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Theorem 7.30. Let \( G \) be a complex spectral family on \( H \) with \( G\left( {t + \mathrm{i}s}\right) = \) \( E\left( t\right) F\left( s\right) \) . | Proof.\n\n(a) For all \( s,{s}^{\prime }, t,{t}^{\prime } \in \mathbb{R} \n\n\[ \nG\left( {t + \mathrm{i}s}\right) G\left( {{t}^{\prime } + \mathrm{i}{s}^{\prime }}\right) = E\left( t\right) F\left( s\right) E\left( {t}^{\prime }\right) F\left( {s}^{\prime }\right) \n\]\n\n\[ \n= E\left( t\right) E\left( {t}^{\prime }\... | Yes |
Theorem 7.31 (The spectral theorem for bounded normal operators). Let \( H \) be a complex Hilbert space, and let \( T \in B\left( H\right) \) be a normal operator. Then there exists exactly one complex spectral family \( G \) for which\n\n\[ T = {\int }_{\mathbb{C}}z\mathrm{\;d}G\left( z\right) \] | Proof. If \( A \) and \( B \) are defined as in the theorem, then it is obvious that \( {A}^{ * } = A,{B}^{ * } = B, T = A + \mathrm{i}B \), and\n\n\[ {AB} = \frac{1}{4\mathrm{i}}\left( {{T}^{2} - T{T}^{ * } + {T}^{ * }T - {T}^{*2}}\right) = \frac{1}{4\mathrm{i}}\left( {{T}^{2} - {T}^{*2}}\right) = {BA}. \]\n\nThen we ... | Yes |
Assume \( T \) is a compact normal operator on the complex Hilbert space \( H,\left\{ {{\lambda }_{1},{\lambda }_{2},\ldots }\right\} \) are its non-zero eigenvalues, \( {P}_{j} \) is the orthogonal projection onto \( N\left( {{\lambda }_{j} - T}\right) ,{\lambda }_{0} = 0 \), and \( {P}_{0} \) is the orthogonal projec... | The proof goes as in the self-adjoint case. | No |
Theorem 7.34. Let \( T \) be a normal operator on a complex Hilbert space, and let \( G \) be the spectral family of \( T \) . (a) \( z \in \sigma \left( T\right) \) if and only if\n\n\[ G\left( {z + \epsilon + \mathrm{i}\epsilon }\right) + G\left( {z - \epsilon - \mathrm{i}\epsilon }\right) - G\left( {z + \epsilon - \... | (a) The proof is analogous to that of Theorem 7.22. Observe that \( \widehat{G}\left( {\chi }_{M}\right) = G\left( {{b}_{1} + \mathrm{i}{b}_{2}}\right) + G\left( {{a}_{1} + \mathrm{i}{a}_{2}}\right) - G\left( {{a}_{1} + \mathrm{i}{b}_{2}}\right) - G\left( {{b}_{1} + \mathrm{i}{a}_{2}}\right) \) for \( M = \left\{ {z \i... | No |
Theorem 7.35. Assume that \( {T}_{n}\left( {n \in \mathbb{N}}\right) \) and \( T \) are bounded normal operators on the complex Hilbert space \( H \) and \( \begin{Vmatrix}{T - {T}_{n}}\end{Vmatrix} \rightarrow 0 \) as \( n \rightarrow \infty \) . Then\n\n\[ \sigma \left( T\right) = \mathop{\lim }\limits_{{n \rightarro... | Proof. If \( z \notin \sigma \left( T\right) \), i.e., \( z \in \rho \left( T\right) \), then (second Corollary to Theorem 5.11) \( z \in \rho \left( {T}_{n}\right) \) for sufficiently large \( n \) and \( \begin{Vmatrix}{{\left( z - {T}_{n}\right) }^{-1} - {\left( z - T\right) }^{-1}}\end{Vmatrix} \rightarrow 0 \) . H... | Yes |
Theorem 7.36. If \( U \) is a unitary operator on a complex Hilbert space, then there exists a real spectral family \( E \) for which \( E\left( t\right) = 0 \) for \( t < 0, E\left( t\right) = I \) for \( t \geq {2\pi } \) and \( U = \int {\mathrm{e}}^{\mathrm{i}t}{dE}\left( t\right) \) (cf. also Exercise 7.46). | Proof. By Section 5.2, Example 2 the spectrum of \( U \) is contained in \( \{ z \in \mathbb{C} : \left| z\right| = 1\} = \left\{ {{\mathrm{e}}^{\mathrm{i}t} : 0 \leq t < {2\pi }}\right\} \), i.e., \( G\left( {\left\{ {{\mathrm{e}}^{\mathrm{i}t} : 0 \leq t < {2\pi }}\right\} = I}\right. \), where \( G \) denotes the co... | No |
Theorem 7.39. Let \( T \) be a self-adjoint operator on the complex Hilbert space \( H \), and let \( M \) be a closed subspace of \( H \). If \( {\mathrm{e}}^{\mathrm{i}{sT}}f \in M \) for all \( f \in M \) and \( s \in \mathbb{R} \), then \( M \) reduces \( T \) and \( {\mathrm{e}}^{\mathrm{i}{sT}}M = M,{\mathrm{e}}^... | Proof. We have \( {\mathrm{e}}^{\mathrm{i}{sT}}M = M \) for all \( s \in \mathbb{R} \), since\n\n\[{\mathrm{e}}^{\mathrm{i}{sT}}M \subset M\text{ for all }s \in \mathbb{R}\]\n\nby assumption, and because every \( f \in M \) can be written in the form\n\n\[f = {\mathrm{e}}^{\mathrm{i}{sT}}\left( {{\mathrm{e}}^{-\mathrm{... | Yes |
Theorem 7.41. Let \( T, S \), and \( T + S \) be self-adjoint operators on the Hilbert space \( H \) . Assume that these operators are bounded from below. Then\n\n\[ \n{\mathrm{e}}^{-t\left( {T + S}\right) } = s - \mathop{\lim }\limits_{{n \rightarrow \infty }}{\left\lbrack {\mathrm{e}}^{-\left( {t/n}\right) T}{\mathrm... | The proof follows that of Theorem 7.40. We consider only nonnegative \( t \) and \( s \) ; the details can be left to the reader. | No |
Theorem 8.1. The defect index \( \beta \left( {T, z}\right) \) is constant on each connected subset of \( \mathbf{\Gamma }\left( T\right) \) . If \( T \) is Hermitian, then the defect index is constant in the upper and lower half-planes. | Proof. It is sufficient to show that \( \beta \left( {T, z}\right) \) is locally constant in \( \Gamma \left( T\right) \), i.e., that for every \( {z}_{0} \in \Gamma \left( T\right) \) there exists an \( \epsilon > 0 \) such that \( \beta \left( {T, z}\right) = \beta \left( {T,{z}_{0}}\right) \) for all \( z \in \Gamma... | Yes |
Theorem 8.2. Let \( T \) be a symmetric operator on the complex Hilbert space \( H \) . The Cayley transform of \( T \) is an isometric mapping of \( R\left( {-\mathrm{i} - T}\right) \) onto \( R\left( {\mathrm{i} - T}\right) \) . The range \( R\left( {I - V}\right) \) is dense in \( H \), and \( T = \mathrm{i}\left( {... | Proof. For every \( g = \left( {-\mathrm{i} - T}\right) f \in R\left( {-\mathrm{i} - T}\right) = D\left( V\right) \) we have\n\n\[ \parallel {Vg}{\parallel }^{2} = {\begin{Vmatrix}\left( \mathrm{i} - T\right) {\left( -\mathrm{i} - T\right) }^{-1}g\end{Vmatrix}}^{2} = \parallel \left( {\mathrm{i} - T}\right) f{\parallel... | Yes |
An operator \( V \) on the complex Hilbert space \( H \) is the Cayley transform of a symmetric operator \( T \) if and only if \( V \) has the following properties:\n\n(i) \( V \) is an isometric mapping of \( D\left( V\right) \) onto \( R\left( V\right) \),\n\n(ii) \( R\left( {I - V}\right) \) is dense in \( H \) .\n... | Proof. If \( V \) is the Cayley transform of \( T \), then \( V \) has properties (i) and (ii) by Theorem 8.2. We also have then that \( T = \mathrm{i}\left( {I + V}\right) {\left( I - V\right) }^{-1} \) . Let \( V \) now be an operator with properties (i) and (ii). Then \( I - V \) is injective, since the equality \( ... | Yes |
Theorem 8.4. Let \( T \) be a symmetric operator on a complex Hilbert space, and let \( V \) denote its Cayley transform.\n\n(a) The following statements are equivalent:\n\n(i) \( T \) is closed,\n\n(ii) \( V \) is closed,\n\n(iii) \( D\left( V\right) = R\left( {\mathrm{i} + T}\right) \) is closed,\n\n(iv) \( R\left( V... | Proof.\n\n(a) (i) is equivalent to (iii) and to (iv): \( T \) is closed if and only if \( ( \pm \mathrm{i} - \) \( T{)}^{-1} \) is closed. The bounded operator \( {\left( \pm \mathrm{i} - T\right) }^{-1} \) is closed if and only if its domain \( D\left( {\left( \mathrm{i} - T\right) }^{-1}\right) = R\left( {\mathrm{i} ... | Yes |
Theorem 8.6. Let \( T \) be a closed symmetric operator on a complex Hilbert space, and let \( V \) denote its Cayley transform.\n\n(a) \( {V}^{\prime } \) is the Cayley transform of a closed symmetric extension \( {T}^{\prime } \) of \( T \) if and only if the following holds: There exist closed subspaces \( {F}_{ - }... | Proof.\n\n(a) If \( {V}^{\prime } \) has the given form, then \( {V}^{\prime } \) is obviously an isometric mapping of \( R\left( {-\mathrm{i} - T}\right) \oplus {F}_{ + } \) onto \( R\left( {\mathrm{i} - T}\right) \oplus {F}_{ - } \) . Consequently, \( {V}^{\prime } \) satisfies assumption (i) of Theorem 8.3. Since \(... | Yes |
Theorem 8.7. Let \( T \) be a symmetric operator on a complex Hilbert space. The operator \( T \) is essentially self-adjoint if and only if \( T \) has exactly one self-adjoint extension. | Proof. If \( T \) is essentially self-adjoint, then \( \bar{T} \) is the only self-adjoint extension of \( T \) by Theorem 5.31(c). We show: If \( T \) is not essentially self-adjoint, i.e., if \( \bar{T} \) is not self-adjoint, then \( T \) has either no or infinitely many self-adjoint extensions. If the defect indice... | No |
Theorem 8.8. Let \( T \) be a symmetric operator on a complex Hilbert space.\n\n(a) If \( \Gamma \left( T\right) \cap \mathbb{R} \neq \varnothing \), then \( T \) has self-adjoint extensions.\n\n(b) If \( T \) is semibounded, then \( T \) has self-adjoint extensions. | Proof.\n\n(a) \( \Gamma \left( T\right) \) is connected, since \( \Gamma \left( T\right) \cap \mathbb{R} \neq \varnothing \) . Then \( {\gamma }_{ + }\left( T\right) = {\gamma }_{ - }\left( T\right) \) by Theorem 8.1.\n\n(b) Let \( T \) be bounded, for example, from below, and let \( c \) be a lower\n\nbound of \( T \)... | Yes |
Theorem 8.9. Let \( H \) be a complex Hilbert space, and let \( K \) be a conjugation on \( H \). If \( T \) is a \( K \)-real symmetric operator on \( H \), then \( T \) possesses self-adjoint extensions. | Proof. It follows from (8.1(b)) and (8.2(a)) that \( D\left( T\right) \supset {KD}\left( T\right) \supset {K}^{2}D\left( T\right) \) \( = D\left( T\right) \). Consequently, \( {KD}\left( T\right) = D\left( T\right) \). If \( f \in R{\left( \mathrm{i} - T\right) }^{ \bot } \), then\n\n\[ \langle {Kf},\left( {-\mathrm{i}... | Yes |
Example 3. The formula\n\n\\[ \nK\\left( {{f}_{1},{f}_{2}}\\right) = \\left( {{f}_{1}^{ * },{f}_{2}^{ * }}\\right) \\;\\text{ for }\\;\\left( {{f}_{1},{f}_{2}}\\right) \\in {L}_{2}\\left( M\\right) \\oplus {L}_{2}\\left( M\\right) \n\\]\n\ndefines a conjugation on \\( {L}_{2}\\left( M\\right) \\oplus {L}_{2}\\left( M\\... | is symmetric, since\n\n\\[ \n\\left\\langle {T\\left( {{f}_{1},{f}_{2}}\\right) ,\\left( {{g}_{1},{g}_{2}}\\right) }\\right\\rangle = \\int {f}_{2}^{\\prime * }{g}_{1}\\mathrm{\\;d}x - \\int {f}_{1}^{\\prime * }{g}_{2}\\mathrm{\\;d}x \n\\]\n\n\\[ \n= - \\int {f}_{2}^{ * }{g}_{1}^{\\prime }\\mathrm{d}x + \\int {f}_{1}^{... | Yes |
Theorem 8.10. Let \( T \) be a closed symmetric operator on the complex Hilbert space \( H \) with equal finite defect indices \( \left( {m, m}\right) \) . If \( {T}_{1} \) and \( {T}_{2} \) are self-adjoint extensions of \( T \), then \( {\left( z - {T}_{1}\right) }^{-1} - {\left( z - {T}_{2}\right) }^{-1} \) is of ra... | Proof. Every \( z \in \rho \left( {T}_{1}\right) \cap \rho \left( {T}_{2}\right) \) obviously belongs to \( \mathbf{\Gamma }\left( T\right) \) ; consequently, \( R{\left( z - T\right) }^{ \bot } \) is \( m \) -dimensional. Since \( {\left( z - {T}_{1}\right) }^{-1}f = {\left( z - {T}_{2}\right) }^{-1}f = {\left( z - T\... | Yes |
Theorem 8.11 (The first formula of von Neumann). Let \( T \) be a closed symmetric operator on a complex Hilbert space. Then\n\n\[ D\left( {T}^{ * }\right) = D\left( T\right) \dot{ + }{N}_{ + }\dot{ + }{N}_{ - }\text{(direct sum),} \]\n\n\[ {T}^{ * }\left( {{f}_{0} + {g}_{ + } + {g}_{ - }}\right) = T{f}_{0} + \mathrm{i... | Proof. Since \( {N}_{ + } \subset D\left( {T}^{ * }\right) \) and \( {N}_{ - } \subset D\left( {T}^{ * }\right) \), we obviously have \( D\left( T\right) + \) \( {N}_{ + } + {N}_{ - } \subset D\left( {T}^{ * }\right) \) . We show that we have equality here, i.e., every \( f \in D\left( {T}^{ * }\right) \) can be writte... | Yes |
Theorem 8.12 (The second formula of von Neumann). Let \( T \) be a closed symmetric operator on a complex Hilbert space.\n\n(a) \( {T}^{\prime } \) is a closed symmetric extension of \( T \) if and only if the following holds: There are closed subspaces \( {F}_{ + } \) of \( {N}_{ + } \) and \( {F}_{ - } \) of \( {N}_{... | Proof. This theorem immediately follows from Theorem 8.6 if we show that the operator \( {T}^{\prime } \) of Theorem 8.6 can be represented in the above form. We have (with \( \widetilde{V} \) as in Theorem 8.6)\n\n\[ D\left( {T}^{\prime }\right) = R\left( {I - {V}^{\prime }}\right) = \left( {I - {V}^{\prime }}\right) ... | Yes |
Theorem 8.13. Let \( T \) be a closed symmetric operator on a complex Hilbert space, and let \( {T}^{\prime } \) be a symmetric extension of \( T \) .\n\n(a) \( {T}^{\prime } \) is an m-dimensional extension if and only if \( {F}_{ + } \) (defined in Theorem 8,12) is \( m \) -dimensional.\n\n(b) If \( T \) has defect i... | Proof.\n\n(a) As \( D\left( {T}^{\prime }\right) = D\left( T\right) + \left( {I + \widehat{V}}\right) {F}_{ + } \) is a direct sum, we have \( \dim D\left( {T}^{\prime }\right) /D\left( T\right) = \dim \left( {I + \widehat{V}}\right) {F}_{ + } \) . Since \( \widehat{V}{F}_{ + } = {F}_{ - } \subset {N}_{ - },\;{F}_{ + }... | Yes |
Consider the operator \( \overline{{T}_{1,0}} \) of Section 6.4 (cf. Theorems 6.29 and 6.31) defined on \( {L}_{2}\left( {0,\infty }\right) \) by \( D\left( \overline{{T}_{1,0}}\right) = \left\{ {f \in {W}_{2,1}\left( {0,\infty }\right) : f\left( 0\right) = 0}\right\} \) and \( \overline{{T}_{1,0}}f = \frac{1}{\mathrm{... | We have \( \overline{{T}_{1,0}^{ * }} = {T}_{1,0}^{ * } = {T}_{1} \), where \( D\left( {T}_{1}\right) = {W}_{2,1}\left( {0,\infty }\right) \text{ and }{T}_{1}f = \frac{1}{\mathrm{i}}{f}^{\prime }\text{ for }f \in D\left( {T}_{1}\right) \). Then \( {N}_{ + } = N\left( {\mathrm{i} - {T}_{1}}\right) \) is the set of those... | Yes |
Consider the operator \( \overline{{T}_{1,0}} \) from Section 6.4 (cf. Theorem 6.31) defined on \( {L}_{2}\left( {a, b}\right) , - \infty < a < b < \infty \) by the formulae\n\n\[ D\left( \overline{{T}_{1,0}}\right) = \left\{ {f \in {W}_{2,1}\left( {a, b}\right) : f\left( a\right) = f\left( b\right) = 0}\right\} \]\n\n... | Proof. For every \( f = {f}_{0} + c{e}_{ + } + c{\mathrm{e}}^{\mathrm{i}\vartheta }{\mathrm{e}}_{ - } \in D\left( {\mathrm{\;S}}_{\vartheta }\right) \) with \( {f}_{0} \in D\left( \overline{{T}_{1,0}}\right) \) we have\n\n\[ \frac{f\left( a\right) }{f\left( b\right) } = \frac{c{e}_{ + }\left( a\right) + c{\mathrm{e}}^{... | Yes |
Theorem 8.16. If \( {T}^{\prime } \) is an \( m \) -dimensional extension of \( T \), then\n\n\[ \dim \left( {N\left( {\lambda - {T}^{\prime }}\right) \ominus N\left( {\lambda - T}\right) }\right) \leq m. \]\n\nIf, in addition, \( n\left( {T,\lambda }\right) < \infty \), then \( n\left( {{T}^{\prime },\lambda }\right) ... | Proof. It is obvious that \( N\left( {\lambda - T}\right) \subset N\left( {\lambda - {T}^{\prime }}\right) \) . The formula\n\n\[ \left( {N\left( {\lambda - {T}^{\prime }}\right) \ominus N\left( {\lambda - T}\right) }\right) \cap D\left( T\right) = \{ 0\} \]\n\nimplies\n\n\[ \left( {N\left( {\lambda - {T}^{\prime }}\ri... | Yes |
Theorem 8.17. Let \( T \) be a closed symmetric operator.\n\n(a) We have \( {S}_{e}\left( T\right) \subset S\left( T\right) \subset \mathbb{R} \) and \( S\left( T\right) \subset \sigma \left( T\right) \). | (a) If \( \lambda \in {S}_{e}\left( T\right) \), then \( \dim N\left( {\lambda - T}\right) = \infty \) or \( {\left( \lambda - {T}_{\lambda }\right) }^{-1} \) is unbounded. It is clear that in both cases \( \lambda \) does not lie in \( \Gamma \left( T\right) \), i.e., \( \lambda \in S\left( T\right) \) . It is also ev... | Yes |
Theorem 8.18. Let \( T \) be a closed symmetric operator on a complex Hilbert space with equal finite defect indices. Then all self-adjoint extensions of \( T \) have the same essential spectrum. If some self-adjoint extension of \( T \) has a pure discrete spectrum, then all self-adjoint extensions of \( T \) do, too. | Proof. The first assertion immediately follows from Theorem 8.17(c) and (d). The second assertion follows from the fact that the spectrum is discrete if and only if the essential spectrum is empty. | "No" |
Theorem 8.19. Let \( T \) be a closed symmetric operator on a complex Hilbert space with equal finite defect indices \( \left( {m, m}\right) \) and assume that \[ \parallel \left( {\lambda - T}\right) f\parallel \geq c\parallel f\parallel \;\text{ for all }\;f \in D\left( T\right) \] with some \( \lambda \in \mathbb{R}... | Proof. By the first proposition after Theorem 7.24 we only have to prove that \( \dim R\left( {E\left( {\lambda + c - }\right) - E\left( {\lambda - c}\right) }\right) \leq m \) for the spectral family \( E \) of \( {T}^{\prime } \) . Assume that \( \dim R\left( {E\left( {\lambda + c - }\right) - E\left( {\lambda - c}\r... | Yes |
Let \( T \) be a closed symmetric operator on a complex Hilbert space with finite defect indices \( \left( {m, m}\right) \), and let \( {T}_{1} \) and \( {T}_{2} \) be self-adjoint extensions of \( T \) . If \( \sigma \left( {T}_{1}\right) \cap \left( {a, b}\right) = \varnothing \), then \( \sigma \left( {T}_{2}\right)... | If \( - \infty < a < b < \infty \), then \( T \) satisfies the assumptions of Theorem 8.19 with \( \lambda = \left( {a + b}\right) /2 \) and \( c = \left( {b - a}\right) /2 \), since for all \( f \in D\left( T\right) \)\n\n\[ \parallel \left( {\lambda - T}\right) f{\parallel }^{2} = {\begin{Vmatrix}\left( \lambda - {T}... | Yes |
Corollary 2. If \( T \) is a closed symmetric operator on a complex Hilbert space, bounded from below with lower bound \( \gamma \) and finite defect indices \( \left( {m, m}\right) \), and \( {T}^{\prime } \) is a self-adjoint extension of \( T \), then \( \sigma \left( {T}^{\prime }\right) \cap \left( {-\infty ,\gamm... | Proof. Theorem 8.19 can be applied with any \( \lambda < \gamma \) and \( c = \gamma - \lambda \), since\n\n\[ \parallel \left( {\lambda - T}\right) f\parallel \geq \langle f,\left( {T - \lambda }\right) f\rangle \parallel f{\parallel }^{-1} \geq \left( {\gamma - \lambda }\right) \parallel f\parallel \]\n\nfor all \( f... | Yes |
Theorem 8.20. Let \( L \) be as in (8.8). The operator \( {T}_{0} \) is symmetric. If \( s = 0 \), then \( {T}_{0} \) has equal defect indices, i.e., \( {T}_{0} \) has self-adjoint extensions. | Proof. The Hermitian character of \( {T}_{0} \) follows by integration by parts. \( D\left( {T}_{0}\right) \) is dense, because \( {C}_{0}^{\infty }\left( {a, b}\right) \subset D\left( {T}_{0}\right) \). Therefore, \( {T}_{0} \) is symmetric. If \( s = 0 \), then \( {T}_{0} \) is \( K \) -real for the natural conjugati... | Yes |
Theorem 8.21. Let \( {L}_{2,0}\left( {a, b, r}\right) \) be the subspace of those functions in \( {L}_{2}\left( {a, b, r}\right) \) that vanish almost everywhere near \( a \) and \( b \) . Then\n\n\[ R\left( {T}_{0}\right) = \left\{ {k \in {L}_{2,0}\left( {a, b, r}\right) : {\int }_{a}^{b}u{\left( x\right) }^{ * }k\lef... | Proof. We denote the subspace on the right hand side by \( R \) . For \( f \in D\left( {T}_{0}\right) \) and for every solution \( u \) of the equation \( {Lu} = 0 \) we obtain via integration\n\n\( {}^{3} \) Concerning the results mentioned here about ordinary differential equations we refer to the textbooks on this s... | No |
Theorem 8.22. We have \( {T}_{0}^{ * } = T \) . The operator \( {T}_{0} \) is essentially self-adjoint if and only if \( T \) is symmetric. Then \( {\bar{T}}_{0} = T \) . | Proof. Integration by parts shows that \( {T}_{0} \) and \( T \) are formal adjoints of each other. To prove that \( {T}_{0}^{ * } = T \), it remains to prove that \( D\left( {T}_{0}^{ * }\right) \subset D\left( T\right) \) . Let \( f \in D\left( {T}_{0}^{ * }\right) \) . Then \( g = {T}_{0}^{ * }f \) is locally integr... | Yes |
Theorem 8.23. The defect index \( {\gamma }_{ + } = {\gamma }_{ + }\left( {T}_{0}\right) \left( {{\gamma }_{ - } = {\gamma }_{ - }\left( {T}_{0}\right) }\right) \) is equal to the number of linearly independent solutions of the equation \( \left( {L + \mathrm{i}}\right) u \) \( = 0\left( {\left( {L - \mathrm{i}}\right)... | Proof. We have \( R{\left( \mathrm{i} - {T}_{0}\right) }^{ \bot } = N\left( {\mathrm{i} + T}\right) \) and \( R{\left( -\mathrm{i} - {T}_{0}\right) }^{ \bot } = N\left( {-\mathrm{i} + T}\right) \) . Furthermore, \( N\left( {\pm \mathrm{i} + T}\right) \) is equal to the set of those solutions of the equation \( \left( {... | Yes |
Theorem 8.25. Let \( L \) be a regular differential form of the kind (8.8). Then we have the following:\n\n(a) For every \( f \in D\left( T\right) \) the functions \( f \) and \( {f}^{\prime } \) are continuously extendible to \( \left\lbrack {a, b}\right\rbrack \) . For \( f, g \in D\left( T\right) \) we have\n\n\[{\l... | Proof.\n\n(a) If \( f \in D\left( T\right) \) and \( g = {Tf} \), then \( f \) can be represented in the form (8.10) with a fundamental system \( {u}_{1},{u}_{2} \) of the equation \( {Lu} = 0 \) . As the functions \( {u}_{j} \) and \( {u}_{j}^{\prime } \) are continuously extendible to \( \left\lbrack {a, b}\right\rbr... | Yes |
Theorem 8.27 (The Weyl alternative). Let \( L \) be a Sturm-Liouville differential form defined on \( \left( {a, b}\right) \), and let \( c \in \left( {a, b}\right) \) . Either every solution \( u \) of the equation \( \left( {L - z}\right) u = 0 \) lies in \( {L}_{2}\left( {c, b, r}\right) \) for every \( z \in \mathb... | Proof. In order to prove the alternative, it is sufficient to show the following: If there exists a \( {z}_{0} \in \mathbb{C} \) such that \( u \in {L}_{2}\left( {c, b, r}\right) \) for every solution \( u \) of the equation \( \left( {L - {z}_{0}}\right) u = 0 \), then this holds for all \( z \in \mathbb{C} \) . Let \... | Yes |
Theorem 8.29. Let \( L \) be a Sturm-Liouville differential form (8.13). Moreover, let \( \lambda \in \mathbb{R} \), and let \( v \) and \( w \) be real solutions of the equation \( \left( {L - \lambda }\right) u = 0 \) . (a) The operator \( {T}_{v, w} \) defined by the formulae\n\n\[ D\left( {T}_{v, w}\right) = \left\... | (a) If we have the limit point case at both boundary points, then\n\n\[ \langle f,{Tg}\rangle - \langle {Tf}, g\rangle = {\left\lbrack f, g\right\rbrack }_{b} - {\left\lbrack f, g\right\rbrack }_{a} = 0\text{ for all }f, g \in D\left( T\right) \]\n\nby Auxiliary theorem 8.28. Consequently, \( T \) is symmetric and thus... | Yes |
Theorem 8.30. If \( S \) is a self-adjoint operator and \( f \) is an analytic vector of \( S \) , then\n\n\[ f \in D\left( {\mathrm{e}}^{zS}\right) \text{ and }{\mathrm{e}}^{zS}f = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{z}^{n}}{n!}{S}^{n}f \]\n\nfor every \( z \in \mathbb{K} \) such that \( \left| z\right| < ... | Proof. Let \( E \) denote the spectral family of \( S \) . Then\n\n\[ {\left\{ {\int }_{-M}^{M}{\left| {\mathrm{e}}^{zs}\right| }^{2}\mathrm{\;d}\parallel E\left( s\right) f{\parallel }^{2}\right\} }^{1/2} = \begin{Vmatrix}{{\int }_{-M}^{M}{\mathrm{e}}^{zs}\mathrm{\;d}E\left( s\right) f}\end{Vmatrix} \]\n\n\[ = \begin{... | Yes |
Let \( {H}_{1},{H}_{2},{T}_{1},{T}_{2}, A \) and \( B \) be as above. (a) \( A \) is different from zero (i.e., there exists an \( f \in D\left( A\right) \) such that \( {Af} \neq 0 \) ) if and only if \( {T}_{1} \) and \( {T}_{2} \) are different from zero. If \( A \) is different from zero, then \( A \) is bounded if... | (a) If \( {T}_{1} \) and \( {T}_{2} \) are different from zero, then there are elements \( {f}_{1} \in D\left( {T}_{1}\right) \) and \( {f}_{2} \in D\left( {T}_{2}\right) \) for which \( {T}_{1}{f}_{1} \neq 0 \) and \( {T}_{2}{f}_{2} \neq 0 \) . Hence \( A\left( {{f}_{1} \otimes {f}_{2}}\right) \neq 0 \) , i.e., \( A \... | Yes |
Theorem 8.35. Let \( {T}_{1} \) and \( {T}_{2} \) be self-adjoint operators on complex Hilbert spaces, and let \( B = {T}_{1} \otimes {I}_{2} + {I}_{1} \otimes {T}_{2} \) . Then\n\n\[ \exp \left( {\mathrm{i}t\bar{B}}\right) = \overline{\exp \left( {\mathrm{i}t{T}_{1}}\right) \otimes \exp \left( {\mathrm{i}t{T}_{2}}\rig... | Proof. For all simple tensors \( f \otimes g \) such that \( f \in D\left( {T}_{1}\right) \) and \( g \in D\left( {T}_{2}\right) \) we have \( f \otimes g \in D\left( \bar{B}\right) \), and thus\n\n\[ \frac{\mathrm{d}}{\mathrm{d}t}\left\lbrack {\exp \left( {\mathrm{i}t\bar{B}}\right) \left( {f \otimes g}\right) }\right... | Yes |
Let \( T \) be self-adjoint and bounded from below with lower bound \( {\gamma }_{T} \) . Let \( V \) be symmetric and \( T \) -bounded with \( T \) -bound \( < 1 \) . Then \( T + V \) is self-adjoint and bounded from below. If \[ \parallel {Vf}\parallel \leq a\parallel f\parallel + b\parallel {Tf}\parallel \;\text{ fo... | Proof. By Corollary 2 to Theorem 7.22 it is sufficient to show that \( \left( {-\infty ,\gamma }\right) \) is contained in \( \rho \left( {T + V}\right) \), i.e., that the operator \( T + V - \lambda = \left( {T - \lambda }\right) + V \) is bijective for every \( \lambda < \gamma \) . By Theorem 5.11, this is surely th... | Yes |
Theorem 9.2. Let \( T \) be self-adjoint and bounded from below, and let \( V \) be symmetric and \( T \) -bounded. If \( T + {\mu V} \) is closed for all \( \mu \in \left\lbrack {0,1}\right\rbrack \), then \( T + V \) is self-adjoint and bounded from below. | Proof. The operator \( T + V \) is self-adjoint by Theorem 5.27. For every \( \mu \in \left\lbrack {0,1}\right\rbrack \) the operator \( V \) is relatively bounded with respect to \( T + {\mu V} \), i.e., there exist \( {a}_{\mu } \geq 0 \) and \( {b}_{\mu } \geq 0 \) for which\n\n\[ \parallel {Vf}\parallel \leq {a}_{\... | Yes |
Theorem 9.3 (Heinz). Assume that \( T \) is self-adjoint, non-negative, \( S \) is symmetric, \( D\left( T\right) \subset D\left( S\right) \), and \( \parallel {Sf}\parallel \leq \parallel {Tf}\parallel \) for all \( f \in D\left( T\right) \) . Then \[ \left| {\langle f,{Sf}\rangle }\right| \leq \langle f,{Tf}\rangle \... | Proof. Theorem 9.1 applies with \( V = {\kappa S} \) for every \( \kappa \in \left( {-1,1}\right) \) if we take \( a = 0, b = \left| \kappa \right| \), and \( {\gamma }_{T} = 0 \) . Then \( \gamma = 0 \) . Consequently, \( T + {\kappa S} \) is self-adjoint and non-negative for every \( \kappa \in \left( {-1,1}\right) \... | Yes |
Theorem 9.4. Let \( S \) and \( T \) be self-adjoint and non-negative.\n\n(a) \( D\left( T\right) \subset D\left( S\right) \) and \( \parallel {Sf}\parallel \leq \parallel {Tf}\parallel \) for all \( f \in D\left( T\right) \) imply \( D\left( {T}^{1/2}\right) \) \( \subset D\left( {S}^{1/2}\right) \) and \( \begin{Vmat... | (a) It follows from Theorem 9.3 that\n\n\[ \n{\begin{Vmatrix}{S}^{1/2}f\end{Vmatrix}}^{2} = \langle f,{Sf}\rangle \leq \langle f,{Tf}\rangle = {\begin{Vmatrix}{T}^{1/2}f\end{Vmatrix}}^{2}\text{ for all }f \in D\left( T\right) .\n\]\n\nLet \( f \in D\left( {T}^{1/2}\right) \) . Since \( D\left( T\right) \) is a core of ... | Yes |
Theorem 9.6. Let \( {T}_{1} \) and \( {T}_{2} \) be self-adjoint operators with the spectral families \( {E}_{1} \) and \( {E}_{2} \) . (a) If \( R\left( {{E}_{1}\left( J\right) }\right) \subset D\left( {T}_{2}\right) \) and \( \left( {{T}_{1} - {T}_{2}}\right) {E}_{1}\left( J\right) \) is compact for every bounded int... | Proof. (a) Assume that \( \lambda \in {\sigma }_{e}\left( {T}_{1}\right) \) . As in the proof of Theorem 7.24 (part (i) implies (ii)) we can show that there exists a singular sequence \( \left( {f}_{n}\right) \) for \( {T}_{1} \) and \( \lambda \) that is contained in \( R\left( {{E}_{1}\left( {\lambda + 1}\right) - {E... | Yes |
Theorem 9.7. Let \( A \) be an operator from \( {H}_{1} \) into \( {H}_{2} \), and let \( B \) be an \( A \) -compact operator from \( {H}_{1} \) into \( {H}_{3} \). If \( A \) or \( B \) is closable, then \( B \) is A-bounded with A-bound zero. | Proof. Let us assume that the \( A \) -bound of \( B \) is positive. Then there exists an \( \epsilon > 0 \) with the property that for every \( n \in \mathbb{N} \) there is an \( {f}_{n} \in D\left( A\right) \) such that \( \begin{Vmatrix}{B{f}_{n}}\end{Vmatrix} > n\begin{Vmatrix}{f}_{n}\end{Vmatrix} + \epsilon \begin... | Yes |
Theorem 9.8. Let \( A \) be a closed operator from \( {H}_{1} \) into \( {H}_{2} \), and let \( B \) be an operator from \( {H}_{1} \) into \( {H}_{3} \) . Then the following assertions are equivalent.\n\n(i) \( B \) is \( A \) -compact.\n\n(ii) \( {f}_{n} \in D\left( A\right) ,{f}_{n}\overset{w}{ \rightarrow }0 \) and... | Proof. (i) implies (ii): If \( {f}_{n}\overset{w}{ \rightarrow }0 \) and \( A{f}_{n}\overset{w}{ \rightarrow }0 \), then\n\n\[{\left\langle {f}_{n}, g\right\rangle }_{A} = \left\langle {{f}_{n}, g}\right\rangle + \left\langle {A{f}_{n},{Ag}}\right\rangle \rightarrow 0\]\n\nfor all \( g \in D\left( A\right) \), i.e., \(... | Yes |
Theorem 9.9. Let \( T \) be a self-adjoint operator on the Hilbert space \( H \), and let \( V \) be a symmetric \( T \) -compact operator. Then \( T + V \) is self-adjoint, \( T \) and \( T + V \) have the same singular sequences, and \( {\sigma }_{e}\left( T\right) = {\sigma }_{e}\left( {T + V}\right) \) . | Proof. By Theorem 9.7 the operator \( V \) is \( T \) -bounded with \( T \) -bound 0 . Therefore, \( T + V \) is self-adjoint by Theorem 5.28. \( V \) is also \( \left( {T + V}\right) \) - compact by the corollary to Theorem 9.7. Now it follows from Theorem 9.8 that \( {T}_{1} = T \) and \( {T}_{2} = T + V \) satisfy t... | Yes |
Theorem 9.10. Let \( T \) be a self-adjoint operator on \( H \) such that \( \rho \left( T\right) \neq \varnothing \), and let \( p > 0 \) . An operator \( V \) is \( {T}^{p} \) -compact (respectively \( {T}^{p} \) -bounded) if and only if \( V{\left( z - T\right) }^{-p} \) is compact (respectively bounded) for some (a... | Proof. We obviously have \( D\left( {T}^{p}\right) = D\left( {\left( z - T\right) }^{p}\right) \) ; the \( {T}^{p} \) -norm and the \( {\left( z - T\right) }^{p} \) -norm are equivalent. Consequently, \( V \) is \( {T}^{p} \) -compact \( \left( {T}^{p}\right. \) - bounded) if and only if it is \( {\left( z - T\right) }... | Yes |
Theorem 9.11. Let \( T \) be a self-adjoint operator with spectral family \( E \), and let \( V \) be a \( T \) -bounded operator. Then\n\n(a) \( V \) is \( {T}^{p} \) -bounded with \( {T}^{p} \) -bound zero for all \( p > 1 \) . | Proof.\n\n(a) There are numbers \( a, b \geq 0 \) such that \( \parallel {Vf}\parallel \leq a\parallel f\parallel + b\parallel {Tf}\parallel \) for all \( f \in D\left( T\right) \) . We have \( D\left( {T}^{p}\right) \subset D\left( T\right) \) for \( p > 1 \), and thus\n\n\[ \parallel {Vf}\parallel < \parallel V\left(... | Yes |
Theorem 9.12. Let \( {T}_{1} \) and \( {T}_{2} \) be self-adjoint operators and assume that \( D\left( {T}_{1}\right) = D\left( {T}_{2}\right) \) . Put \( V = {T}_{2} - {T}_{1} \) . (a) \( V \) is \( {T}_{1}^{2} \) -compact if and only if it is \( {T}_{2}^{2} \) -compact. | Proof. Write \( {R}_{j} = {\left( z - {T}_{j}\right) }^{-1} \) for \( z \in \rho \left( {T}_{1}\right) \cap \rho \left( {T}_{2}\right) \) . Then the operators \( V{R}_{j} \) are bounded for \( j = 1,2 \) . (If \( H \) is real and \( \sigma \left( {T}_{1}\right) \cup \sigma \left( {T}_{2}\right) = \mathbb{R} \), then \(... | Yes |
Theorem 9.13. Let \( T \) be a self-adjoint operator on \( H \), and denote its spectral family by \( E \) . Assume that \( V \) is symmetric, \( D\left( T\right) \subset D\left( V\right) \), and \( T + V \) is self-adjoint. Assume, furthermore, that \( {VE}\left( J\right) \) is compact for every bounded interval \( J ... | Proof. It follows from the assumptions that \( V \) is \( {T}^{2} \) -compact, and thus also \( {\left( T + V\right) }^{2} \) -compact by Theorem 9.12. Therefore \( V{E}^{\prime }\left( J\right) \) is also compact for every bounded interval \( J \), where \( {E}^{\prime } \) denotes the spectral family of \( T + V \) .... | Yes |
Theorem 9.15. Let \( {T}_{n}\left( {n \in \mathbb{N}}\right) \) and \( T \) be self-adjoint operators on the complex Hilbert space \( H \) . If \( {\left( {z}_{0} - {T}_{n}\right) }^{-1}\overset{s}{ \rightarrow }{\left( {z}_{0} - T\right) }^{-1} \) for some \( {z}_{0} \in \mathbb{C} \smallsetminus \mathbb{R} \), then \... | Proof. If \( z \in \mathbb{C} \) and \( \left| {z - {z}_{0}}\right| < \left| {\operatorname{Im}{z}_{0}}\right| \), then by Theorem 5.14 we have\n\n\[{\left( z - {T}_{n}\right) }^{-1}f - {\left( z - T\right) }^{-1}f = \mathop{\sum }\limits_{{k = 0}}^{\infty }{\left( {z}_{0} - z\right) }^{k}\left\lbrack {{\left( {z}_{0} ... | Yes |
Theorem 9.16. Let \( {T}_{n}\left( {n \in \mathbb{N}}\right) \) and \( T \) be self-adjoint operators on the complex Hilbert space \( H \) . The sequence \( \left( {T}_{n}\right) \) converges to \( T \) in the sense of the strong resolvent convergence if one of the following assumptions is satisfied:\n\n(i) There is a ... | Proof.\n\n(i) We have\n\n\[ {\left( \mathrm{i} - {T}_{n}\right) }^{-1}f - {\left( \mathrm{i} - T\right) }^{-1}f \]\n\n\[ = {\left( \mathrm{i} - {T}_{n}\right) }^{-1}\left( {{T}_{n} - T}\right) {\left( \mathrm{i} - T\right) }^{-1}f \rightarrow 0\;\text{ as }\;n \rightarrow \infty \]\n\nfor all \( f \in H \) such that \(... | Yes |
Theorem 9.18. Let \( {T}_{n}\left( {n \in \mathbb{N}}\right) \) and \( T \) be self-adjoint operators on the complex Hilbert space \( H \) . Assume that \( {\left( \mathrm{i} - {T}_{n}\right) }^{-1}\overset{s}{ \rightarrow }{\left( \mathrm{i} - T\right) }^{-1} \) .\n\n(a) \( {\mathrm{e}}^{\mathrm{i}t{T}_{n}}\overset{s}... | Proof.\n\n(a) The function \( s \mapsto {\mathrm{e}}^{\mathrm{i}{ts}} \) is continuous and bounded on \( \mathbb{R} \) . The assertion therefore follows from Theorem 9.17.\n\n(b) With \( u\left( s\right) = {\mathrm{e}}^{-{ts}} \) for \( s \geq \gamma \) and \( u\left( s\right) = {\mathrm{e}}^{-{t\gamma }} \) for \( s <... | Yes |
Theorem 10.1. We have \( {F}_{0}S\left( {\mathbb{R}}^{m}\right) \subset S\left( {\mathbb{R}}^{m}\right) \) . For every \( f \in S\left( {\mathbb{R}}^{m}\right) \) and every multiindex \( \alpha \)\n\n\[ \n{\mathrm{D}}^{\alpha }{F}_{0}f = {\left( -1\right) }^{\left| \alpha \right| }{F}_{0}{M}_{\alpha }f,\;{M}_{\alpha }{... | Proof. It is easy to see that the function\n\n\[ \n\left( {{F}_{0}f}\right) \left( x\right) = {\left( 2\pi \right) }^{-m/2}\int {\mathrm{e}}^{-\mathrm{i}{xy}}f\left( y\right) \mathrm{d}y \]\n\nis arbitrarily many times continuously differentiable. The differentiation can be done under the integral sign, i.e.,\n\n\[ \n{... | Yes |
Theorem 10.2. The function \( \vartheta : {\mathbb{R}}^{m} \rightarrow \mathbb{R} \) defined by the equality\n\n\[ \vartheta \left( x\right) = \exp \left( {-\frac{1}{2}{\left| x\right| }^{2}}\right) \;\text{ for all }\;x \in {\mathbb{R}}^{m} \]\n\nis in \( S\left( {\mathbb{R}}^{m}\right) \) . We have \( {F}_{0}\varthet... | Proof. The reader can easily verify that \( \vartheta \in \mathcal{S}\left( {\mathbb{R}}^{m}\right) \) . To prove that \( {F}_{0}\vartheta = \vartheta \) , we first consider the case \( m = 1 \) . In this case we obviously have the first order differential equation\n\n\[ {\vartheta }^{\prime }\left( x\right) + {x\varth... | No |
We have \( \begin{Vmatrix}{{F}_{0}f}\end{Vmatrix} = \parallel f\parallel \) and \( \begin{Vmatrix}{{F}_{0}^{-1}f}\end{Vmatrix} = \parallel f\parallel \) for all \( f \in S\left( {\mathbb{R}}^{m}\right) \) (here \( \parallel \) . II denotes the norm in \( {L}_{2}\left( {\mathbb{R}}^{m}\right) ).{F}_{0} \) and \( {F}_{0}... | For \( f, g \in \mathcal{S}\left( {\mathbb{R}}^{m}\right) \) we have\n\n\[ \langle f, g\rangle = \int f{\left( x\right) }^{ * }\left( {{F}_{0}^{-1}{F}_{0}g}\right) \left( x\right) \mathrm{d}x \]\n\n\[ = \int f{\left( x\right) }^{ * }{\left( 2\pi \right) }^{-m}\left\{ {\int {\mathrm{e}}^{\mathrm{i}{xy}}\left\lbrack {\in... | Yes |
The mappings \( {F}_{1} \) and \( {\widetilde{F}}_{1} \) of \( {L}_{1}\left( {\mathbb{R}}^{m}\right) \) into the space \( {C}_{\infty }\left( {\mathbb{R}}^{m}\right) \) of continuous bounded functions defined on \( {\mathbb{R}}^{m} \) are injective. | Proof. Take an \( f \) from \( {L}_{1}\left( {\mathbb{R}}^{m}\right) \) for which \( {F}_{1}f = 0 \) (i.e., \( \left( {{F}_{1}f}\right) \left( x\right) = 0 \) for all \( x \in {\mathbb{R}}^{m} \) ). We have to prove that \( f = 0 \) . It follows from the equality \( {F}_{1}f = 0 \) that\n\n\[ \n\int f\left( x\right) \l... | Yes |
Theorem 10.6. For all \( f \in {L}_{2}\left( {\mathbb{R}}^{m}\right) \)\n\n\[ \left( {Ff}\right) \left( x\right) = \mathop{\lim }\limits_{{n \rightarrow \infty }}{\left( 2\pi \right) }^{-m/2}{\int }_{K\left( n\right) }{\mathrm{e}}^{-\mathrm{i}{xy}}f\left( y\right) \mathrm{d}y. \] | Proof. The functions \( {\chi }_{n}f \) belong to \( {L}_{1}\left( {\mathbb{R}}^{m}\right) \cap {L}_{2}\left( {\mathbb{R}}^{m}\right) \), and \( {\chi }_{n}f \rightarrow f \) in \( {L}_{2}\left( {\mathbb{R}}^{m}\right) \) . Therefore, \( F\left( {\chi, f}\right) \rightarrow {Ff} \) in \( {L}_{2}\left( {\mathbb{R}}^{m}\... | Yes |
Theorem 10.9. Suppose \( f \in {W}_{2, s}\left( {\mathbb{R}}^{m}\right) \) and \( \alpha \) is a multiindex such that \( \left| \alpha \right| < s \) . (a) We have \( \left\langle {{\mathrm{D}}^{\alpha }f, g}\right\rangle = \left\langle {f,{\mathrm{D}}^{\alpha }g}\right\rangle \) for all \( g \in {W}_{2,\left| \alpha \... | Proof. (a) If \( g \in {W}_{2,\left| \alpha \right| }\left( {\mathbb{R}}^{m}\right) \), then \( {Fg} \in {L}_{2,\left| \alpha \right| }\left( {\mathbb{R}}^{m}\right) \), and thus \( {Fg} \) belongs to the domain of the operator of multiplication by \( {x}^{\alpha } \) . Therefore, \[ \left\langle {{\mathrm{D}}^{\alpha ... | Yes |
(a) For every \( s > 0 \) the set \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{m}\right) \) is dense in \( S\left( {\mathbb{R}}^{m}\right) \) with respect to the norm || . ||,. | (a) Suppose \( r \in {\mathbb{N}}_{0} \) and \( r \geq s \) . We show that \( {C}_{0}^{\infty }\left( {\mathbb{R}}^{m}\right) \) is dense in \( \mathcal{S}\left( {\mathbb{R}}^{m}\right) \) with respect to \( \parallel \cdot {\parallel }_{r,0} \) ; since \( \parallel \cdot {\parallel }_{s} \leq C\parallel \cdot {\parall... | Yes |
Theorem 10.12. The following assertions are equivalent:\n\n(i) All coefficients \( {c}_{\alpha } \) of \( P \) are real.\n\n(ii) \( {T}_{0} \) is symmetric.\n\n(iii) \( {T}_{0} \) is essentially self-adjoint.\n\n(iv) \( T \) is self-adjoint. | The proof immediately follows from Theorem 10.11(a). | No |
Theorem 10.13. Let \( T \) be a self-adjoint differential operator with constant coefficients induced by \( P \), and let \( E \) denote the spectral family of \( T \). (a) For all \( s \in \mathbb{R} \n\[ \nE\left( s\right) = {F}^{-1}M{\chi }_{\left\{ x \in {\mathbf{R}}^{m} : P\left( x\right) < s\right\} }F.\n\]\n(b) ... | Proof.\n(a) The first assertion is clear, since\n\[ \nF\left( s\right) = M{\chi }_{\left\{ x \in {\mathbb{R}}^{m} : P\left( x\right) < s\right\} }\;\text{ for }\;s \in \mathbb{R}\n\]\nis the spectral family of \( {M}_{P} \).\n(b) We have\n\[ \nE\left( t\right) - E\left( s\right) = {F}^{-1}M{\chi }_{\left\{ x \in {\math... | Yes |
Theorem 10.14. Let \( P \) be a polynomial of degree \( r \), and let \( T \) be the maximal differential operator induced by \( P \). Then the following statements are equivalent:\n\n(i) \( P \) is elliptic.\n\n(ii) The principal part of \( P \) vanishes only for \( x = 0 \).\n\n(iii) \( D\left( T\right) = {W}_{2, r}\... | Proof. (i) implies (ii): Let us assume that there is an \( {x}_{0} \in {\mathbb{R}}^{m},{x}_{0} \neq 0 \) such that \( {P}_{r}\left( {x}_{0}\right) = 0 \). Then we also have \( {P}_{r}\left( {s{x}_{0}}\right) = 0 \) for all \( s \in \mathbb{R} \). Therefore,\n\n\[ \left| {P\left( {s{x}_{0}}\right) }\right| = \left| {\m... | Yes |
Theorem 10.15. Let \( T \) be a self-adjoint elliptic differential operator with constant coefficients on \( {L}_{2}\left( {\mathbb{R}}^{m}\right) \). (a) If \( m > 1 \), then \( T \) is semibounded. | (a) Since \( T \) is self-adjoint, \( P \) is real-valued. As \( T \) is elliptic, \( \left| {P\left( x\right) }\right| \rightarrow \infty \) as \( \left| x\right| \rightarrow \infty \). Consequently, \( \left| {P\left( x\right) }\right| > 0 \) for all \( \left| x\right| \geq {c}_{0} \). Because of the continuity of \(... | Yes |
Theorem 10.16. Let \( 0 \leq s < r \) (not necessarily integral). Then for every \( \eta > 0 \) there exists a \( {C}_{\eta } \geq 0 \) such that\n\n\[ \parallel f{\parallel }_{s} \geq \eta \parallel f{\parallel }_{r} + {C}_{\eta }\parallel f\parallel \;\text{ for all }\;f \in {W}_{2, r}\left( {\mathbb{R}}^{m}\right) .... | Proof. The assertion is equivalent to the inequality\n\n\[ \parallel f{\parallel }_{\left( s\right) } \leq \eta \parallel f{\parallel }_{\left( r\right) } + {C}_{\eta }\parallel f\parallel \;\text{ for all }\;f \in {L}_{2, r}\left( {\mathbb{R}}^{m}\right) . \]\n\nFor all \( N > 0 \) we have\n\n\[ \parallel f{\parallel ... | Yes |
Theorem 10.18. Assume \( r \in \mathbb{N} \) and \( T \) is a closed operator on \( {L}_{2}\left( {\mathbb{R}}^{m}\right) \) such that \( D\left( T\right) \subset {W}_{2, r}\left( {\mathbb{R}}^{m}\right) \) . Let \( V \) be an operator on \( {L}_{2}\left( {\mathbb{R}}^{m}\right) \) such that\n\n\[ D\left( V\right) \sup... | Proof. If \( {M}_{{k}_{r}} \) is the operator of multiplication by \( {k}_{r} = {\left( 1 + {\left| \cdot \right| }^{2}\right) }^{r/2} \) on \( {L}_{2}\left( {\mathbb{R}}^{m}\right) \) and \( {T}_{r} = {F}^{-1}{M}_{{k}_{r}}F \), then \( {T}_{r} \) is a self-adjoint operator on \( {L}_{2}\left( {\mathbb{R}}^{m}\right) \... | Yes |
Theorem 10.21. Let \( r, T \), and \( V \) be defined as in Theorem 10.18.\n\n(a) If \( {q}_{\alpha } = 0 \) for \( \left| \alpha \right| = r \) and\n\n\[ \n{N}_{{q}_{\alpha }}\left( x\right) \rightarrow 0\;\text{ as }\;\left| x\right| \rightarrow \infty \;\text{ for }\;\left| \alpha \right| < r, \n\]\n\nthen \( V \) i... | Proof. The mappings\n\n\[ \n{W}_{2, r}\left( {\mathbb{R}}^{m}\right) \rightarrow {W}_{2, r - \left| \alpha \right| }\left( {\mathbb{R}}^{m}\right) ,\;f \mapsto {\mathrm{D}}^{\alpha }f \n\]\nare bounded, and by Theorem 10.20 the mappings\n\n\[ \n{W}_{2, r - \left| \alpha \right| }\left( {\mathbb{R}}^{m}\right) \rightarr... | Yes |
Theorem 10.22. Let the operator \( T \) be defined as in (10.3). Assume that \( {b}_{1},\ldots ,{b}_{m} \in {C}^{1}\left( {\mathbb{R}}^{m}\right) \) are bounded with bounded derivatives, and \( q \in {M}_{\rho }\left( {\mathbb{R}}^{m}\right) \) for some \( \rho < 4 \) . Then \( T \) is essentially self-adjoint and \( D... | The proof can be obtained immediately from Theorems 5.28, 10.18, and 9.1 if we consider \( - \Delta \) as the unperturbed operator (cf. the representation (10.4) of \( T \) ). | No |
(a) If \( {q}_{ - } \in {M}_{\rho }\left( {\mathbb{R}}^{m}\right) \) for some \( \rho < 4 \), then \( S \) is bounded from below (here \( \left. {{q}_{ - }\left( x\right) = \max \left( {-q\left( x\right) ,0}\right) }\right) \) . If \( q \geq 0 \), then \( S \) is non-negative. | (a) By Auxiliary theorem 10.25 with \( \eta = 1 \) we have for all \( f \in {C}_{0}^{\infty }\left( {\mathbb{R}}^{m}\right) \) that\n\n\[ \langle f,{Sf}\rangle = \mathop{\sum }\limits_{{j = 1}}^{m}{\begin{Vmatrix}\left( {\mathrm{D}}_{j} - {b}_{j}\right) f\end{Vmatrix}}^{2} + \langle f,{qf}\rangle \]\n\n\[ > \mathop{\su... | Yes |
Theorem 10.30. Assume that \( {b}_{j} = 0 \) for \( j = 1,2,\ldots, m, q \in {M}_{\rho }\left( {\mathbb{R}}^{m}\right) \) for some \( \rho < 4 \), and\n\n\[ q\left( {ax}\right) = {a}^{-\gamma }q\left( x\right) \;\text{ for }\;x \in {\mathbb{R}}^{m} \smallsetminus \{ 0\} \]\n\nwith some \( \gamma \in \left( {0,2}\right)... | Proof. If \( f \) is an eigenelement of \( S \) belonging to the eigenvalue \( \lambda \), then\n\n\[ - {\Delta f}\left( x\right) = {\lambda f}\left( x\right) - q\left( x\right) f\left( x\right) . \]\n\n(10.11)\n\nIt follows from this for every \( a > 0 \) using the notation \( {f}_{a}\left( x\right) = f\left( {ax}\rig... | Yes |
Theorem 10.31. Let \( S \) be as in Theorem \( {10.29}\left( b\right) \) . Assume further that there exist constants \( C \geq 0,\epsilon > 0 \) and \( r \geq 0 \) such that\n\n\[ q\left( x\right) \leq - C{\left| x\right| }^{-2 + \epsilon }\;\text{ for all }\;x \in {\mathbb{R}}^{m}\;\text{ with }\;\left| x\right| \geq ... | Proof. We only have to prove that \( S \) has infinitely many negative eigenvalues. According to Theorem 7.26(b) it is sufficient to find an infinite-dimensional subspace \( M \) of \( D\left( S\right) \) with \( \langle f,{Sf}\rangle < 0 \) for every nonvanishing \( f \in M \) . Let \( \vartheta \in {C}_{0}^{\infty }\... | Yes |
Theorem 10.32. Let \( A \in B\left( {{L}_{2}\left( {\mathbb{R}}^{m}\right) }\right) \) be a real \( {}^{3} \) positivity improving selfadjoint operator. Assume that \( \parallel A\parallel \) is an eigenvalue of \( A \) . Then the multiplicity of the eigenvalue \( \parallel A\parallel \) equals 1 and there is an \( f >... | Proof. Assume that \( f \neq 0 \) and \( {Af} = \parallel A\parallel f \) . Since \( A \) is real, we may assume that \( f \) is real (otherwise we could replace \( f \) by \( \operatorname{Re}f \) or \( \operatorname{Im}f \), because \( A\left( {\operatorname{Re}f}\right) = A\left( {f + {Kf}}\right) /2 = \left( {{Af} ... | Yes |
Theorem 10.33. Let \( S \) be defined as above with \( {b}_{j} = 0\left( {j = 1,2,\ldots, m}\right) \) , \( q \in {M}_{\rho ,\text{ loc }}\left( {\mathbb{R}}^{m}\right) \) and \( {q}_{ - } \in {M}_{\rho }\left( {\mathbb{R}}^{m}\right) \) for some \( \rho < 4 \) . Then \( S \) is bounded from below. If the lowest point ... | Proof. By Theorem 10.29(a) the operators \( S \) and \( {S}_{0} - {q}_{ - } \) are bounded from below. The lower bound of \( {S}_{0} - {q}_{ - } \) is, at the same time, a lower bound of the operators \( {S}_{n} \) and \( S - {Q}_{n} \) used in steps 2 and 3 . These operators therefore have a common lower bound, so tha... | Yes |
Theorem 10.36. Assume that \( q = {q}_{1} + {q}_{2} \), where \( {q}_{1} \) and \( {q}_{2} \) are measurable Hermitian \( 4 \times 4 \) matrix-valued functions such that\n\n\[ \left| {{q}_{1}\left( x\right) }\right| \leq C\frac{1}{\left| x\right| }\text{ and }{q}_{2}\left( .\right) \in {M}_{\rho }\left( {\mathbb{R}}^{3... | Proof. By Auxiliary theorem 10.35\n\n\[ {\begin{Vmatrix}{q}_{1}f\end{Vmatrix}}^{2} \leq {C}^{2}\int {\left| x\right| }^{-2}\mathop{\sum }\limits_{{j = 1}}^{4}{\left| {f}_{j}\left( x\right) \right| }^{2}\mathrm{\;d}x \leq 4{C}^{2}\mathop{\sum }\limits_{{k = 1}}^{3}\mathop{\sum }\limits_{{j = 1}}^{4}\int {\left| \frac{\p... | Yes |
Theorem 10.37. Let \( q \) be as in Theorem 10.36. Assume, moreover, that \( {N}_{\left| q\right| }\left( x\right) \rightarrow 0 \) as \( \left| x\right| \rightarrow \infty \) . Then \( Q \) is \( {T}^{2} \) -compact. If \( C < 1/2 \), then \( {\sigma }_{e}\left( S\right) = \) \( {\sigma }_{e}\left( T\right) = \left( {... | Proof. We have \( {T}^{2} = {F}^{-1}{M}_{P}^{2}F \) . Therefore,\n\n\[ D\left( {T}^{2}\right) = {F}^{-1}D\left( {M}_{P}^{2}\right) = {F}^{-1}{L}_{2,2}{\left( {\mathbb{R}}^{3}\right) }^{4} = {W}_{2,2}{\left( {\mathbb{R}}^{3}\right) }^{4}. \]\n\nSince \( \left| q\right| = \left| {{q}_{1} + {q}_{2}}\right| \in {M}_{\rho }... | Yes |
Theorem 10.38. Let \( q \) be as in Theorem 10.36 with some \( C < 1/2 \) . Moreover, assume that \( q\left( {ax}\right) = q\left( x\right) /a \) for all \( a > 0 \) and \( x \in {\mathbb{R}}^{3} \smallsetminus \{ 0\} \) . Then \( S = T + Q \) has no eigenvalue in \( \left( {-\infty , - 1}\right) \cup \left( {1,\infty ... | Proof. If \( \left( {\lambda - S}\right) f = 0 \), then the function \( {f}_{a}\left( x\right) = f\left( {ax}\right) \) obviously belongs to \( D\left( S\right) \) and\n\n\[ \left( {S{f}_{a}}\right) \left( x\right) = \mathop{\sum }\limits_{{j = 1}}^{3}{\alpha }_{j}{D}_{j}f\left( {ax}\right) + \left( {\beta + q\left( x\... | Yes |
Theorem 10.39. Assume that \( q = {q}_{1} + {q}_{2} \), where \( {q}_{1} \) and \( {q}_{2} \) are measurable Hermitian \( 4 \times 4 \) matrix-valued functions such that\n\n\[ \left| {{q}_{1}\left( x\right) }\right| \leq C\frac{1}{\left| x\right| }\;\text{ for some }\;C < \frac{1}{2}, \]\n\n\[ \left| {{q}_{2}\left( .\r... | Proof. Let \( {\varphi }_{n} \) be as in the proof of Theorem 10.23, let \( {\widetilde{q}}_{n} = {\varphi }_{3n}q \), and let \( {S}_{n} = T + {Q}_{n} \), where \( {Q}_{n} \) is the operator of multiplication by \( {\widetilde{q}}_{n} \). The function \( {\widetilde{q}}_{n} \) satisfies the assumption of Theorem 10.36... | Yes |
Theorem 11.2. Assume that \( {T}_{1} \) and \( {T}_{2} \) are self-adjoint operators on the complex Hilbert space \( H,{M}_{1} \subset {}_{{T}_{1}}D\left( {{\Omega }_{ + }\left( {{T}_{2},{T}_{1}}\right) }\right) ,{P}_{1} \) is the orthogonal projection onto \( {M}_{1} \), and \( {M}_{2} = R\left( {{W}_{ + }\left( {{T}_... | Proof. Since \( {M}_{1} \) is closed and is contained in \( D\left( {{\Omega }_{ + }\left( {{T}_{2},{T}_{1}}\right) }\right) \), the subspace \( {M}_{2} = {\Omega }_{ + }\left( {{T}_{2},{T}_{1}}\right) {M}_{1} \) is also closed. By Theorem 11.1\n\n\[ \n{\mathrm{e}}^{\mathrm{i}s{T}_{2}}{W}_{ + } = {\mathrm{e}}^{\mathrm{... | Yes |
Theorem 11.3. Assume that \( {T}_{1} \) and \( {T}_{2} \) are self-adjoint operators on the complex Hilbert space \( H,{M}_{1} \subset {}_{{T}_{1}}D\left( {{\Omega }_{ + }\left( {{T}_{2},{T}_{1}}\right) }\right) ,{P}_{1} \) is the orthogonal projection onto \( {M}_{1},{M}_{2} = R\left( {{W}_{ + }\left( {{T}_{2},{T}_{1}... | Proof. (11.1) follows from the definition of \( {W}_{ + } \) by multiplication by \( {\mathrm{e}}^{-\mathrm{i}t{T}_{2}} \) . (11.2) follows from (11.1) by multiplication by \( {\mathrm{e}}^{\mathrm{i}t{T}_{1}} \) . If we multiply (11.2) by \( {W}_{ + }^{ * } \) from the right, then we obtain (11.3). Relation (11.4) fol... | Yes |
Theorem 11.4. Assume that \( {T}_{1},{T}_{2} \), and \( {T}_{3} \) are self-adjoint operators on the complex Hilbert space \( H,{M}_{1} \subset {}_{{T}_{1}}D\left( {{\Omega }_{ + }\left( {{T}_{2},{T}_{1}}\right) }\right) ,{M}_{2} \subset {}_{{T}_{2}}D\left( {{\Omega }_{ + }\left( {{T}_{3},{T}_{2}}\right) }\right) ,{P}_... | Proof. For \( f \in {M}_{1} \)\n\n\[ \n{\mathrm{e}}^{\mathrm{i}t{T}_{3}}{\mathrm{e}}^{-\mathrm{i}t{T}_{1}}f = \left( {{\mathrm{e}}^{\mathrm{i}t{T}_{3}}{\mathrm{e}}^{-\mathrm{i}t{T}_{2}}}\right) \left( {{\mathrm{e}}^{\mathrm{i}t{T}_{2}}{\mathrm{e}}^{-\mathrm{i}t{T}_{1}}}\right) f \n\]\n\n\[ \n= {\mathrm{e}}^{\mathrm{i}t... | Yes |
Theorem 11.5. Assume that \( {T}_{1} \) and \( {T}_{2} \) are self-adjoint operators on the complex Hilbert space \( H,{M}_{1} \) and \( {M}_{2} \) are closed subspaces, \( {P}_{1} \) and \( {P}_{2} \) are the orthogonal projections onto \( {M}_{1} \) and \( {M}_{2} \), respectively. If \( {M}_{1} \subset {}_{{T}_{1}}D... | Proof. We obviously have \( {W}_{ + }\left( {{T}_{1},{T}_{1},{P}_{1}}\right) = {P}_{1} \) and \( {W}_{ + }\left( {{T}_{2},{T}_{2},{P}_{2}}\right) = {P}_{2} \) . It therefore follows from Theorem 11.4 that\n\n\[ \n{W}_{ + }\left( {{T}_{1},{T}_{2},{P}_{2}}\right) {W}_{ + }\left( {{T}_{2},{T}_{1},{P}_{1}}\right) = {P}_{1}... | Yes |
Theorem 11.7. Let \( {T}_{1} \) and \( {T}_{2} \) be self-adjoint operators on the complex Hilbert space \( H \) . If \( {\mathrm{e}}^{-{itT}}f \in D\left( {T}_{1}\right) \cap D\left( {T}_{2}\right) \) for all \( t \in \mathbb{R} \), the function \[ \mathbb{R} \rightarrow H, t \mapsto \left( {{T}_{2} - {T}_{1}}\right) ... | Proof. Since \( {\mathrm{e}}^{-\mathrm{i}t{T}_{1}}f \in D\left( {T}_{2}\right) \cap D\left( {T}_{1}\right) \), the function \( \Omega \left( t\right) f = {\mathrm{e}}^{\mathrm{i}t{T}_{2}}{\mathrm{e}}^{-\mathrm{i}t{T}_{1}}f \) is differentiable for all \( t \in \mathbb{R} \), and its derivative is \[ \frac{\mathrm{d}}{\... | Yes |
Theorem 11.9. Let \( {T}_{1} \) and \( {T}_{2} \) be self-adjoint operators on the complex Hilbert space \( H \), and let \( E \) be the spectral family of \( {T}_{1} \). (a) If \( J \) is a bounded interval, \( R\left( {E\left( J\right) {P}_{1,{ac}}}\right) \subset D\left( {T}_{2}\right) \), and \( \left( {{T}_{2} - }... | Proof. (a) If \( f \in R\left( {E\left( J\right) {P}_{1,{ac}}}\right) \) then \( {H}_{f} \subset R\left( {E\left( J\right) {P}_{1,{ac}}}\right) \), and thus \( \left( {{T}_{2} - {T}_{1}}\right) {P}_{f} \in \) \( {B}_{1}\left( H\right) \) . By Theorem 11.8 \( {H}_{f} \subset D\left( {{\Omega }_{ \pm }\left( {{T}_{2},{T}... | Yes |
Theorem 11.10. Let \( {T}_{1} \) and \( {T}_{2} \) be self-adjoint operators on the complex Hilbert space \( \mathrm{H} \). The wave operators \( {W}_{ \pm }\left( {{T}_{2},{T}_{1},{P}_{1,{ac}}}\right) \) and \( {W}_{ \pm }\left( {{T}_{1},{T}_{2},{P}_{2,{ac}}}\right) \) exist and are complete provided that \( R\left( {... | Proof. If \( \left( {{T}_{2} - {T}_{1}}\right) {E}_{j}\left( J\right) \in {B}_{1}\left( H\right) \) for \( j = 1,2 \) and for every bounded interval \( J \), then the wave operators \( {W}_{ \pm }\left( {{T}_{2},{T}_{1},{P}_{1,{ac}}}\right) \) and \( {W}_{ \pm }\left( {{T}_{1},{T}_{2},{P}_{2,{ac}}}\right) \) exist by T... | Yes |
Theorem 11.11. Let \( {T}_{1} \) and \( {T}_{2} \) be self-adjoint operators on the complex Hilbert space \( H \) . Assume that \( D\left( {T}_{1}\right) = D\left( {T}_{2}\right) \) and \( \left( {{T}_{2} - {T}_{1}}\right) {E}_{1}\left( J\right) \in {B}_{1}\left( H\right) \) for every bounded interval \( J \) . Then th... | Proof of THEOREM 11.11. The existence of \( {W}_{ \pm }\left( {{T}_{2},{T}_{1}}\right) \) follows from Theorem 11.9(b), because \( R\left( {{E}_{1}\left( J\right) }\right) \subset D\left( {T}_{1}\right) = D\left( {T}_{2}\right) \) for every bounded interval \( J \) . It therefore remains to prove the existence of\n\n\[... | No |
Theorem 11.16. Let \( {T}_{1} \) and \( {T}_{2} \) be self-adjoint operators on the complex Hilbert space \( H \) . Assume that \( {T}_{1} \geq \gamma ,{T}_{2} \geq \gamma \), and \( {\left( {T}_{1} - \lambda \right) }^{-p} - {\left( {T}_{2} - \lambda \right) }^{-p} \in \) \( {B}_{1}\left( H\right) \) for some \( \lamb... | Proof. Let \( \vartheta : \mathbb{R} \rightarrow \mathbb{R} \) be a twice continuously differentiable function such that \( \vartheta \left( \mathrm{t}\right) = \lambda + {t}^{-1/p} \) for \( t \geq {\left( \gamma - \lambda \right) }^{-p} \) and \( {\vartheta }^{\prime }\left( t\right) < 0 \) for all \( t \in \mathbb{R... | Yes |
Theorem 11.18. Let us assume that with a closed set \( A \subset {\mathbb{R}}^{m} \) of measure zero we have\n\n\[ h \in {C}^{\infty }\left( {{\mathbb{R}}^{m} \smallsetminus A}\right) \;\text{ and }\;\operatorname{grad}h\left( x\right) \neq 0\;\text{ for }\;x \notin A. \]\n\nLet \( V \) be a symmetric operator on \( {L... | In the following we shall not prove this theorem but a somewhat more general one that also considers operators on \( {L}_{2}{\left( {\mathbb{R}}^{m}\right) }^{M} \) (for example, Dirac operators). | No |
Theorem 11.23. Let \( {T}_{1} \) be equal to \( - \Delta \), with \( D\left( {T}_{1}\right) = {W}_{2,2}\left( {\mathbb{R}}^{m}\right) \). Let \( V \) be symmetric such that \( D\left( V\right) \supset D\left( {T}_{1}\right) \), and let \( {T}_{2} = {T}_{1} + V \) be self-adjoint. Assume that there exist an \( s \geq 0,... | Proof. Since \( D{\left( {T}_{1}\right) }^{n} = {W}_{2,{2n}}\left( {\mathbb{R}}^{m}\right) \), by Auxiliary theorem 11.22 \( V{\left( \mathrm{i} - {T}_{1}\right) }^{-n} \) \( \in {B}_{1}\left( {{L}_{2}\left( {\mathbb{R}}^{m}\right) }\right) \) for every \( n \in \mathbb{N} \) such that \( {2n} - s > m \), and thus\n\n\... | Yes |
Theorem 1.2. Let \( f \) be the power series associated with \( {\alpha }_{ * }\mu \) . Let\n\n\[ \n{Z}_{n + 1}^{ * } = \mathbf{Z}\left( d\right) \times \mathbf{Z}{\left( {p}^{n + 1}\right) }^{ * }\n\]\n\nThen\n\n\[ \nf\left( X\right) \equiv \mathop{\sum }\limits_{{a \in {Z}_{n + 1}^{ * }}}{\mu }_{n + 1}\left( a\right)... | Proof. By the definition of the associated power series, we have\n\n\[ \nf\left( X\right) \equiv \mathop{\sum }\limits_{{r = 0}}^{{{p}^{n} - 1}}\left( {{\alpha }_{ * }\mu }\right) \left( r\right) {\left( 1 + X\right) }^{r}.\n\]\n\nBut letting char denote the characteristic function, we have:\n\n\[ \n\left( {{\alpha }_{... | Yes |
Let \( \psi \) be a nontrivial character of \( 1 + p{\mathbf{Z}}_{p} \), with conductor \( {p}^{n + 1} \). Define \( \psi \left( a\right) = \psi \left( {\langle a\rangle }\right) \). Let\n\n\[ \psi \left( \gamma \right) = \zeta = \text{ primitive }{p}^{n}\text{-th root of unity. } \]\n\nLet \( f \) be the power series ... | Proof. We have\n\n\[ {\int }_{{Z}^{ * }}{\psi d\mu } = {\int }_{{\mathbf{Z}}_{p}}\psi \left( {\gamma }^{x}\right) d\left( {{\alpha }_{ * }\mu }\right) \left( x\right) \;\text{ (by Theorem 1.1) } \]\n\n\[ = {\int }_{{\mathbf{Z}}_{p}}{\zeta }^{x}d\left( {{\alpha }_{ * }\mu }\right) \left( x\right) \]\n\n\[ = f\left( {\ze... | Yes |
Corollary 2. There exists a positive integer \( {n}_{0} \) (depending only on \( f \) ) such that if \( n \geq {n}_{0} \) and cond \( \psi = {p}^{n} \), then\n\n\[ B\left( {\psi ,\mu }\right) \sim {p}^{m}{\left( \zeta - 1\right) }^{\lambda } \]\n\nwhere \( \zeta \) is a primitive \( {p}^{n} \) -th root of unity. | Proof. As \( n \rightarrow \infty \), the values \( \left| {\zeta - 1}\right| \) approach 1, and so the term \( {c}_{\lambda }{\left( \zeta - 1\right) }^{\lambda } \) dominates in the power series \( f\left( {\zeta - 1}\right) \) above. | No |
For some constant \( c = c\\left( f\\right) \), we have\n\n\\[ \n{\\operatorname{ord}}_{p}\\mathop{\\prod }\\limits_{\\substack{{\\text{ cond }\\psi = {p}^{t}} \\\\ {{n}_{0} \\leq t \\leq n} }}B\\left( {\\psi ,\\mu }\\right) = m{p}^{n} + {\\lambda n} + c\\left( f\\right) \n\\] | Proof. Since\n\n\\[ \n\\mathop{\\prod }\\limits_{\\substack{{\\zeta {p}^{n} = 1} \\\\ {\\zeta \\neq 1} }}\\left( {\\zeta - 1}\\right) = {p}^{n} \n\\]\n\nthe formula is immediate, since the product taken for \( {n}_{0} \\leq t \\leq n \) differs by only a finite number of factors (depending on \( {n}_{0} \) ) from the p... | Yes |
Theorem 1.3. Let \( n \) be an integer \( \geq 0 \) such that \( {c}_{r}^{\left( n\right) } \) is a p-unit for some integer \( r \) with\n\n\[ 0 \leq r \leq {p}^{n} - 1 \]\n\nThen the exponential Iwasawa invariant \( m \) of \( \mu \) is equal to 0, and we have \( \lambda \leq {p}^{n} \) | Proof. Some coefficient \( {a}_{r}^{\left( n\right) } \) must also be a \( p \) -unit with \( r \) in the same range, and we can write\n\n\[ f\left( X\right) = \mathop{\sum }\limits_{{r = 0}}^{{{p}^{n} - 1}}{a}_{r}^{\left( n\right) }{X}^{r} + {g}_{1}\left( X\right) {X}^{{p}^{n}} + p{g}_{2}\left( X\right) ,\]\n\nwhere \... | Yes |
Theorem 1.4. Let \( {m}_{s},{\lambda }_{s} \) be the Iwasawa invariants of \( {\mu }^{\left( s\right) } \) . If \( {m}_{s} = 0 \) for some \( s \), then \( {m}_{s} = 0 \) for all \( s \) . Suppose this is the case, and let \( n \) be the positive integer such that | \[ {p}^{n - 1} \leq {\lambda }_{0} < {p}^{n} \] Then we also have \[ {p}^{n - 1} \leq {\lambda }_{s} < {p}^{n} \] for all s. | No |
Theorem 2.1. For every integer \( k \geq 1 \) and character \( \chi \) of conductor \( d{p}^{n} \) with \( n \geq 0 \), we have \[ {L}_{p}\left( {1 - k,\chi }\right) = - \left( {1 - {\chi }_{k}\left( p\right) {p}^{k - 1}}\right) \frac{1}{k}{B}_{k,{\chi }_{k}}. \] | Proof. We have: \[ - \left( {1 - {\chi }_{k}\left( c\right) {c}_{p}^{k}}\right) {L}_{p}\left( {1 - k,\chi }\right) = {\int }_{{Z}^{ * }}{\chi }_{k}\left( a\right) {a}_{p}^{k - 1}d{E}_{1, c}\left( a\right) . \] Write \[ {\int }_{{Z}^{ * }} = {\int }_{Z} - {\int }_{pZ} \] Let \( N = d{p}^{n + 1} \) . Then \[ {\int }_{pZ}... | Yes |
Theorem 2.2 (Iwasawa congruences). Let \( d \) be an integer \( \geq 1 \) and prime to \( p \) . Let \( \theta \) be an even character \( \neq 1 \) of conductor \( d \) or \( {dp} \) . If no coefficient of \( {f}_{\theta, k} \) is a p-unit, then we have the congruences (independent of \( k \) ):\n\n\[ \mathop{\sum }\li... | Proof. We have proved the assertion when \( \alpha \) lies in \( 1 + p{\mathbf{Z}}_{p} \) . However, for any fixed \( {\eta }_{0} \in {\mathbf{\mu }}_{p - 1} \) we can make the change of variables\n\n\[ \eta \mapsto \eta {\eta }_{0} \]\n\nleading to the congruences as stated above. | Yes |
Theorem 2.4. Let \( \\theta \\neq 1 \) be an even character of conductor \( p \) . Then the Iwasawa congruences imply that there exists an odd integer\n\n\[ v \\not\\equiv - 1{\\;\\operatorname{mod}\\;}p - 1 \]\n\nsuch that, for all \( \\alpha \\in {\\mathbf{Z}}_{p}^{ * } \) and all integers \( n \\geq 1 \) we have\n\n... | Proof. We have\n\n\[ {s}_{n}\\left( {\\alpha \\eta }\\right) = {s}_{n + 1}\\left( {\\alpha \\eta }\\right) - {t}_{n + 1}\\left( {\\alpha \\eta }\\right) {p}^{n + 1}. \]\n\nFurthermore\n\n\[ {s}_{n + 1}\\left( {\\alpha \\eta }\\right) \\equiv {\\alpha \\eta }{\\;\\operatorname{mod}\\;{p}^{n + 2}} \]\n\nand\n\n\[ \\matho... | Yes |
Theorem 2.5. Let \( \theta \neq 1 \) be an even character of conductor \( d \) or \( {dp} \) with \( d > 1 \) prime to p. Let \( {\theta }_{1} = \theta {\omega }^{-1} \) . Then the Iwasawa congruences imply that for all \( \alpha \in {\mathbf{Z}}_{p}^{ * } \) and all \( n \geq 0 \) we have\n\n\[ \mathop{\sum }\limits_{... | Proof. In Theorem 2.2 we may rewrite the congruence in the form\n\n\[ \frac{1}{d{p}^{n + 1}}\sum a{\theta }_{1}\left( a\right) \equiv 0{\;\operatorname{mod}\;\mathfrak{p}} \]\n\nwhere the sum is taken over \( a \) prime to \( {dp} \), such that\n\n\[ 0 < a < d{p}^{n + 1}\text{ and }\langle a{\rangle }_{p} \equiv \langl... | Yes |
Lemma 1. Let \( k \) be an integer with \( 1 \leq k \leq p - 1 \) . Then\n\n\[ \frac{1}{k}{B}_{k,{\omega }^{-k}} \equiv - \frac{1}{kp}{\;\operatorname{mod}\;{\mathbf{Z}}_{p}}. \] | Proof. The proof is entirely similar to that of the Von Staudt congruence, Corollary 2 of Theorem 2.3, Chapter 2, combined with the expression for the Bernoulli number as an integral in Theorem 2.4 of Chapter 2. We leave it to the reader. | No |
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