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Lemma 4.46. Let \( Z = \left\{ {{z}_{1},{z}_{2},\cdots }\right\} \) and \( {z}_{0} \notin Z \) . We have\n\n\[ \n{N}_{p}\left( {Z \cup \left\{ {z}_{0}\right\} }\right) \leq \frac{1 + 2{N}_{p}\left( Z\right) }{{\rho }_{p}\left( {{z}_{0}, Z}\right) }\n\]\n\nfor all \( 0 < p \leq \infty \) . | Proof. We may assume that \( {N}_{p}\left( Z\right) < \infty \), that is, \( Z \) is an interpolating sequence for \( {F}_{\alpha }^{p} \) . Given a sequence of values \( \left\{ {{v}_{0},{v}_{1},{v}_{2},\cdots }\right\} \) with the \( {l}^{p} \) norm of\n\n\[ \n\left\{ {{v}_{0}{\mathrm{e}}^{-\frac{\alpha }{2}{\left| {... | Yes |
Lemma 4.48. Given positive constants \( {l}_{0} \) and \( \alpha \), there is a constant \( C = C\left( {{l}_{0},\alpha }\right) > \) 0 such that if \( {N}_{p}\left( {Z,\alpha }\right) \leq {l}_{0} \), then\n\n\[ \n{\int }_{Q}\log {\rho }_{p}\left( {z, Z}\right) \mathrm{d}A\left( z\right) \geq - C{\left| Q\right| }^{2}... | Proof. By the proof of Lemma 4.8, there exists a point \( {z}_{0} \in Q \) and a positive constant \( \delta = \delta \left( {\alpha ,{l}_{0}}\right) \) such that \( d\left( {{z}_{0}, Z}\right) \geq \delta \) . By translation invariance, we may assume that \( {z}_{0} = 0 \) . It then follows from Lemma 4.47 that there ... | Yes |
Corollary 5.2. Suppose \( 0 < p \leq \infty \) and \( \left\{ {z}_{n}\right\} \) is the zero sequence of some \( f \in {F}_{\alpha }^{p} \) with \( f\left( 0\right) \neq 0 \) . Then\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }\frac{1}{{\left| {z}_{n}\right| }^{r}} < \infty \]\n\nfor every \( r > 2 \) . | The function\n\n\[ f\left( z\right) = \frac{\sin \left( {\delta {z}^{2}}\right) }{\delta {z}^{2}} \]\n\nused in the proof of Theorem 5.4 shows that the estimate in Theorem 5.1 is best possible. More specifically, we can find a positive constant \( C \) in this case such that\n\n\[ {C}^{-1}\sqrt{n} \leq \left| {z}_{n}\r... | No |
Theorem 5.4. Suppose \( \alpha > 0 \) and \( 0 < p \leq \infty \) . There exist two zero sequences for \( {F}_{\alpha }^{p} \) whose union is no longer a zero sequence for \( {F}_{\alpha }^{p} \) . | Proof. Fix \( \delta \in \left( {{\pi \alpha }/8,\alpha /2}\right) \) and consider the sequence\n\n\[ Z = \left\{ {{\mathrm{e}}^{{k\pi }\mathrm{i}/2}\sqrt{{n\pi }/\delta } : k = 0,1,2,3;n = 1,2,3,\cdots }\right\} . \]\n\nIt is easy to see that \( Z \) is the zero sequence of the entire function\n\n\[ f\left( z\right) =... | Yes |
Theorem 5.5. Let \( \alpha > 0 \) and \( 0 < p \leq \infty \) . There exists an \( {F}_{\alpha }^{p} \) zero sequence \( \left\{ {z}_{n}\right\} \) and a subsequence \( \left\{ {z}_{{n}_{k}}\right\} \) which is not an \( {F}_{\alpha }^{p} \) zero sequence. | Proof. Fix a positive constant \( \delta \) such that \( \delta < \alpha /2 \) and consider the following entire function:\n\n\[ f\left( z\right) = \frac{{\mathrm{e}}^{\mathrm{i}\delta {z}^{2}} - 1}{\mathrm{i}\delta {z}^{2}}. \]\n\nIt is easy to check that \( f \in {F}_{\alpha }^{p} \) . Thus, its zero set\n\n\[ \left\... | Yes |
Lemma 5.6. Let \( 0 < {\alpha }_{1} < \alpha < {\alpha }_{2} < \infty \) . We have:\n\n(a) \( {\sigma }_{\alpha } \in {F}_{{\alpha }_{2}}^{p} \) for all \( 0 < p \leq \infty \) .\n\n(b) \( {\sigma }_{\alpha } \notin {F}_{{\alpha }_{1}}^{p} \) for any \( 0 < p \leq \infty \) .\n\n(c) \( {\sigma }_{\alpha } \in {F}_{\alp... | Proof. It follows from the quasiperiodicity of \( {\sigma }_{\alpha } \) that if \( z = {\omega }_{mn} + w \) and \( w \in {\Omega }_{\alpha } \) , then\n\n\[ \left| {{\sigma }_{\alpha }\left( z\right) }\right| {\mathrm{e}}^{-\frac{\alpha }{2}{\left| z\right| }^{2}} = \left| {{\sigma }_{\alpha }\left( w\right) }\right|... | Yes |
Lemma 5.7. Suppose \( 0 < p < \infty \) and \( f \in {F}_{\alpha }^{p} \) . If \( f\left( z\right) = 0 \) for all \( z \in {\Lambda }_{\alpha } \), then \( f \) is identically zero. | Proof. By the Weierstrass factorization theorem, we can write \( f = h{\sigma }_{\alpha } \), where \( h \) is an entire function. In view of the quasiperiodicity of \( {\sigma }_{\alpha } \), we have\n\n\[{\int }_{\mathbb{C}}{\left| f\left( z\right) {\mathrm{e}}^{-\frac{\alpha }{2}{\left| z\right| }^{2}}\right| }^{p}\... | Yes |
Theorem 5.8. Suppose \( 0 < p \leq \infty ,0 < q \leq \infty \), and \( {\alpha }_{1} \neq {\alpha }_{2} \) . Then \( {F}_{{\alpha }_{1}}^{p} \) and \( {F}_{{\alpha }_{2}}^{q} \) have different zero sets. | Proof. Without loss of generality, let us assume that \( {\alpha }_{1} < \alpha < {\alpha }_{2} \) . By Lemma 5.6, the Weierstrass function \( {\sigma }_{\alpha } \) belongs to \( {F}_{{\alpha }_{2}}^{q} \), so its zero sequence \( {\Lambda }_{\alpha } \) is a zero set for \( {F}_{{\alpha }_{2}}^{q} \) . On the other h... | Yes |
Proposition 5.11. There exists an interpolating sequence for \( {F}_{\alpha }^{p} \) that is not a zero set for \( {F}_{\alpha }^{p} \) . | Proof. Fix some \( \delta > 2/\sqrt{\alpha } \) . For any positive integer \( k \), let \( {Z}_{k} \) denote the set of \( k + 1 \) points evenly spaced in the first quadrant on the circle \( \left| z\right| = {k\delta } \), including the end-points \( {k\delta } \) and \( {k\delta } \) i. Let\n\n\[ Z = \mathop{\bigcup... | Yes |
Proposition 6.1. For any complex numbers \( a \) and \( b \), and for any bounded functions \( \varphi \) and \( \psi \), we have:\n\n(i) \( {T}_{{a\varphi } + {b\psi }} = a{T}_{\varphi } + b{T}_{\psi } \) .\n\n(ii) \( {T}_{\bar{\varphi }} = {T}_{\varphi }^{ * } \) .\n\n(iii) \( {T}_{\varphi } \geq 0 \) if \( \varphi \... | Proof. These follow easily from the definitions. We omit the routine details. | No |
Theorem 6.2. Suppose \( \varphi \) is Lebesgue measurable on \( \mathbb{C} \) and \( S \) is a bounded linear operator on \( {F}_{\alpha }^{2} \). If\n\n(1) \( \varphi \) satisfies condition \( \left( {I}_{2}\right) \),\n\n(2) \( {T}_{\varphi } \) is bounded on \( {F}_{\alpha }^{2} \),\n\n(3) \( {T}_{\varphi }S \) is t... | Proof. By hypothesis (1), each function \( \bar{\varphi }K\left( {\cdot, z}\right) \) is in \( {L}_{\alpha }^{2} \), and by (6.4), we can write\n\n\[ {K}_{{T}_{\varphi }}\left( {\cdot, z}\right) = \bar{\varphi }K\left( {\cdot, z}\right) - H\left( {\cdot, z}\right) ,\]\n\nwhere \( H\left( {\cdot, z}\right) \bot {F}_{\al... | Yes |
Corollary 6.3 Suppose \( \varphi \) satisfies condition \( \left( {I}_{2}\right) \) . If \( {T}_{\varphi } \) is in the trace class and \( \varphi \in {L}^{1}\left( {\mathbb{C},\mathrm{d}A}\right) \), then\n\n\[ \operatorname{tr}\left( {T}_{\varphi }\right) = {\int }_{\mathbb{C}}\varphi \left( z\right) K\left( {z, z}\r... | Proof. When \( S \) is the identity operator, we have \( {K}_{S}\left( {z, w}\right) = K\left( {z, w}\right) \), so the condition\n\n\[ {\int }_{\mathbb{C}}{\int }_{\mathbb{C}}\left| {\varphi \left( z\right) }\right| \left| {K\left( {w, z}\right) }\right| \left| {{K}_{S}\left( {w, z}\right) }\right| \mathrm{d}{\lambda ... | Yes |
Corollary 6.4 Suppose \( \\varphi \) is bounded and compactly supported in \( \\mathbb{C} \) . Then for any bounded linear operator \( S \) on \( {F}_{\\alpha }^{2} \), the operator \( {T}_{\\varphi }S \) is trace class and\n\n\[ \n\\operatorname{tr}\\left( {{T}_{\\varphi }S}\\right) = {\\int }_{\\mathbb{C}}\\varphi \\... | Proof. It is easy to see that hypotheses (1)-(3) of Theorem 6.2 are satisfied. To check hypothesis (4) of Theorem 6.2, we write\n\n\[ \nI = {\\int }_{\\mathbb{C}}\\left| {\\varphi \\left( z\\right) }\\right| \\mathrm{d}{\\lambda }_{\\alpha }\\left( z\\right) {\\int }_{\\mathbb{C}}\\left| {{K}_{S}\\left( {w, z}\\right) ... | Yes |
Theorem 6.5. Let \( \mathcal{C} \) denote the set of all Toeplitz operators \( {T}_{\varphi } \), where \( \varphi \) is continuous and has compact support in \( \mathbb{C} \) . Then:\n\n(1) \( \mathcal{C} \) is trace-norm dense in the trace class \( \mathcal{T} \) of \( {F}_{\alpha }^{2} \) .\n\n(2) \( \mathcal{C} \) ... | Proof. Let \( \mathcal{L} \) denote the space of all bounded linear operators on \( {F}_{\alpha }^{2} \) . Then, it is well known that \( {\mathcal{T}}^{ * } = \mathcal{L} \) and \( {\mathcal{K}}^{ * } = \mathcal{T} \), with the duality pairing given by \( \langle S, T\rangle = \operatorname{tr}\left( {ST}\right) \) .\... | No |
Lemma 6.6. For nonnegative integers \( m \) and \( n \), let\n\n\[ \n{I}_{mn} = {\int }_{\mathbb{R}}{H}_{n}\left( x\right) {H}_{m}\left( x\right) {\mathrm{e}}^{-{x}^{2}}\mathrm{\;d}x.\n\]\n\nThen \( {I}_{mn} = 0 \) for \( m \neq n \) and \( {I}_{\mathrm{{nn}}} = {2}^{n}n!\sqrt{\pi } \). | Proof. For any polynomial \( f \), we use integration by parts \( n \) times to get\n\n\[ \n{\int }_{\mathbb{R}}{H}_{n}\left( x\right) f\left( x\right) {\mathrm{e}}^{-{x}^{2}}\mathrm{\;d}x = {\left( -1\right) }^{n}{\int }_{\mathbb{R}}f\left( x\right) \frac{{\mathrm{d}}^{n}}{\mathrm{\;d}{x}^{n}}{\mathrm{e}}^{-{x}^{2}}\m... | Yes |
Theorem 6.7. For any nonnegative integer \( n \), let\n\n\[ \n{h}_{n}\left( x\right) = {\left( \frac{2\alpha }{\pi }\right) }^{\frac{1}{4}}\frac{1}{\sqrt{{2}^{n}n!}}{\mathrm{e}}^{-\alpha {x}^{2}}{H}_{n}\left( {\sqrt{2\alpha }x}\right) .\n\]\n\nThen \( \left\{ {h}_{n}\right\} \) is an orthonormal basis of \( {L}^{2}\lef... | Proof. It follows from a change of variables and Lemma 6.6 that \( \left\{ {h}_{n}\right\} \) is an orthonormal set. In particular, for any positive integer \( N \), the functions\n\n\[ \n{h}_{0}\left( x\right) ,{h}_{1}\left( x\right) ,\cdots ,{h}_{N}\left( x\right)\n\]\n\nare linearly independent. It follows that the ... | Yes |
For any positive \( \alpha \), the Bargmann transform is an isometry from \( {L}^{2}\left( {\mathbb{R},\mathrm{d}x}\right) \) onto \( {F}_{\alpha }^{2} \) . | It suffices for us to show that for any nonnegative integer \( n \), we have \( {\mathcal{B}}_{\alpha }{h}_{n} = \) \( {e}_{n} \), where\n\n\[ \n{e}_{n}\left( z\right) = \sqrt{\frac{{\alpha }^{n}}{n!}}{z}^{n}.\n\]\n\nTo this end, first observe that if \( u = x - z \), where \( x \) is fixed, then \( \mathrm{d}/\mathrm{... | Yes |
Proposition 6.9. The inverse of the Bargmann transform is given by\n\n\[ \n\left\lbrack {{\mathcal{B}}_{\alpha }^{-1}f}\right\rbrack \left( x\right) = {\left( \frac{2\alpha }{\pi }\right) }^{\frac{1}{4}}{\int }_{\mathbb{C}}f\left( z\right) {\mathrm{e}}^{{2\alpha x}\bar{z} - \alpha {x}^{2} - \frac{\alpha }{2}{\bar{z}}^{... | Proof. Fix any polynomial \( f \in {F}_{\alpha }^{2} \) and any function \( g \in {L}^{2}\left( {\mathbb{R},\mathrm{d}x}\right) \) that is compactly supported. Since\n\n\[ \n{\mathcal{B}}_{\alpha } : {L}^{2}\left( {\mathbb{R},\mathrm{d}x}\right) \rightarrow {F}_{\alpha }^{2} \subset {L}_{\alpha }^{2}\n\]\n\nis an isome... | Yes |
Proposition 6.10. Let \( a = r + \mathrm{i}s \in \mathbb{C} \) and \( {k}_{a} \) be the normalized reproducing kernel of \( {F}_{\alpha }^{2} \) at point a. Then\n\n\[ \left\lbrack {{\mathcal{B}}_{\alpha }^{-1}{k}_{a}}\right\rbrack \left( x\right) = {\left( \frac{2\alpha }{\pi }\right) }^{\frac{1}{4}}{\mathrm{e}}^{-{2\... | Proof. Let \( c = {\left( 2\alpha /\pi \right) }^{1/4} \) . By Proposition 6.9 and the reproducing property in \( {F}_{\alpha }^{2} \),\n\n\[ \left\lbrack {{\mathcal{B}}_{\alpha }^{-1}{k}_{a}}\right\rbrack \left( x\right) = c{\int }_{\mathbb{C}}{\mathrm{e}}^{{2\alpha x}\bar{z} - \alpha {x}^{2} - \frac{\alpha }{2}{\bar{... | Yes |
Lemma 6.11. We have\n\n\[ \n{\int }_{\mathbb{R}}{\mathrm{e}}^{-{2\pi }\mathrm{i}{zx} - \pi {x}^{2}}\mathrm{\;d}x = {\mathrm{e}}^{-\pi {z}^{2}} \n\]\n\nfor all complex numbers \( z \) . | Proof. Recall that\n\n\[ \n{h}_{0}\left( x\right) = {\left( \frac{2\alpha }{\pi }\right) }^{\frac{1}{4}}{\mathrm{e}}^{-\alpha {x}^{2}} \n\]\n\nis the first vector in the orthonormal basis \( \left\{ {h}_{n}\right\} \) of \( {L}^{2}\left( {\mathbb{R},\mathrm{d}x}\right) \) . By Theorem 6.8 and its proof, \( {\mathcal{B}... | Yes |
Lemma 6.13. For any positive \( \alpha \), we have\n\n\[{\mathcal{B}}_{\alpha }^{-1}A{\mathcal{B}}_{\alpha } = X + \mathrm{i}D = Z,\;{\mathcal{B}}_{\alpha }^{-1}{A}^{ * }{\mathcal{B}}_{\alpha } = X - \mathrm{i}D = {Z}^{ * },\]\n\nwhere \( X, D \), and \( Z \) are the (unbounded) operators on \( {L}^{2}\left( {\mathbb{R... | Proof. Let \( {C}_{c}\left( \mathbb{R}\right) \) denote the space of continuous functions on \( \mathbb{R} \) having compact support. Then \( {C}_{c}\left( \mathbb{R}\right) \) is dense in \( {L}^{2}\left( {\mathbb{R},\mathrm{d}x}\right) \). Given \( f \in {C}_{c}\left( \mathbb{R}\right) \), we differentiate\n\n\[{\mat... | Yes |
Theorem 6.14. Let\n\n\\[ \n\\sigma \\left( z\\right) = \\sigma \\left( {z,\\bar{z}}\\right) = \\sum {c}_{nm}{z}^{n}{\\bar{z}}^{m} \n\\]\n\nbe real analytic and\n\n\\[ \n\\sigma \\left( {Z,{Z}^{ * }}\\right) = \\sum {c}_{nm}{Z}^{n}{Z}^{*m} \n\\]\n\nbe the anti-Wick pseudodifferential operator on \\( {L}^{2}\\left( {\\ma... | Proof. By Lemma 6.13, we have\n\n\\[ \n{\\mathcal{B}}_{\\alpha }\\sigma \\left( {Z,{Z}^{ * }}\\right) {\\mathcal{B}}_{\\alpha }^{-1} = \\sum {c}_{nm}{A}^{n}{A}^{*m}.\n\\]\n\nThus, for \\( f \\in {F}_{\\alpha }^{2} \\), we have\n\n\\[ \n{\\mathcal{B}}_{\\alpha }\\sigma \\left( {Z,{Z}^{ * }}\\right) {\\mathcal{B}}_{\\alp... | Yes |
Theorem 6.16. Suppose \( \varphi \) satisfies condition \( \left( {I}_{2}\right) \) and \( {T}_{\varphi } \) is bounded on \( {F}_{\alpha }^{2} \) . Then \( {B}_{\beta }\varphi \) is bounded for all \( \beta \) with \( 0 < \beta < {2\alpha } \) . | Proof. Let \( \beta = \alpha \left( {1 - t}\right) \) with \( - 1 < t < 1 \) . The condition \( - 1 < t < \sqrt{2} - 1 \) is equivalent to \( \alpha \left( {2 - \sqrt{2}}\right) < \beta < {2\alpha } \) . Also, according to the trace-norm estimate in (6.21), we have\n\n\[ \n{\begin{Vmatrix}{T}_{a}^{\left( t\right) }\end... | Yes |
Theorem 6.17. Suppose \( g \) satisfies condition \( \left( {I}_{2}\right) \) and \( \sigma \left( {D, X}\right) \) is the pseudodifferential operator on \( {L}^{2}\left( {\mathbb{R},\mathrm{d}x}\right) \) with symbol\n\n\[ \sigma \left( {\zeta, x}\right) = \sigma \left( z\right) = {B}_{2\alpha }g\left( \bar{z}\right) ... | Proof. Let \( T = {\mathcal{B}}_{\alpha }\sigma \left( {D, X}\right) {\mathcal{B}}_{\alpha }^{-1} \) . By Theorem 6.12, we have\n\n\[ \widetilde{T}\left( z\right) = {B}_{2\alpha }\sigma \left( \bar{z}\right) = {B}_{2\alpha }{B}_{2\alpha }g\left( z\right) . \]\n\nBy the semigroup property (Corollary 3.15), we have\n\n\[... | Yes |
Theorem 6.18. Let \( g \) be a symbol function on \( \mathbb{C} \) that satisfies condition \( \left( {I}_{2}\right) \) . If there exists some \( \beta \in \left( {{2\alpha },\infty }\right) \) such that \( {B}_{\beta }g \in {L}^{\infty }\left( \mathbb{C}\right) \), then \( {T}_{g} \) is bounded on \( {F}_{\alpha }^{2}... | Proof. Let \( \sigma \left( z\right) = {B}_{2\alpha }g\left( \bar{z}\right) \) . In view of Theorem 6.17, the Toeplitz operator \( {T}_{g} \) on \( {F}_{\alpha }^{2} \) is unitarily equivalent to the pseudodifferential operator \( \sigma \left( {D, X}\right) \) on \( {L}^{2}\left( {\mathbb{R},\mathrm{d}x}\right) \) . W... | Yes |
Theorem 6.19. Suppose \( \varphi \geq 0 \) satisfies condition \( \left( {I}_{1}\right) \) . Then the following conditions are equivalent:\n\n(a) \( {T}_{\varphi } \) is bounded on \( {F}_{\alpha }^{2} \) .\n\n(b) \( \widetilde{\varphi } = {B}_{\alpha }\varphi \in {L}^{\infty }\left( \mathbb{C}\right) \) .\n\n(c) \( {B... | Proof. The equivalences of (a), (b), and (d) follow from the characterization of Fock-Carleson measures in Sect. 3.4. In fact, when \( \varphi \) is nonnegative, we have\n\n\[{\left\langle {T}_{\varphi }f, f\right\rangle }_{\alpha } = {\int }_{\mathbb{C}}{\left| f\right| }^{2}\varphi \mathrm{d}{\lambda }_{\alpha }\n\nT... | Yes |
Theorem 6.20. Suppose \( \varphi \in {\mathrm{{BMO}}}^{1} \) . Then the following conditions are equivalent:\n\n(a) \( {T}_{\varphi } \) is bounded on \( {F}_{\alpha }^{2} \) .\n\n(b) \( \widetilde{\varphi } = {B}_{\alpha }\varphi \in {L}^{\infty }\left( \mathbb{C}\right) \) .\n\n(c) \( {B}_{\beta }\varphi \in {L}^{\in... | Proof. By (3.22) of Theorem 3.34, there exists a constant \( C > 0 \) such that\n\n\[{\begin{Vmatrix}\varphi \circ {\varphi }_{z} - \widetilde{\varphi }\left( z\right) \end{Vmatrix}}_{{L}^{1}\left( {\mathrm{\;d}{\lambda }_{\alpha }}\right) } \leq C\]\n\nfor all \( z \in \mathbb{C} \), where \( {\varphi }_{z}\left( w\ri... | Yes |
Theorem 6.21. Suppose \( \varphi \) satisfies condition \( \left( {I}_{2}\right) \) and \( {T}_{\varphi } \) is compact on \( {F}_{\alpha }^{2} \) . Then \( {B}_{\beta }\varphi \in {C}_{0}\left( \mathbb{C}\right) \) for all \( \beta \in \left( {0,{2\alpha }}\right) \) . | Proof. Recall from Theorem 6.16 and its proof that, for any \( \beta \in \left( {0,{2\alpha }}\right) \), there exists a positive constant \( C = C\left( \beta \right) \) such that \( {\begin{Vmatrix}{B}_{\beta }f\end{Vmatrix}}_{\infty } \leq C\begin{Vmatrix}{T}_{f}\end{Vmatrix} \) whenever \( {T}_{f} \) is bounded on ... | Yes |
Theorem 6.23. Suppose \( \varphi \) is nonnegative and satisfies condition \( \left( {I}_{1}\right) \) . Then, the following conditions are equivalent:\n\n(a) \( {T}_{\varphi } \) is compact on \( {F}_{\alpha }^{2} \) .\n\n(b) \( \widetilde{\varphi } \in {C}_{0}\left( \mathbb{C}\right) \) .\n\n(c) \( {B}_{\beta }\varph... | Proof. The equivalence of (a), (b), and (d) follow from the characterization of vanishing Fock-Carleson measures in Sect. 3.4. See the proof of Theorem 6.19 for the connection to Fock-Carleson measures. The equivalence of (b) and (c) follows from Theorem 3.23. | Yes |
Lemma 6.24. Suppose \( f \in {\mathrm{{BMO}}}^{1} \) and \( \widetilde{f} = {B}_{\alpha }f \) is bounded. Then\n\n\[ \n{T}_{f}{K}_{z} = {K}_{z}\left\lbrack {P\left( {f \circ {\varphi }_{z}}\right) }\right\rbrack \circ {\varphi }_{z} \n\]\n\nfor all \( z \in \mathbb{C} \), where \( P : {L}_{\alpha }^{2} \rightarrow {F}_... | Proof. Since \( {\mathrm{{BMO}}}^{1} \) and the Berezin transform are both translation invariant, we see that for any \( z \in \mathbb{C} \), we have\n\n\[ \nf \circ {\varphi }_{z} \in {\mathrm{{BMO}}}^{1},\;{B}_{\alpha }\left( {f \circ {\varphi }_{z}}\right) \in {L}^{\infty }\left( \mathbb{C}\right) .\n\]\n\nIn partic... | Yes |
Lemma 6.25. Suppose \( f \in {\mathrm{{BMO}}}^{1} \) and \( \widetilde{f} \) is bounded. Then there exists a positive constant \( C \) such that\n\n\[ \mathop{\sup }\limits_{{z \in \mathbb{C}}}\left| {P\left( {f \circ {\varphi }_{z}}\right) \left( w\right) }\right| \leq C{\mathrm{e}}^{\alpha {\left| w\right| }^{2}/4} \... | Proof. Recall from the proof of Theorem 6.20 that if \( f \in {\mathrm{{BMO}}}^{1} \) and \( \widetilde{f} \) is bounded, then \( \widetilde{\left| f\right| } \) is bounded as well. By translation invariance of \( {\mathrm{{BMO}}}^{1} \) and the Berezin transform, there exists a positive constant \( C \) such that\n\n\... | Yes |
Corollary 6.28 Let \( f \in {\mathrm{{BMO}}}^{1},\alpha > 0 \), and \( \beta > 0 \) . Then \( {B}_{\alpha }f \in {C}_{0}\left( \mathbb{C}\right) \) if and only if \( {B}_{\beta }f \in {C}_{0}\left( \mathbb{C}\right) \) . | Proof. Without loss of generality, assume that \( 0 < \alpha < \beta \) . If \( {B}_{\beta }f \in {C}_{0}\left( \mathbb{C}\right) \), then by Proposition 3.21, \( {B}_{\alpha }f \in {C}_{0}\left( \mathbb{C}\right) \) . We do not need the assumption that \( f \in {\mathrm{{BMO}}}^{1} \) here.\n\nIf \( {B}_{\alpha }f \in... | Yes |
Proposition 6.29. Suppose \( \mu \) is a positive Borel measure on \( \mathbb{C} \) and satisfies condition (M). Then \( {T}_{\mu } \) is in the trace-class \( {S}_{1} \) if and only if \( \mu \) is finite on \( \mathbb{C} \) . Moreover, \( \operatorname{tr}\left( {T}_{\mu }\right) = \left( {\alpha /\pi }\right) \mu \l... | Proof. Since all integrands below are nonnegative, we use Fubini's theorem to obtain\n\n\[ \operatorname{tr}\left( {T}_{\mu }\right) = \frac{\alpha }{\pi }{\int }_{\mathbb{C}}\widetilde{\mu }\left( z\right) \mathrm{d}A\left( z\right) \]\n\n\[ = \frac{\alpha }{\pi }{\int }_{\mathbb{C}}{\mathrm{e}}^{\alpha {\left| z\righ... | Yes |
Lemma 6.30. If \( p \geq 1 \) and \( \varphi \in {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) \), then \( {T}_{\varphi } \in {S}_{p} \) . | Proof. If \( \varphi \in {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) \), then \( \varphi \circ {t}_{a} \in {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) \) by a simple change of variables. It follows that \( \varphi \circ {t}_{a} \in {L}^{p}\left( {\mathbb{C},\mathrm{d}{\lambda }_{\alpha }}\right) \) for every \( a \... | Yes |
Lemma 6.31. Suppose \( r > 0,\mu \) is a positive Borel measure on \( \mathbb{C} \), and\n\n\[ \n{\widehat{\mu }}_{r}\left( z\right) = \frac{\mu \left( {B\left( {z, r}\right) }\right) }{\pi {r}^{2}},\;z \in \mathbb{C}.\n\]\n\nIf \( {\widehat{\mu }}_{r} \) is in \( {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) \) for so... | Proof. Let\n\n\[ \nC = {\int }_{\mathbb{C}}\mu {\left( B\left( z, r\right) \right) }^{p}\mathrm{\;d}A\left( z\right) < \infty .\n\]\n\nFor any \( a \in \mathbb{C} \), we have\n\n\[ \n{\int }_{B\left( {a, r/2}\right) }\mu {\left( B\left( z, r\right) \right) }^{p}\mathrm{\;d}A\left( z\right) \leq C.\n\]\n\nWhen \( z \in ... | Yes |
Lemma 6.35. Suppose \( \mu \geq 0,0 < p \leq 1 \), and \( \mu \) satisfies condition (M). If the function \( \widetilde{\mu } \) belongs to \( {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) \), then the operator \( {T}_{\mu } \) belongs to \( {S}_{p} \) . | Proof. Since \( \widetilde{\mu } \) belongs to \( {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) \) and \( \widetilde{\mu } \) dominates \( {\widehat{\mu }}_{r} \), Lemma 6.31 shows that \( {T}_{\mu } \) is bounded. Thus, \( \widetilde{{T}_{\mu }} = \widetilde{\mu } \) and the desired result follows from Proposition 3.6... | No |
Lemma 6.39. Suppose \( \mu \) is a finite complex Borel measure on \( \mathbb{C} \) with compact support. If \( {T}_{\mu } \) has rank less than \( n \), then\n\n\[ \det \left( \begin{matrix} \mu \left( {{f}_{1}{\bar{g}}_{1}}\right) & \cdots & \mu \left( {{f}_{n}{\bar{g}}_{1}}\right) \\ \vdots & \vdots & \vdots \\ \mu ... | Proof. Given one-variable polynomials \( {f}_{1},\cdots ,{f}_{n} \), the functionals \( {T}_{\mu }\left( {f}_{1}\right) ,\cdots ,{T}_{\mu }\left( {f}_{n}\right) \) are linearly dependent because \( {T}_{\mu } \) has rank less than \( n \) . So there are coefficients \( {c}_{1},\cdots ,{c}_{n} \), not all 0, such that\n... | Yes |
Lemma 6.40. Suppose \( \mu \) is a finite complex Borel measure on \( \mathbb{C} \) with compact support. If \( {T}_{\mu } \) has rank less than \( n \) and\n\n\[ \n\mathrm{d}{\mu }_{n}\left( {{z}_{1},\cdots ,{z}_{n}}\right) = {\mathrm{e}}^{-\alpha \left( {{\left| {z}_{1}\right| }^{2} + \cdots + {\left| {z}_{n}\right| ... | Proof. Since the determinant is linear in each column, we can rephrase (6.40) as follows:\n\n\[ \n{\int }_{{\mathbb{C}}^{n}}{f}_{1}\left( {z}_{1}\right) \cdots {f}_{n}\left( {z}_{n}\right) \overline{\Delta \left( {{g}_{1},\cdots ,{g}_{n}}\right) \left( z\right) }\mathrm{d}{\mu }_{n}\left( z\right) = 0, \n\] \n\nwhere \... | Yes |
Lemma 6.41. Let \( K \) be a permutation invariant compact set in \( {\mathbb{C}}^{n} \), let \( {\Phi }_{s} \) denote the algebra consisting of all finite linear combinations of functions of the form \( \psi \bar{\varphi } \) , where \( \psi \) and \( \varphi \) are symmetric polynomials in \( P\left( {\mathbb{C}}^{n}... | Proof. It is clear that \( {\Phi }_{s} \) is an algebra that contains the constant functions and is closed under complex conjugation. If it also separated points in \( K \), the desired result would then follow from the Stone-Weierstrass approximation theorem. But it is easy to see that \( {\Phi }_{s} \) does not separ... | Yes |
Theorem 6.42. Suppose \( \mu \) is a compactly supported finite complex Borel measure on \( \mathbb{C} \) such that the rank of \( {T}_{\mu } \) is less than \( n \), where \( n \) is a positive integer. Then \( \mu \) is supported on less than \( n \) points in \( \mathbb{C} \) . | Proof. Recall that for \( z = \left( {{z}_{1},\cdots ,{z}_{n}}\right) \) ,\n\n\[ V\left( z\right) = \det \left( \begin{matrix} 1 & 1 & \cdots & 1 \\ {z}_{1} & {z}_{2} & \cdots & {z}_{n} \\ \vdots & \vdots & \vdots & \vdots \\ {z}_{1}^{n - 1} & {z}_{2}^{n - 1} & \cdots & {z}_{n}^{n - 1} \end{matrix}\right) = \mathop{\pr... | Yes |
Corollary 6.43 Let \( \varphi \) be a compactly supported and locally integrable function on \( \mathbb{C} \) . Then the Toeplitz operator \( {T}_{\varphi } \) on \( {F}_{\alpha }^{2} \) has finite rank if and only if \( \varphi = 0 \) . | In the rest of this section, we present an example to show that it is necessary to assume that the measure \( \mu \) in Theorem 6.42 and \( \varphi \) in Corollary 6.43 are compactly supported. These results will be false without this assumption. To better understand the intricacy of the problem, we note that if \( \va... | No |
Theorem 7.2. Suppose \( \varphi \in {F}_{\alpha }^{2} \) . Then \( {h}_{\bar{\varphi }} \) is compact on \( {F}_{\alpha }^{2} \) if and only if \( f \in {f}_{\alpha /2}^{\infty } \) , that is,\n\n\[ \mathop{\lim }\limits_{{z \rightarrow \infty }}{\mathrm{e}}^{-\alpha {\left| z\right| }^{2}/4}\varphi \left( z\right) = 0... | Proof. First, assume that \( \varphi \) is an entire function that satisfies condition (7.2). Then there exists a sequence of polynomials \( \left\{ {p}_{k}\right\} \) such that\n\n\[ \mathop{\lim }\limits_{{k \rightarrow \infty }}{\begin{Vmatrix}{p}_{k} - \varphi \end{Vmatrix}}_{{F}_{\alpha /2}^{\infty }} = 0 \]\n\nBy... | Yes |
Corollary 7.3. Suppose \( f \) is an entire function. Then \( f = {P}_{\alpha }\left( g\right) \) for some \( g \in {L}^{\infty }\left( \mathbb{C}\right) \) if and only if \( f \in {F}_{\alpha /2}^{\infty } \). Similarly, \( f = {P}_{\alpha }\left( g\right) \) for some \( g \in {C}_{0}\left( \mathbb{C}\right) \) if and... | Proof. If \( f = {P}_{\alpha }\left( g\right) \) for some \( g \in {L}^{\infty }\left( \mathbb{C}\right) \), then \( {h}_{\bar{f}} = {h}_{\bar{g}} \) is bounded, so by Theorem 7.1, \( f \in {F}_{\alpha /2}^{\infty } \).\n\nIf \( f = {P}_{\alpha }\left( g\right) \) for some \( g \in {C}_{0}\left( \mathbb{C}\right) \), t... | Yes |
Lemma 7.6. Let \( P\left( \mathbb{C}\right) \) denote the ring of all complex polynomials of the variable \( z \) . If \( J \) is an ideal in \( P\left( \mathbb{C}\right) \) containing at least one nonzero polynomial, then there are a finite number of complex numbers \( {a}_{k},1 \leq k \leq N \), and for each \( k \),... | Proof. By a well-known fact in abstract algebra (see [146] for example), every ideal \( J \neq \left( 0\right) \) of \( P\left( \mathbb{C}\right) \) is generated by a polynomial, that is, there exists a polynomial \( q \) such that \( J = \{ {pq} : p \in P\left( \mathbb{C}\right) \} \) . If \( {a}_{1},\cdots ,{a}_{N} \... | Yes |
Theorem 7.7. A bounded small Hankel operator has finite rank if and only if it can be written as \( {h}_{\varphi } \), where\n\n\[ \varphi \left( z\right) = \mathop{\sum }\limits_{{k = 1}}^{N}\mathop{\sum }\limits_{{i = 0}}^{{N}_{k}}{c}_{ki}{\varphi }_{ki}\left( z\right) \]\n\n(7.5)\n\nHere, \( {\varphi }_{ki}\left( z\... | Proof. We have already proved that \( {h}_{\varphi } \) has finite rank if \( \varphi \) is given by (7.5).\n\nTo prove the other direction, we write the small Hankel operator as \( {h}_{\varphi } \), where \( \varphi \) is conjugate analytic. If \( {h}_{\varphi } \) has finite rank, then the restriction of \( {h}_{\va... | Yes |
Lemma 8.1. If there exists a positive constant \( C \) such that\n\n\[ \left| {\varphi \left( z\right) - \varphi \left( w\right) }\right| \leq C\left| {z - w}\right| \]\n\nfor all complex numbers \( z \) and \( w \) . Then \( \varphi \) satisfies condition \( \left( {I}_{1}\right) \) and \( \begin{Vmatrix}{H}_{\varphi ... | Proof. It is easy to check that any Lipschitz function satisfies condition \( \left( {I}_{1}\right) \) and hence induces a well-defined Hankel operator. To estimate the norm of \( {H}_{\varphi } \), consider the integrals\n\n\[ I\left( z\right) = {\int }_{\mathbb{C}}\left| {z - w}\right| \left| {K\left( {z, w}\right) }... | Yes |
Lemma 8.2. Suppose \( f \) satisfies condition \( \left( {I}_{2}\right) \) . Then the operators \( {T}_{f} \) and \( {H}_{f} \) are both densely defined on \( {F}_{\alpha }^{2} \) . Moreover, we have\n\n\[ \n{T}_{f}{k}_{z} = {U}_{z}P\left( {f \circ {\varphi }_{z}}\right) = P\left( {f \circ {\varphi }_{z}}\right) \circ ... | Proof. Since each \( {U}_{z} \) commutes with the projection \( P \), we have\n\n\[ \n{T}_{f}{k}_{z} = P\left( {f{k}_{z}}\right) = P{U}_{z}\left( {f \circ {\varphi }_{z}}\right) = {U}_{z}P\left( {f \circ {\varphi }_{z}}\right) .\n\nThis proves the desired results. | No |
Theorem 8.4. Suppose \( \varphi \) satisfies condition \( \left( {I}_{2}\right) \) . Then the following two conditions are equivalent:\n\n1. Both \( {H}_{\varphi } \) and \( {H}_{\bar{\varphi }} \) are bounded on \( {F}_{\alpha }^{2} \) .\n\n2. The function \( \varphi \) belongs to \( {\mathrm{{BMO}}}^{2} \) . | Proof. First, assume that \( \varphi \in {\mathrm{{BMO}}}^{2} \) . Then, by Corollary 3.37, we can write \( \varphi = \) \( {\varphi }_{1} + {\varphi }_{2} \), where the function \( {\varphi }_{1} \) satisfies the Lipschitz estimate\n\n\[ \left| {{\varphi }_{1}\left( z\right) - {\varphi }_{1}\left( w\right) }\right| \l... | Yes |
Theorem 8.5. Suppose \( \varphi \) satisfies condition \( \left( {I}_{2}\right) \) . Then the following two conditions are equivalent:\n\n1. Both \( {H}_{\varphi } \) and \( {H}_{\bar{\varphi }} \) are compact on \( {F}_{\alpha }^{2} \) .\n\n2. The function \( \varphi \) belongs to \( {\mathrm{{VMO}}}^{2} \) . | Proof. If \( \varphi \in {\mathrm{{VMO}}}^{2} \), then \( {\begin{Vmatrix}\varphi - {\varphi }_{r}\end{Vmatrix}}_{{\mathrm{{BMO}}}^{2}} \rightarrow 0 \) as \( r \rightarrow \infty \), where \( {\varphi }_{r} \) is \( \varphi \) times the characteristic function of the Euclidean ball \( B\left( {0, r}\right) \) . It is ... | Yes |
Lemma 8.7. If \( f \in {L}^{\infty }\left( \mathbb{C}\right) \), then \( \left| {{Pf}\left( z\right) }\right| \leq \parallel f{\parallel }_{\infty }{\mathrm{e}}^{\alpha {\left| z\right| }^{2}/4} \) for all \( z \in \mathbb{C} \) . | Proof. This follows directly from Corollary 2.5. | No |
Lemma 8.8. Suppose \( F\left( {w, z}\right) \) is a nonnegative measurable function on \( \mathbb{C} \times \mathbb{C} \) with the property that there is a constant \( B > 0 \) such that\n\n\[ F\left( {w, z}\right) \leq B{\mathrm{e}}^{\frac{\alpha }{4}{\left| z\right| }^{2}},\;z, w \in \mathbb{C}. \]\n\nThen there exis... | Proof. We make an obvious change of variables to rewrite the integral on the left-hand side as\n\n\[ \frac{\alpha }{\pi }{\mathrm{e}}^{\frac{\alpha }{2}{\left| w\right| }^{2}}{\int }_{\mathbb{C}}F\left( {w, u}\right) {\mathrm{e}}^{-\frac{\alpha }{2}{\left| u\right| }^{2}}\mathrm{\;d}A\left( u\right) . \]\n\nDenote the ... | Yes |
Lemma 8.9. Suppose \( f \in {L}^{\infty }\left( \mathbb{C}\right) \) . For any \( z \in \mathbb{C} \), we have\n\n\[{\int }_{\mathbb{C}}\left| {P\left( {f \circ {\varphi }_{w}}\right) \left( {{\varphi }_{w}\left( z\right) }\right) }\right| \left| {{K}_{w}\left( z\right) }\right| {K}_{w}{\left( w\right) }^{\frac{1}{2}}\... | Proof. It follows from (8.1) that\n\n\[ \left| {P\left( {f \circ {\varphi }_{w}}\right) \left( {{\varphi }_{w}\left( z\right) }\right) }\right| \left| {{K}_{w}\left( z\right) }\right| = \left| {P\left( {f{K}_{w}}\right) \left( z\right) }\right| \]\n\n\[ = \left| {{\int }_{\mathbb{C}}f\left( u\right) {K}_{w}\left( u\rig... | Yes |
Lemma 8.12. If \( f \in {L}^{\infty }\left( \mathbb{C}\right) \) and \( {H}_{f} \) is compact, then both \( {H}_{\widetilde{f}} \) and \( {T}_{f - \widetilde{f}} \) are compact. | Proof. By Theorem 8.10,\n\n\[ \mathop{\lim }\limits_{{a \rightarrow \infty }}\begin{Vmatrix}{f \circ {\varphi }_{a} - P\left( {f \circ {\varphi }_{a}}\right) }\end{Vmatrix} = 0 \]\n\nwhich, according to Lemma 8.11, implies that\n\n\[ \mathop{\lim }\limits_{{a \rightarrow \infty }}\begin{Vmatrix}{\widetilde{f} \circ {\v... | Yes |
Theorem 8.13. Suppose \( f \in {L}^{\infty }\left( \mathbb{C}\right) \) . Then \( {H}_{f} \) is compact if and only if \( {H}_{\bar{f}} \) is compact. | Proof. Let \( g = \bar{f} \) and assume that \( {H}_{g} \) is compact. By Theorem 8.10,\n\n\[\n\mathop{\lim }\limits_{{a \rightarrow \infty }}\begin{Vmatrix}{g \circ {\varphi }_{a} - P\left( {g \circ {\varphi }_{a}}\right) }\end{Vmatrix} = 0.\n\]\n\nCombining this with Lemma 8.11, we see that\n\n\[\n\mathop{\lim }\limi... | Yes |
Lemma 8.14. Let \( 2 \leq p < \infty \) . If \( {H}_{f} \) and \( {H}_{\bar{f}} \) are both in the Schatten class \( {S}_{p} \), then \( {MO}\left( f\right) \in {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) \) . | Proof. If \( {H}_{f} \) is in \( {S}_{p} \), then \( {\left( {H}_{f}^{ * }{H}_{f}\right) }^{p/2} \) is in the trace class \( {S}_{1} \), so by Proposition 3.3,\n\n\[ \n{\int }_{\mathbb{C}}\left\langle {{\left( {H}_{f}^{ * }{H}_{f}\right) }^{p/2}{k}_{z},{k}_{z}}\right\rangle \mathrm{d}A\left( z\right) < \infty , \n\] \n... | Yes |
Lemma 8.15. Let \( 0 < p \leq 2 \) . If \( {MO}\left( f\right) \in {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) \), then both \( {H}_{f} \) and \( {H}_{\bar{f}} \) are in the Schatten class \( {S}_{p} \) . | Proof. By Proposition 8.3, the condition \( {MO}\left( f\right) \in {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) \) implies that the function \( z \mapsto \begin{Vmatrix}{{H}_{f}{k}_{z}}\end{Vmatrix} \) is in \( {L}^{p}\left( {\mathbb{C},\mathrm{d}A}\right) \) . This, along with Proposition 3.3 and Lemma 3.4, shows th... | Yes |
Lemma 8.16. Suppose \( 2 \leq p < \infty \) and \( T \) is the integral operator defined by\n\n\[ \n{Tf}\left( z\right) = {\int }_{\mathbb{C}}G\left( {z, w}\right) K\left( {z, w}\right) f\left( w\right) \mathrm{d}{\lambda }_{\alpha }\left( w\right) ,\n\]\n\nwhere \( G \) is a measurable function on \( \mathbb{C} \times... | Proof. The case \( p = 2 \) follows from the classical characterization of Hilbert-Schmidt integral operators on \( {L}^{2} \) spaces; see [113]. If \( G \in {L}^{\infty }\left( {\mathbb{C} \times \mathbb{C}}\right) \), then \( T \) is dominated by the bounded operator \( {Q}_{\alpha } \) considered in Sect. 2.2, so th... | No |
Lemma 8.19. Suppose \( f \) is locally square integrable and \( v \in {\mathbb{Z}}^{2}/N \) . Then\n\n\[ \n{\int }_{{S}_{N}}{\left| f \circ {t}_{v} - {f}_{{S}_{N}}\right| }^{2}\mathrm{\;d}A \leq \left( {{N}^{2} + \frac{4{N}^{4}\left| {\gamma \left( v\right) }\right| }{9}}\right) \mathop{\sum }\limits_{{a \in \gamma \le... | Proof. The case \( v = 0 \) is trivial. If \( v \neq 0 \), we write\n\n\[ \n\gamma \left( v\right) = \left\{ {{a}_{0},{a}_{1},\ldots ,{a}_{l}}\right\}\n\]\n\nin the order in which \( \gamma \left( v\right) \) is defined, where \( l + 1 = \left| {\gamma \left( v\right) }\right| \) is the length of the path \( \gamma \le... | Yes |
Lemma 8.20. Suppose \( f \) satisfies condition \( \left( {I}_{2}\right) \) . There exists a positive constant \( C = {C}_{N} \) (depending on \( N \) ) such that\n\n\[ \mathop{\sup }\limits_{{z \in {S}_{N}}}{MO}{\left( f\right) }^{2}\left( z\right) \leq C\mathop{\sum }\limits_{{v \in {\mathbb{Z}}^{2}/N}}\mathop{\sum }... | Proof. For any constant \( c \), we have\n\n\[ {\int }_{\mathbb{C}}{\left| f \circ {t}_{z} - c\right| }^{2}\mathrm{\;d}{\lambda }_{\alpha } = \widetilde{{\left| f\right| }^{2}}\left( z\right) - \bar{c}\widetilde{f}\left( z\right) - c\overline{\widetilde{f}\left( z\right) } + {\left| c\right| }^{2} \]\n\n\[ = {\left| \w... | Yes |
Lemma 8.21. Suppose \( f \) satisfies condition \( \left( {I}_{2}\right) \) . If \( 0 < p \leq 2 \), then there exists a positive constant \( C = {C}_{N} \), depending on \( N \) and \( p \) but not on \( f \), such that\n\n\[ \n{\int }_{\mathbb{C}}{\left\lbrack MO\left( f\right) \left( z\right) \right\rbrack }^{p}\mat... | Proof. Let us consider the integral\n\n\[ \nI = {\int }_{\mathbb{C}}{\left\lbrack MO\left( f\right) \left( z\right) \right\rbrack }^{p}\mathrm{\;d}A\left( z\right)\n\]\n\nIt is clear that\n\n\[ \n\mathbb{C} = \bigcup \left\{ {{S}_{N} + u : u \in \frac{{\mathbb{Z}}^{2}}{N}}\right\}\n\]\n\nand this is a disjoint union. I... | Yes |
Lemma 8.23. Suppose \( 0 < p < \infty \) and \( f \) satisfies condition \( \left( {I}_{2}\right) \) . Then the Hankel operators \( {H}_{f} \) and \( {H}_{\bar{f}} \) both belong to the Schatten class \( {S}_{p} \) if and only if the commutator \( \left\lbrack {{M}_{f}, P}\right\rbrack = {M}_{f}P - P{M}_{f} \) belongs ... | Proof. It is easy to see that\n\n\[ \left\lbrack {{M}_{f}, P}\right\rbrack = \left\lbrack {{M}_{f}, P}\right\rbrack P + \left\lbrack {{M}_{f}, P}\right\rbrack \left( {I - P}\right) = {H}_{f} - {H}_{\bar{f}}^{ * }.\]\n\nSo the simultaneous membership of \( {H}_{f} \) and \( {H}_{\bar{f}} \) in \( {S}_{p} \) implies that... | Yes |
Theorem 8.26. Suppose \( 0 < p < \infty, r > 0, N \) is any positive integer, and \( f \) satisfies condition \( \left( {I}_{2}\right) \) . Then the following conditions are equivalent:\n\n(a) The operators \( {H}_{f} \) and \( {H}_{\bar{f}} \) both belong to the Schatten class \( {S}_{p} \) .\n\n(b) The function\n\n\[... | Proof. That (a) implies (b) follows from Lemmas 8.14 and 8.25. Lemmas 8.15 and 8.18 show that condition (b) implies (a). So (a) and (b) are equivalent.\n\nBy the double integral representations for \( {MO}\left( f\right) \) and \( M{O}_{r}\left( f\right) \), it is easy for us to find a positive constant \( C = C\left( ... | Yes |
Lemma 1. Let \( f : X \rightarrow {\mathbb{R}}_{\infty } \) be convex, and let \( C \) be a convex set such that, for certain positive constants \( \delta \) and \( N \), we have \( \left| {f\left( x\right) }\right| \leq N\;\forall x \in C + {\delta B} \) . Then \( f \) is Lipschitz on \( C \) of rank \( {2N}/\delta \)... | To prove the lemma, let us fix two distinct points \( x \) and \( y \) in \( C \) . The point \( z \) defined by \( z = y + \delta \left( {y - x}\right) /\parallel y - x\parallel \) belongs to \( C + {\delta B} \), and satisfies\n\n\[ y = \frac{\delta }{\delta + \parallel y - x\parallel }x + \frac{\parallel y - x\paral... | Yes |
Lemma 2. Let \( {x}_{0} \) be a point such that, for certain numbers \( M \) and \( \varepsilon > 0 \), we have \( f\left( x\right) \leq M\forall x \in B\left( {{x}_{0},\varepsilon }\right) \) . Then, for any \( x \in \operatorname{int}\operatorname{dom}f \), there exists a neighborhood \( V \) of \( x \) and \( N \geq... | Without loss of generality, we prove the lemma for \( {x}_{0} = 0 \) . Let \( x \in \operatorname{int}\operatorname{dom}f \) . There exists \( r \in \left( {0,1}\right) \) such that \( x/r \in \operatorname{dom}f \) . Then\n\n\[ V \mathrel{\text{:=}} x + \left( {1 - r}\right) B\left( {0,\varepsilon }\right) = B\left( {... | Yes |
Lemma 1. There is a positive constant \( c \) such that\n\n\[ H\left( {x, u, p}\right) \leq c{\left| p - \beta \left( x\right) \right| }^{{r}_{ * }} - \alpha \left( x\right) \forall \left( {x, u}\right) \in Q, p \in {\mathbb{R}}^{m}, \]\n\nwhere \( {r}_{ * } \) is the conjugate exponent to \( r \) . | Proof. Observe that, by the inequality in Hypothesis 6.37 (c), we have:\n\n\[ H\left( {x, u, p}\right) = \mathop{\sup }\limits_{w}\left\{ {\langle p, w\rangle - F\left( {x, u, w}\right) - \delta {\left| w\right| }^{r}/r}\right\} \]\n\n\[ \leq \mathop{\sup }\limits_{w}\left\{ {\langle p, w\rangle - \alpha \left( x\right... | Yes |
Lemma 2. Fix \( \left( {x, u}\right) \in Q \) and \( p \in {\mathbb{R}}^{n} \) . Then:\n\n(a) The function \( H\left( {x, u, \cdot }\right) \) is continuous at \( p \) .\n\n(b) The function \( v \mapsto H\left( {x, v, p}\right) \) is upper semicontinuous at \( u \) in the following sense:\n\n\[ \left( {x,{v}_{i}}\right... | Proof. Since the function \( p \mapsto H\left( {x, u, p}\right) \) is convex and finite on \( {\mathbb{R}}^{n} \) (by Lemma 1), we know it to be continuous, as affirmed in (a). We now fix \( x \) and \( p \) and turn to assertion (b).\n\nLet \( {v}_{i} \) be a sequence converging to \( u \) for which \( \mathop{\lim }\... | Yes |
Lemma 3. Let \( u : \Omega \rightarrow {\mathbb{R}}^{\ell } \) be a measurable function having \( \left( {x, u\left( x\right) }\right) \in Q \) a.e., and let \( p : \Omega \rightarrow {\mathbb{R}}^{n} \) be measurable. Then the function \( x \mapsto H\left( {x, u\left( x\right), p\left( x\right) }\right) \) is measurab... | Proof. Note that the function \( w \mapsto F\left( {x, u, w}\right) \) is continuous, since it is convex and finite. It follows that if \( \left\{ {w}_{i}\right\} \) is a countable dense set in \( {\mathbb{R}}^{n} \), we have (almost everywhere)\n\n\[ H\left( {x, u\left( x\right), p\left( x\right) }\right) = \mathop{\s... | Yes |
Lemma 1.\n\n\\[ \n{f}^{ \circ }\left( {x;v}\right) \leq \mathop{\max }\limits_{{q \in Q\left( x\right) }}\left\langle {{g}_{x}^{\prime }\left( {x, q}\right), v}\right\rangle \n\\] | We may suppose \( v \neq 0 \) . Let \( {x}_{i} \) and \( {t}_{i} \) be sequences realizing \( {f}^{ \circ }\left( {x;v}\right) \), and let \( {q}_{i} \) belong to \( Q\left( {{x}_{i} + {t}_{i}v}\right) \) . Invoking the compactness of \( Q \), we may suppose that \( {q}_{i} \rightarrow q \) ; it follows that \( q \in Q... | No |
Lemma 2.\n\n\[ \mathop{\max }\limits_{{q \in Q\left( x\right) }}\left\langle {{g}_{x}^{\prime }\left( {x, y}\right), v}\right\rangle \leq \mathop{\liminf }\limits_{{t \downarrow 0}}\frac{f\left( {x + {tv}}\right) - f\left( x\right) }{t}. \] | To see this, let \( q \in Q\left( x\right) \) . Then\n\n\[ \frac{f\left( {x + {tv}}\right) - f\left( x\right) }{t} \geq \frac{g\left( {x + {tv}, q}\right) - g\left( {x, q}\right) }{t} = \left\langle {{g}_{x}^{\prime }\left( {z, q}\right), v}\right\rangle \]\n\nfor some \( z \in \left( {x, x + {tv}}\right) \) . Taking l... | Yes |
Lemma 1. The weak decrease of \( \left( {\varphi, F}\right) \) in \( \Omega \) is equivalent to the weak invariance of the system \( \left( {S,{F}_{ + }}\right) \) . | Proof. Let \( \left( {\varphi, F}\right) \) be weakly decreasing in \( \Omega \), and let \( \left( {\alpha, r}\right) \in S \) . (Thus, \( \varphi \left( \alpha \right) \leq r \) .) If \( \alpha \in \partial \Omega \), then \( {F}_{ + }\left( {\alpha, r}\right) \) contains \( \left( {0,0}\right) \), so that the consta... | Yes |
Lemma 2. The condition (1) and the following condition (2) are equivalent:\n\n\[ \n{h}_{F}\left( {x,\zeta }\right) \leq 0\forall \zeta \in {\partial }_{P}\varphi \left( x\right) \forall x \in \Omega .\n\]\n\n(2) | Assume first that (1) holds, and let \( \zeta \in {\partial }_{P}\varphi \left( x\right) \), where \( x \in \Omega \) . Then, by Theorem 11.32, we have \( \left( {\zeta , - 1}\right) \in {N}_{\text{epi }\varphi }^{P}\left( {x,\varphi \left( x\right) }\right) \) . It follows from (1) that \( {h}_{F}\left( {x,\zeta }\rig... | Yes |
Lemma 2. The following are equivalent:\n\n\[ \n{H}_{F}\left( {x,\zeta }\right) \leq 0\forall \zeta \in {\partial }_{P}\varphi \left( x\right) ,\forall x \in \Omega \n\] \n\n(1) \n\n\[ \n{H}_{F}\left( {x,\zeta }\right) \leq 0\forall \left( {\zeta , - \lambda }\right) \in {N}_{\text{epi }\varphi }^{P}\left( {x, r}\right)... | To prove this, assume first that (2) holds, and let \( \zeta \in {\partial }_{P}\varphi \left( x\right) \), where \( x \in \Omega \) . Then, by Theorem 11.32, we have \( \left( {\zeta , - 1}\right) \in {N}_{\text{epi }\varphi }^{P}\left( {x,\varphi \left( x\right) }\right) \) . It follows from (2) that \( {H}_{F}\left(... | Yes |
Lemma 1. There is a measurable function \( {w}_{i} \) such that \( {w}_{i}\left( t\right) \) is (almost everywhere) a point at which the minimum defining \( {\Lambda }_{i}\left( {t,{y}_{i}\left( t\right) ,{y}_{i}^{\prime }\left( t\right) }\right) \) is achieved: | The proof of the lemma uses the multifunction \( \Gamma \) defined by\n\n\[ \Gamma \left( t\right) = \left\{ {w \in {\mathbb{R}}^{n} : {\Lambda }_{i}\left( {t,{y}_{i}\left( t\right) ,{y}_{i}^{\prime }\left( t\right) }\right) = \Lambda \left( {t, w,{y}_{i}^{\prime }\left( t\right) }\right) + {n}_{i}k\left( t\right) {\le... | Yes |
Lemma 2. \( \mathop{\lim }\limits_{{i \rightarrow \infty }}{I}_{i} = J\left( 0\right) \) . | To see this, let \( {w}_{i} \) be the function provided by Lemma 1 . The equality defining it, together with the Lipschitz hypothesis (LH), gives rise to the following estimate: \( \left| {{w}_{i}\left( t\right) - {y}_{i}\left( t\right) }\right| \leq {n}_{i}^{-1} \) a.e. With the help of this, we now calculate\n\n\[ \n... | Yes |
Lemma 1. There exists \( L \) such that, for any \( x \) on \( \left\lbrack {0,\tau }\right\rbrack \) with \( x\left( \tau \right) = \beta \), for any \( \delta \in (0,\tau \rbrack \) \[ {\int }_{\tau - \delta }^{\tau }\Lambda \left( {x,{x}^{\prime }}\right) {dt} \leq u\left( {\tau ,\beta }\right) + \mu \Rightarrow \le... | The lemma, whose relevance will become apparent, follows from elementary arguments (using Hölder’s inequality and the coercivity of \( \Lambda \) ) that lead to a uniform estimate of the type \[ {\int }_{t}^{\tau }\left| {{x}^{\prime }\left( s\right) }\right| {ds} \leq C. \] | No |
There exists an arc \( x \) on the interval \( \left\lbrack {0,\tau }\right\rbrack \) satisfying \( x\left( \tau \right) = \beta \) and\n\n\[ u\left( {t, x\left( t\right) }\right) + {\int }_{t}^{\tau }\Lambda \left( {x\left( s\right) ,{x}^{\prime }\left( s\right) }\right) {ds} \leq u\left( {\tau ,\beta }\right) \forall... | Proof. We define the compact \ | No |
Theorem 22.2. The transversality condition. In the context of Theorem 22.2, suppose that the target set \( E \) is defined by\n\n\[ E = \{ x \in S : g\left( x\right) \leq 0, h\left( x\right) = 0\} ,\]\n\nwhere \( g \) and \( h \) are continuously differentiable functions with values in \( {\mathbb{R}}^{{k}_{1}} \) and ... | Proof. Suppose first that the natural constraint qualification for the set \( E \) is satisfied at \( {x}_{ * }\left( b\right) \) (see Exer. 11.40). Then we simply invoke Theorem 22.2 directly, and the transversality condition of that theorem yields the explicit transversality condition, in view of the available charac... | No |
Lemma 2. Let \( C \in \mathcal{C} \) . Then there exists an element \( \left( {\eta, p}\right) \in Z \) such that, for every \( {u}_{i} \in C \), we have\n\n\[ \left\langle {p\left( t\right), f\left( {t,{x}_{ * }\left( t\right) ,{u}_{i}\left( t\right) }\right) }\right\rangle \leq \left\langle {p\left( t\right), f\left(... | Proof. The idea is to call upon Theorem 25.1 for a \ | No |
Lemma 1. There exists \( {\varepsilon }_{1} > 0 \) and \( M \) such that\n\n\[ t \in \left\lbrack {a, b}\right\rbrack ,\left| {x - {x}_{ * }\left( t\right) }\right| \leq {\varepsilon }_{1}, u \in U,\left| {u - {u}_{ * }\left( t\right) }\right| \leq r,\varphi \left( {t, x, u}\right) \in \Phi ,\]\n\n\[ \lambda \in {N}_{\... | Proof. We argue by contradiction. If the lemma is false, there exist sequences\n\n\[ {t}_{i} \in \left\lbrack {a, b}\right\rbrack ,{\varepsilon }_{i} \downarrow 0,{x}_{i} \in B\left( {{x}_{ * }\left( {t}_{i}\right) ,{\varepsilon }_{i}}\right) ,{u}_{i} \in U \cap B\left( {{u}_{ * }\left( {t}_{i}\right), r}\right) ,{\lam... | Yes |
For \( i \) sufficiently large, there exist \( \left( {{\gamma }_{i},{\tau }_{i},{\lambda }_{i}}\right) \) with \( {\lambda }_{i} \in {N}_{\Phi }^{L}\left( {\varphi \left( {{t}_{i},{x}_{i},{u}_{i}}\right) }\right) \) and \( {\tau }_{i} \in {N}_{U}^{L}\left( {u}_{i}\right) \) such that\n\n\[ \left\lbrack {{\alpha }_{i},... | Proof. It is a matter of showing that Theorem 11.38 applies. Let \( i \) be large enough to ensure that \( x \mathrel{\text{:=}} {x}_{i} \) satisfies \( \left| {x - {x}_{ * }\left( {t}_{i}\right) }\right| < {\varepsilon }_{1} \), where \( {\varepsilon }_{1} \) is provided by Lemma 1 . Suppose that, for some \( \lambda ... | Yes |
Proposition 1.1. (Grave, Herglotz,...) Let \( \Gamma \) be a nonsingular analytic Jordan arc in \( {\mathbf{R}}^{2} \cong \mathbb{C} \). Then there is a neighborhood \( \Omega \) of \( \Gamma \) and a (uniquely determined) analytic function \( S \) on \( \Omega \) such that\n\n(1.1)\n\n\[ S\left( z\right) = \bar{z}, z ... | Proof: We’ll work with definition (ii) above. We’ll give one proof now, and another in \( §{1.5} \). Clearly it is sufficient to construct \( S \) on a neighborhood of an arbitrary point \( {z}_{0} \in \Gamma \) since (1.1) then ensures that these local functions fit together to one analytic on a neighborhood of \( \Ga... | Yes |
Proposition 1.3. Let \( D \) be a plane domain part of whose boundary is a non-singular analytic arc \( \sigma \), and suppose that \( f \), continuous on the closure of \( D \), maps it conformally on a domain \( E \) part of whose boundary is the non-singular analytic arc \( \tau \), and \( f\left( \sigma \right) = \... | Proof: Clearly we can define a function \( \widetilde{f} \) for points \( z \) near \( \sigma, z \notin D \) by\n\n\[ \widetilde{f}\left( z\right) = {R}_{\tau }\left( {f\left( {{R}_{\sigma }\left( z\right) }\right) }\right) \]\n\nThis function is easily checked to be analytic and continuously extendable to points of \(... | Yes |
Proposition 3.2. Let \( \varphi \) be a rational function without poles in \( \overline{\mathbf{D}} \) (where D denotes the open unit disk) and injective in \( \overline{\mathbf{D}} \) . Then, there is a function \( S \) analytic on \( \Omega \mathrel{\text{:=}} \varphi \left( \mathbf{D}\right) \) except for poles, con... | Proof of Prop. 3.2: We can define a function \( S \) from \( \bar{\Omega } \) to the extended complex numbers by\n\n(3.5)\n\n\[ S\left( {\varphi \left( w\right) }\right) = {\varphi }^{ * }\left( {1/w}\right) ,\;w \in \overline{\mathbf{D}} \]\n\nwhere \( {\varphi }^{ * } \) denotes the (rational) function defined by \( ... | Yes |
Proposition 6.1. Let \( D \) be a plane domain part of whose boundary is a non-singular analytic arc \( \sigma \), and suppose that the real-valued harmonic function \( u \), continuous on the closure of \( D \), vanishes on \( \sigma \) . Then, there is an open set \( N \subset {\mathbf{R}}^{2} \) containing \( \sigma... | It is easy to deduce Prop. 1.3 from this. Indeed, in proving Prop. 1.3 we may assume \( D, E \) are simply connected and map each of these conformally on the upper half-plane. This leads easily to a reduction of Prop. 1.3 to its special case where \( E = f\left( D\right) \) is the upper half-plane. But then \( u = \ope... | Yes |
Proposition 6.2. Let \( u \) be in \( {C}^{1}\left( B\right) \), where \( B \) is an open ball in \( {\mathbf{R}}^{n} \) , and harmonic on \( B \smallsetminus \Gamma \), where \( \Gamma \) is a non-singular \( {C}^{1} \) hypersurface. Then \( u \) is harmonic in \( B \) . | This proposition can be proved in the same way as Prop. 1.4, so we'll just sketch the argument. The crucial thing is that the analog of Lemma 1.5 holds in the setting of Prop. 6.2. This is because a function in \( {C}^{1}\left( \Omega \right) \) which, together with its first partial derivatives, extends continuously u... | No |
Proposition 6.3. If \( u \) is harmonic in a sufficiently large neighborhood of an arc of an irreducible algebraic curve \( \Gamma \) in \( {\mathbf{R}}^{2} \) and vanishes on \( \Gamma \) , then at a pair of points of \( {\overline{\mathbf{R}}}^{2} \) which are opposite vertices of a Study quadrilateral, with the rema... | This is Study's version of the reflection principle. Let us hasten to add that it is not necessary that \( \Gamma \) be algebraic (and indeed, Study does not assume this), we have only made this assumption for simplicity (in discussing irreducibility). In a local version, it is easy to check that it suffices for \( \Ga... | No |
Proposition 7.1. With the above-stated regularity assumptions concerning \( \Gamma \) and \( F, F \) can be expressed as \( {F}_{i} - {F}_{e} \), where \( {F}_{i},{F}_{e} \) are in \( C\left( \Gamma \right) \), and:\n\n(a) \( {F}_{i} \) is the boundary value of a function holomorphic in \( \Omega \) .\n\n(b) \( {F}_{e}... | We have only to check the uniqueness and, in view of linearity it suffices to consider \( F = 0 \) . But, if \( {F}_{i} - {F}_{e} = 0 \), then the functions in \( {\Omega }_{i},{\Omega }_{e} \) whose boundary values are \( {F}_{i},{F}_{e} \) respectively combine to form a single entire function which, vanishing at \( \... | Yes |
Proposition 7.2. [Kerzman and Stein, 1978]. The Szegö projector can be expressed in terms of the Hilbert projector by\n\n\[ S = H{\left\lbrack I + \left( H - {H}^{ * }\right) \right\rbrack }^{-1} \]\n\nwhere \( I \) denotes the identity operator on \( {L}^{2}\left( \Gamma \right) \) . | Proof: We have the immediate relations\n\n\[ {HS} = S,\;{SH} = H \]\n\n\[ S{H}^{ * } = S,\;{H}^{ * }S = {H}^{ * }.\]\n\nHence \( S\left( {H - {H}^{ * }}\right) = H - S \), whence \( S\left\lbrack {I + \left( {H - {H}^{ * }}\right) }\right\rbrack = H \) . Since the bracketed operator has its spectrum on the line \( \{ \... | Yes |
Lemma 1.1 A linear subspace \( \mathcal{E} \) of \( {\mathcal{H}}_{1} \oplus {\mathcal{H}}_{2} \) is the graph of a linear operator \( T \) from \( {\mathcal{H}}_{1} \) into \( {\mathcal{H}}_{2} \) if and only if \( \left( {0, y}\right) \in \mathcal{E} \) for \( y \in {\mathcal{H}}_{2} \) implies that \( y = 0 \) . | The operator \( T \) is then uniquely determined by \( \mathcal{E} \) ; it acts by \( {Tx} = y \) for \( \left( {x, y}\right) \in \mathcal{E} \), and its domain is \( \mathcal{D}\left( T\right) = \left\{ {x \in {\mathcal{H}}_{1} : }\right. \) There exists \( y \in {\mathcal{H}}_{2} \) such that \( \left( {x, y}\right) ... | No |
Lemma 1.3 Let \( S \) and \( T \) be linear operators such that \( S \subseteq T \) . If \( S \) is surjective and \( T \) is injective, then \( S = T \) . | Proof Let \( x \in \mathcal{D}\left( T\right) \) . Since \( S \) is surjective, there is a \( y \in \mathcal{D}\left( S\right) \) such that \( {Tx} = {Sy} \) . From \( S \subseteq T \) we get \( {Tx} = {Ty} \), so \( x = y \), because \( T \) is injective. Hence, we have \( x = y \in \mathcal{D}\left( S\right) \) . Thu... | Yes |
Proposition 1.4 The following statements are equivalent:\n\n(i) \( T \) is closed.\n\n(ii) If \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) is a sequence of vectors \( {x}_{n} \in \mathcal{D}\left( T\right) \) such that \( \mathop{\lim }\limits_{n}{x}_{n} = x \) in \( {\mathcal{H}}_{1} \) and \( \mathop{\lim }\limi... | Proof (i) is equivalent to (ii), because (ii) is only a reformulation of the closedness of the graph \( \mathcal{G}\left( T\right) \) in \( {\mathcal{H}}_{1} \oplus {\mathcal{H}}_{2} \) .\n\n(i) \( \leftrightarrow \) (iii): By the definition (1.4) of the graph norm, the map \( x \rightarrow \left( {x,{Tx}}\right) \) of... | Yes |
Proposition 1.5 The following are equivalent:\n\n(i) \( T \) is closable.\n\n(ii) If \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) is a sequence of vectors \( {x}_{n} \in \mathcal{D}\left( T\right) \) such that \( \mathop{\lim }\limits_{n}{x}_{n} = 0 \) in \( {\mathcal{H}}_{1} \) and \( \mathop{\lim }\limits_{n}T\l... | Proof (i) \( \rightarrow \) (iii): Let \( S \) be a closed operator such that \( T \subseteq S \) . Then \( \mathcal{G}\left( T\right) \subseteq \mathcal{G}\left( S\right) \) , and so \( \overline{\mathcal{G}\left( T\right) } \subseteq \mathcal{G}\left( S\right) \) . From Lemma 1.1 it follows that \( \overline{\mathcal... | Yes |
Example 1.1 (Nonclosable operators) Let \( \mathcal{D} \) be a linear subspace of a Hilbert space \( \mathcal{H} \), and let \( e \neq 0 \) be a vector of \( \mathcal{H} \) . Let \( F \) be a linear functional on \( \mathcal{D} \) which is not continuous in the Hilbert space norm. Define the operator \( T \) by \( \mat... | Proof Since \( F \) is not continuous, there exists a sequence \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) from \( \mathcal{D} \) such that \( \mathop{\lim }\limits_{n}{x}_{n} = 0 \) in \( \mathcal{H} \) and \( \left( {F\left( {x}_{n}\right) }\right) \) does not converge to zero. By passing to a subsequence if ne... | Yes |
Example 1.2 (Example 1.1 continued) Suppose that \( \mathcal{D}\left( T\right) \) is dense in \( \mathcal{H} \) . Since the functional \( F \) is discontinuous, the map \( x \rightarrow \langle {Tx}, y\rangle = F\left( x\right) \langle e, y\rangle \) is continuous if and only if \( y \bot e \) . | Hence, \( \mathcal{D}\left( {T}^{ * }\right) = {e}^{ \bot } \) in \( \mathcal{H} \) and \( {T}^{ * }y = 0 \) for \( y \in \mathcal{D}\left( {T}^{ * }\right) \) . | No |
Statement \( {\left( {M}_{\varphi }\right) }^{ * } = {M}_{\bar{\varphi }} \) . | Proof From the relation\n\n\[ \left\langle {{M}_{\varphi }f, g}\right\rangle = {\int }_{\mathcal{J}}{\varphi f}\bar{g}{dx} = {\int }_{\mathcal{J}}f\overline{\bar{\varphi }g}{dx} = \left\langle {f,{M}_{\bar{\varphi }}g}\right\rangle \]\n\nfor \( f, g \in \mathcal{D}\left( {M}_{\varphi }\right) = \mathcal{D}\left( {M}_{\... | Yes |
Proposition 1.6 Let \( S \) and \( T \) be linear operators from \( {\mathcal{H}}_{1} \) into \( {\mathcal{H}}_{2} \) such that \( \mathcal{D}\left( T\right) \) is dense in \( {\mathcal{H}}_{1} \) . Then:\n\n(i) \( {T}^{ * } \) is a closed linear operator from \( {\mathcal{H}}_{2} \) into \( {\mathcal{H}}_{1} \) . | Proof (i): Let \( {\left( {y}_{n}\right) }_{n \in \mathbb{N}} \) be a sequence from \( \mathcal{D}\left( {T}^{ * }\right) \) such that \( \mathop{\lim }\limits_{n}{y}_{n} = y \) in \( {\mathcal{H}}_{2} \) and \( \mathop{\lim }\limits_{n}{T}^{ * }{y}_{n} = v \) in \( {\mathcal{H}}_{1} \) . For \( x \in \mathcal{D}\left(... | Yes |
Proposition 1.7 Let \( T : {\mathcal{H}}_{1} \rightarrow {\mathcal{H}}_{2} \) and \( S : {\mathcal{H}}_{2} \rightarrow {\mathcal{H}}_{3} \) be linear operators such that \( \mathcal{D}\left( {ST}\right) \) is dense in \( {\mathcal{H}}_{1} \). (i) If \( \mathcal{D}\left( S\right) \) is dense in \( {\mathcal{H}}_{2} \), ... | Proof (i): Note that \( \mathcal{D}\left( T\right) \supseteq \mathcal{D}\left( {ST}\right) \) ; hence, \( \mathcal{D}\left( T\right) \) is dense in \( {\mathcal{H}}_{2} \) . Suppose that \( y \in \mathcal{D}\left( {{T}^{ * }{S}^{ * }}\right) \) . Let \( x \in \mathcal{D}\left( {ST}\right) \) . Then we have \( {Tx} \in ... | Yes |
Corollary 1.9 If \( T \) is a self-adjoint operator such that \( \mathcal{N}\left( T\right) = \{ 0\} \), then \( {T}^{-1} \) is also a self-adjoint operator. | Proof Since \( T = {T}^{ * } \), we have \( \mathcal{R}{\left( T\right) }^{ \bot } = \mathcal{N}\left( T\right) = \{ 0\} \) . Hence, \( \mathcal{R}\left( T\right) \) is dense, and the assertion follows from Theorem 1.8(iv). | Yes |
Lemma 1.10 For any densely defined linear operator \( T \) of \( {\mathcal{H}}_{1} \) into \( {\mathcal{H}}_{2} \), we have\n\n\[ \mathcal{G}\left( {T}^{ * }\right) = V{\left( \mathcal{G}\left( T\right) \right) }^{ \bot } = V\left( {\mathcal{G}{\left( T\right) }^{ \bot }}\right) . \] | Proof Let \( x \in \mathcal{D}\left( T\right) \) and \( y \in \mathcal{D}\left( {T}^{ * }\right) \) . By (1.5),\n\n\[ \langle V\left( {x,{Tx}}\right) ,\left( {y,{T}^{ * }y}\right) \rangle = \langle \left( {-{Tx}, x}\right) ,\left( {y,{T}^{ * }y}\right) \rangle = \langle - {Tx}, y{\rangle }_{2} + {\left\langle x,{T}^{ *... | Yes |
For each \( z \in \overline{\mathbb{C}} \), we have \( {\left( {S}_{z}\right) }^{ * } = {S}_{{\bar{z}}^{-1}} \), where we set \( {0}^{-1} \mathrel{\text{:=}} \infty \) and \( {\bar{\infty }}^{-1} \mathrel{\text{:=}} 0 \) . In particular, the operator \( {S}_{z} \) is self-adjoint if and only if \( \left| z\right| = 1 \... | We carry out the proof for \( z \in \mathbb{C}, z \neq 0 \) ; the cases \( z = 0 \) and \( z = \infty \) are treated similarly. For \( f \in \mathcal{D}\left( {S}_{z}\right) \) and \( g \in {H}^{1}\left( {a, b}\right) \), we use (1.13) to compute\n\n\[ \left\langle {{S}_{z}f, g}\right\rangle - \left\langle {f, - \mathr... | Yes |
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