( \left\{ {e}^{in\theta }\r... | Yes |
Corollary 4.5.2. If \( \phi \) and \( \psi \) are in \( {\mathbf{L}}^{\infty } \), then\n\n\[ \n{H}_{\phi }{H}_{\psi } = {T}_{\breve{\phi }\psi } - {T}_{{e}^{i\theta }\breve{\phi }}{T}_{{e}^{-{i\theta }}\psi }.\n\] | Proof. By the previous theorem,\n\n\[ \n{H}_{{e}^{i\theta }\breve{\alpha }}{H}_{{e}^{i\theta }\beta } = {T}_{\alpha \beta } - {T}_{\alpha }{T}_{\beta }\n\]\n\nfor \( \alpha \) and \( \beta \) in \( {\mathbf{L}}^{\infty } \) . Let \( \alpha = {e}^{i\theta }\breve{\phi } \) and \( \beta = {e}^{-{i\theta }}\psi \) . Makin... | Yes |
Corollary 4.5.3. If the product of two Hankel operators is Toeplitz, then at least one of the Hankel operators is 0. | Proof. If \( {H}_{\phi }{H}_{\psi } \) is a Toeplitz operator, then since the sum of two Toeplitz operators is Toeplitz, it follows from the previous corollary that \( {T}_{{e}^{i\theta }\breve{\phi }}{T}_{{e}^{-{i\theta }}\psi } \) is a Toeplitz operator. Thus either \( {e}^{-{i\theta }}\psi \) is analytic or \( {e}^{... | Yes |
Theorem 4.5.4. Let \( \phi \) and \( \psi \) be in \( {\mathbf{L}}^{\infty } \) . Then\n\n\[ \n{T}_{\breve{\phi }}{H}_{{e}^{i\theta }\psi } + {H}_{{e}^{i\theta }\phi }{T}_{\psi } = {H}_{{e}^{i\theta }{\phi \psi }}.\n\] | Proof. This follows from a computation similar to that in the proof of Theorem 4.5.1. Using \( J{M}_{\phi }J = {M}_{\breve{\phi }} \) and \( {JPJ} = {M}_{{e}^{i\theta }}\left( {I - P}\right) {M}_{{e}^{-{i\theta }}} \), we get\n\n\[ \n{T}_{\breve{\phi }}{H}_{{e}^{i\theta }\psi } = \left( {P{M}_{\breve{\phi }}}\right) \l... | Yes |
Corollary 4.5.5. (i) If \( \psi \) is in \( {\widetilde{\mathbf{H}}}^{\infty } \), then \( {H}_{\phi }{T}_{\psi } = {H}_{\phi \psi } \). (ii) If \( \psi \) is in \( {\widetilde{\mathbf{H}}}^{\infty } \), then \( {T}_{\breve{\psi }}{H}_{\phi } = {H}_{\psi \phi } \). | Proof. Recall from the previous theorem that, for \( \alpha \) and \( \beta \) in \( {\mathbf{L}}^{\infty } \), \[ {T}_{\breve{\alpha }}{H}_{{e}^{i\theta }\beta } + {H}_{{e}^{i\theta }\alpha }{T}_{\beta } = {H}_{{e}^{i\theta }{\alpha \beta }}. \] Taking \( \alpha = {e}^{-{i\theta }}\phi \) and \( \beta = \psi \) gives ... | Yes |
Theorem 5.1.2. If \( {C}_{\phi } \) and \( {C}_{\psi } \) are composition operators then \( {C}_{\phi }{C}_{\psi } = {C}_{\psi \circ \phi } \) . | Proof. Note that\n\n\[ \left( {{C}_{\phi }{C}_{\psi }f}\right) \left( z\right) = \left( {{C}_{\phi }\left( {f \circ \psi }\right) }\right) \left( z\right) = \left( {f \circ \psi \circ \phi }\right) \left( z\right) = \left( {{C}_{\psi \circ \phi }f}\right) \left( z\right) ,\]\n\nand thus\n\n\[ {C}_{\phi }{C}_{\psi } = {... | Yes |
Let \( \phi \left( z\right) = {z}^{2} \). If \( f \) is in \( {\mathbf{H}}^{2} \), then \( {C}_{\phi }f \) is in \( {\mathbf{H}}^{2} \) and, in fact, \( \begin{Vmatrix}{{C}_{\phi }f}\end{Vmatrix} = \parallel f\parallel \). Therefore \( {C}_{\phi } \) is an isometry mapping \( {\mathbf{H}}^{2} \) into itself. | Proof. If \( f \) has power series \( f\left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{n} \), then \( {C}_{\phi }f \) has power series \( \left( {{C}_{\phi }f}\right) \left( z\right) = \mathop{\sum }\limits_{{n = 0}}^{\infty }{a}_{n}{z}^{2n} \). Thus \( {C}_{\phi }f \in {\mathbf{H}}^{2} \) and al... | Yes |
Lemma 5.1.4. If \( f \in {\mathbf{H}}^{2} \), then, for \( r{e}^{it} \in \mathbb{D} \), we have\n\n\[ \n{\left| f\left( r{e}^{it}\right) \right| }^{2} \leq \frac{1}{2\pi }{\int }_{0}^{2\pi }{P}_{r}\left( {\theta - t}\right) {\left| f\left( {e}^{i\theta }\right) \right| }^{2}{d\theta }. \n\] | Proof. Recall that, by the Poisson integral formula (Theorem 1.1.21), we have\n\n\[ \nf\left( {r{e}^{it}}\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }{P}_{r}\left( {\theta - t}\right) f\left( {e}^{i\theta }\right) {d\theta }. \n\]\n\nIf we define the measure \( {d\mu } \) by \( {d\mu }\left( \theta \right) = \frac{1}{2\... | Yes |
Corollary 5.1.6. If \( {C}_{\phi } \) is a composition operator such that \( \phi \left( 0\right) = 0 \) then \( \begin{Vmatrix}{C}_{\phi }\end{Vmatrix} = 1 \) . | Proof. By the previous theorem and \( \phi \left( 0\right) = 0 \), we have \( \begin{Vmatrix}{C}_{\phi }\end{Vmatrix} \leq 1 \) . By the above observation, \( \begin{Vmatrix}{C}_{\phi }\end{Vmatrix} \geq 1 \) . | Yes |
Corollary 5.1.8 (Littlewood’s Subordination Theorem). If \( f \) in \( {\mathbf{H}}^{2} \) is subordinate to \( g \) in \( {\mathbf{H}}^{2} \), then \( \parallel f\parallel \leq \parallel g\parallel \) . | Proof. Apply the previous corollary to \( f = {C}_{\phi }g \) . | No |
Lemma 5.1.9. If \( {C}_{\phi } \) is a composition operator and \( {k}_{\lambda } \) is a reproducing kernel function, then \( {C}_{\phi }^{ * }{k}_{\lambda } = {k}_{\phi \left( \lambda \right) } \) . | Proof. For each \( f \) in \( {\mathbf{H}}^{2} \) , \n\n\[ \n\left( {f,{C}_{\phi }^{ * }{k}_{\lambda }}\right) = \left( {{C}_{\phi }f,{k}_{\lambda }}\right) = \left( {f \circ \phi ,{k}_{\lambda }}\right) = f\left( {\phi \left( \lambda \right) }\right) . \n\] \n\nBut also \n\n\[ \n\left( {f,{k}_{\phi \left( \lambda \rig... | Yes |
Theorem 5.1.10. For every composition operator \( {C}_{\phi } \) , \[ \frac{1}{\sqrt{1 - {\left| \phi \left( 0\right) \right| }^{2}}} \leq \begin{Vmatrix}{C}_{\phi }\end{Vmatrix} \leq \frac{2}{\sqrt{1 - {\left| \phi \left( 0\right) \right| }^{2}}}. \] | Proof. Using the previous lemma with \( \lambda = 0 \) yields \[ {C}_{\phi }^{ * }{k}_{0} = {k}_{\phi \left( 0\right) } \] Recall (Theorem 1.1.8) that \[ {\begin{Vmatrix}{k}_{\lambda }\end{Vmatrix}}^{2} = \frac{1}{1 - {\left| \lambda \right| }^{2}} \] and therefore \( \begin{Vmatrix}{k}_{0}\end{Vmatrix} = 1 \) and \( \... | Yes |
Corollary 5.1.11. The norm of the composition operator \( {C}_{\phi } \) is 1 if and only if \( \phi \left( 0\right) = 0 \) . | Proof. As indicated, we have already established (Corollary 5.1.6) that \( \begin{Vmatrix}{C}_{\phi }\end{Vmatrix} = \) 1 if \( \phi \left( 0\right) = 0 \) . Conversely, if \( \begin{Vmatrix}{C}_{\phi }\end{Vmatrix} = 1 \), then the inequality\n\n\[ \n\frac{1}{\sqrt{1 - {\left| \phi \left( 0\right) \right| }^{2}}} \leq... | Yes |
Theorem 5.1.12. An operator \( A \) on \( {\mathbf{H}}^{2} \) is a composition operator if and only if \( {A}^{ * } \) maps the set of reproducing kernels into itself. | Proof. We showed above that \( {A}^{ * }{k}_{\lambda } = {k}_{\phi \left( \lambda \right) } \) when \( A = {C}_{\phi } \) . Conversely, suppose that for each \( \lambda \in \mathbb{D},{A}^{ * }{k}_{\lambda } = {k}_{{\lambda }^{\prime }} \) for some \( {\lambda }^{\prime } \in \mathbb{D} \) . Define \( \phi : \mathbb{D}... | Yes |
Corollary 5.1.14. The operator \( A \) on \( {\mathbf{H}}^{2} \) is a composition operator if and only if it is multiplicative in the sense that \( \left( {Af}\right) \left( {Ag}\right) = A\left( {fg}\right) \) whenever \( f, g \) , and \( {fg} \) are all in \( {\mathbf{H}}^{2} \) . | Proof. It is clear that composition operators have the stated multiplicative property.\n\nConversely, if \( A \) has the multiplicative property then, in particular, \( A{e}_{n} = \) \( {\left( A{e}_{1}\right) }^{n} \) for all \( n \), so the fact that \( A \) is a composition operator follows from Theorem 5.1.13. | Yes |
Theorem 5.1.15. The composition operator \( {C}_{\phi } \) is normal if and only if there exists \( \lambda \in \mathbb{C} \) such that \( \phi \left( z\right) = {\lambda z} \) and \( \left| \lambda \right| \leq 1 \) . | Proof. First note that \( \phi \left( z\right) = {\lambda z} \) implies that \( {C}_{\phi }{e}_{n} = {\lambda }^{n}{e}_{n} \) for all positive integers \( n \) . Hence \( {C}_{\phi } \) is a diagonal operator with respect to the canonical basis of \( {\mathbf{H}}^{2} \), and it is obvious that every diagonal operator i... | Yes |
Theorem 5.1.16. If there exists a positive number \( s < 1 \) so that \( \left| {\phi \left( z\right) }\right| < s \) for every \( z \in \mathbb{D} \), then \( {C}_{\phi } \) is compact. | Proof. We show that \( {C}_{\phi } \) is compact by exhibiting a sequence of operators of finite rank that converge in norm to \( {C}_{\phi } \) . Observe that, since \( \left| {\phi \left( z\right) }\right| < s \) for all \( z \in \mathbb{D} \), we have, for each natural number \( k \), \[ \begin{Vmatrix}{\phi }^{k}\e... | Yes |
Theorem 5.1.17. If \( {C}_{\phi } \) is compact, then \( \left| {\widetilde{\phi }\left( {e}^{i\theta }\right) }\right| < 1 \) a.e. | Proof. The sequence \( \left\{ {e}_{n}\right\} \) converges weakly to 0 . So, \( \left\{ {{C}_{\phi }{e}_{n}}\right\} \) converges to 0 in norm if \( {C}_{\phi } \) is compact (e.g.,[27, p. 95] or [12, p. 173]).\n\nIf \( \left| {\widetilde{\phi }\left( {e}^{i\theta }\right) }\right| \) was not less than 1 a.e., there w... | Yes |
Theorem 5.2.1. The composition operator \( {C}_{\phi } \) is invertible if and only if \( \phi \) is a conformal mapping of \( \mathbb{D} \) onto itself. In this case, \( {C}_{\phi }^{-1} = {C}_{{\phi }^{-1}} \) . | Proof. If \( \phi \) is a conformal map, let \( {\phi }^{-1} \) be the inverse conformal map. Then \( {C}_{{\phi }^{-1}}{C}_{\phi } = {C}_{\phi }{C}_{{\phi }^{-1}} = I \) by Theorem 5.1.2. Hence \( {C}_{\phi }^{-1} = {C}_{{\phi }^{-1}} \) . To establish the converse, suppose that \( {C}_{\phi } \) is an invertible comp... | Yes |
For a fixed \( \lambda \in \mathbb{D} \), define the function \( {\phi }_{\lambda } \) by\n\n\[ \n{\phi }_{\lambda }\left( z\right) = \frac{\lambda - z}{1 - \bar{\lambda }z}\n\]\n\nThen \( {\left( {C}_{{\phi }_{\lambda }}\right) }^{2} = I \) (i.e., \( {C}_{{\phi }_{\lambda }}^{-1} = {C}_{{\phi }_{\lambda }} \) ). | Proof. As indicated above, each such \( {\phi }_{\lambda } \) is a conformal mapping of \( \mathbb{D} \) into itself. An easy computation shows that \( {\phi }_{\lambda }\left( {{\phi }_{\lambda }\left( z\right) }\right) = z \) for all \( z \in \mathbb{D} \), from which the result follows. | No |
Theorem 5.2.3. If the function \( \phi \) has a fixed point in \( \mathbb{D} \), then the operator \( {C}_{\phi } \) is similar to a composition operator \( {C}_{\psi } \) with the property that \( \psi \left( 0\right) = 0 \) . | Proof. Let \( \phi \left( \lambda \right) = \lambda \) for some \( \lambda \in \mathbb{D} \) . Let\n\n\[{\phi }_{\lambda }\left( z\right) = \frac{\lambda - z}{1 - \bar{\lambda }z}\]\n\nThen \( {C}_{{\phi }_{\lambda }}^{-1} = {C}_{{\phi }_{\lambda }} \) by Example 5.2.2, so\n\n\[{C}_{{\phi }_{\lambda }}^{-1}{C}_{\phi }{... | Yes |
Theorem 5.2.5. If \( \phi \) has a fixed point in \( \mathbb{D} \), then the spectral radius of \( {C}_{\phi } \) is 1 . | Proof. By Theorem 5.2.3, \( {C}_{\phi } \) is similar to a composition operator whose defining function fixes the point 0 . Since similar operators have the same spectra, and therefore equal spectral radii, we may and do assume that \( \phi \left( 0\right) = 0 \) . By the spectral radius formula (Theorem 1.2.4), \[ r\l... | Yes |
Corollary 5.3.2. If \( {C}_{\phi } \) is a composition operator and \( \phi \left( a\right) = a \) for some \( a \in \mathbb{D} \), then\n\n\[ \sigma \left( {C}_{\phi }\right) \supset \{ 1\} \cup \mathop{\bigcup }\limits_{{k = 1}}^{\infty }\left\{ {\left( {\phi }^{\prime }\left( a\right) \right) }^{k}\right\} . \] | Proof. This follows immediately from the previous theorem and Theorem 1.2.4. | No |
Theorem 5.3.3. If \( \phi \) is a nonconstant analytic function mapping the disk into itself and satisfying \( \phi \left( a\right) = a \) for some \( a \in \mathbb{D} \), and if there exists a function \( f \) analytic on \( \mathbb{D} \) that is not identically zero and satisfies the Schröder equation\n\n\[ f\left( {... | Proof. The equations \( \phi \left( a\right) = a \) and \( f\left( {\phi \left( z\right) }\right) = {\lambda f}\left( z\right) \) yield \( f\left( a\right) = {\lambda f}\left( a\right) \) . If \( f\left( a\right) \neq 0 \), then clearly \( \lambda = 1 \) and the theorem is established in that case.\n\nSuppose \( f\left... | Yes |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.