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(i) If \( f \in {H}^{1}\left( {0, + \infty }\right) \), then \( \mathop{\lim }\limits_{{b \rightarrow + \infty }}f\left( b\right) = 0 \) . | Proof We prove (i). The proof of (ii) is similar. Formula (1.13) yields\n\n\[ \n{\int }_{0}^{b}\left( {f\left( t\right) \overline{{f}^{\prime }\left( t\right) } + {f}^{\prime }\left( t\right) \overline{f\left( t\right) }}\right) {dt} = {\left| f\left( b\right) \right| }^{2} - {\left| f\left( 0\right) \right| }^{2} \n\]... | Yes |
Lemma 1.12 Let \( \mathcal{J} \) be an open interval, and let \( c \) be in the closure of \( \mathcal{J} \) . For any \( \varepsilon > 0 \), there is a constant \( {b}_{\varepsilon } > 0 \) such that\n\n\[ \left| {f\left( c\right) }\right| \leq \varepsilon \begin{Vmatrix}{f}^{\prime }\end{Vmatrix} + {b}_{\varepsilon }... | Proof For notational simplicity, assume that \( c \) is not the left end point of \( \mathcal{J} \) . Take \( a \in \mathcal{J}, a < c \), and a function \( \eta \in {C}_{0}^{\infty }\left( \mathbb{R}\right) \) such that \( \eta \left( c\right) = 1,\eta \left( a\right) = 0 \), and \( \left| {\eta \left( t\right) }\righ... | Yes |
Statement \( \mathcal{D}\left( {T}^{ * }\right) = {H}^{1}\left( {0, + \infty }\right) \) and \( {T}^{ * }g = - \mathrm{i}{g}^{\prime } \) for \( g \in \mathcal{D}\left( {T}^{ * }\right) \) . | Proof Suppose that \( g \in {H}^{1}\left( {0, + \infty }\right) \) and let \( f \in \mathcal{D}\left( T\right) \) . Since \( f\left( 0\right) = 0 \), formula (1.19) yields \( \left\langle {{f}^{\prime }, g}\right\rangle + \left\langle {f,{g}^{\prime }}\right\rangle = 0 \) . Hence, \( \langle {Tf}, g\rangle = \left\lang... | Yes |
Lemma 1.13 \( \langle \mathcal{L}f, g{\rangle }_{{L}^{2}\left( \Omega \right) } = {\left\langle f,{\mathcal{L}}^{ + }g\right\rangle }_{{L}^{2}\left( \Omega \right) } \) for \( f, g \in {C}_{0}^{\infty }\left( \Omega \right) \) . | Proof Since \( f, g \in {C}_{0}^{\infty }\left( \Omega \right) \), we can choose a bounded open subset \( \widetilde{\Omega } \) of \( \Omega \) with \( {C}^{\infty } \) -boundary such that \( \operatorname{supp}f \subseteq \widetilde{\Omega } \) and \( \operatorname{supp}g \subseteq \widetilde{\Omega } \) . Then we co... | Yes |
Proposition 1.14 The operators \( {\left( {L}^{ + }\right) }_{0},{\left( {L}^{ + }\right) }_{\min } \), and \( {L}_{\max } \) satisfy the relations\n\n\[{\left( {\left( {L}^{ + }\right) }_{0}\right) }^{ * } = {\left( {\left( {L}^{ + }\right) }_{\min }\right) }^{ * } = {L}_{\max }\;\text{ and }\;{\left( {L}_{\max }\righ... | Proof Let \( f \in \mathcal{D}\left( {L}_{\max }\right) \) . Then Eq. (1.21) can be written as \( \left\langle {{L}_{\max }f,\varphi }\right\rangle = \langle \mathcal{L}f,\varphi \rangle = \) \( \left\langle {f,{\left( {L}^{ + }\right) }_{0}\varphi }\right\rangle \) for \( \varphi \in \mathcal{D}\left( {\left( {L}^{ + ... | Yes |
Proposition 1.15 Let \( {P}_{0} \) be the projection of \( \mathcal{H} \) on the closed linear subspace \( {\mathcal{H}}_{0} \) . The following statements are equivalent:\n\n(i) \( {\mathcal{H}}_{0} \) is a reducing subspace for \( T \) .\n\n(ii) \( {\mathcal{H}}_{0}^{ \bot } \) is a reducing subspace for \( T \) .\n\n... | Proof All assertions are easily derived from the corresponding definitions. As a sample, we prove that (iv) implies (iii). By (1.23) we have \( {P}_{0}\mathcal{D}\left( T\right) \subseteq \mathcal{D}\left( T\right) \) . Let \( {x}_{0} \in \mathcal{D}\left( T\right) \cap {\mathcal{H}}_{0} \) . Then \( T{x}_{0} = T{P}_{0... | Yes |
Proposition 1.17 Let \( T \) be a closed symmetric operator on \( \mathcal{H} \). Let \( {\mathcal{D}}_{0} \) be a dense linear subspace of a closed subspace \( {\mathcal{H}}_{0} \) of \( \mathcal{H} \) such that \( {\mathcal{D}}_{0} \subseteq \mathcal{D}\left( T\right) \) and \( T{\mathcal{D}}_{0} \subseteq {\mathcal{... | Proof Let \( u \in \mathcal{D}\left( T\right) \). Using the facts that \( T \) is symmetric and \( {T}_{0}v \in {\mathcal{H}}_{0} \), we derive\n\n\[ \left\langle {{T}_{0}v,{P}_{0}u}\right\rangle = \left\langle {{P}_{0}{T}_{0}v, u}\right\rangle = \left\langle {{T}_{0}v, u}\right\rangle = \langle v,{Tu}\rangle = \left\l... | Yes |
Proposition 2.1 Let \( T \) be a linear operator on \( \mathcal{H} \), and \( \lambda \in \mathbb{C} \). (i) \( \lambda \in \pi \left( T\right) \) if and only if \( T - {\lambda I} \) has a bounded inverse \( {\left( T - \lambda I\right) }^{-1} \) defined on \( \mathcal{R}\left( {T - {\lambda I}}\right) \) . In this ca... | Proof (i): First suppose that \( \lambda \in \pi \left( T\right) \) . Then \( \mathcal{N}\left( {T - {\lambda I}}\right) = \{ 0\} \) by (2.1), so the inverse \( {\left( T - \lambda I\right) }^{-1} \) exists. Let \( y \in \mathcal{D}\left( {\left( T - \lambda I\right) }^{-1}\right) = \mathcal{R}\left( {T - {\lambda I}}\... | Yes |
Lemma 2.3 If \( \mathcal{F} \) and \( \mathcal{G} \) are closed linear subspaces of a Hilbert space \( \mathcal{H} \) such that \( \dim \mathcal{F} < \dim \mathcal{G} \), then there exists a nonzero vector \( y \in \mathcal{G} \cap {\mathcal{F}}^{ \bot } \) . | Proof In this proof we denote by \( \left| M\right| \) the cardinality of a set \( M \) . First, we suppose that \( k = \dim \mathcal{F} \) is finite. We take a \( \left( {k + 1}\right) \) -dimensional subspace \( {\mathcal{G}}_{0} \) of \( \mathcal{G} \) and define the mapping \( \Phi : {\mathcal{G}}_{0} \rightarrow \... | Yes |
Proposition 2.4 Suppose that \( T \) is a closable linear operator on \( \mathcal{H} \) . Then the defect number \( {d}_{\lambda }\left( T\right) \) is constant on each connected component of the open set \( \pi \left( T\right) \) . | Proof By Proposition 2.1(iii), we can assume without loss of generality that \( T \) is closed. Then \( \mathcal{R}\left( {T - {\mu I}}\right) \) is closed for all \( \mu \in \pi \left( T\right) \) by Proposition 2.1(iv). Therefore, setting \( {\mathcal{K}}_{\mu } \mathrel{\text{:=}} \mathcal{R}{\left( T - \mu I\right)... | Yes |
Lemma 2.5 Let \( T \) be a linear operator on \( \mathcal{H} \). If \( \lambda \in \mathbb{C} \) is not in the closure of \( \Theta \left( T\right) \), then \( \lambda \in \pi \left( T\right) \). | Proof Set \( {\gamma }_{\lambda } \mathrel{\text{:=}} \operatorname{dist}\left( {\lambda ,\Theta \left( T\right) }\right) > 0 \). For \( x \in \mathcal{D}\left( T\right) ,\parallel x\parallel = 1 \), we have\n\n\[ \parallel \left( {T - {\lambda I}}\right) x\parallel \geq \left| {\langle \left( {T - {\lambda I}}\right) ... | Yes |
(i) \( \rho \left( T\right) = \left\{ {\lambda \in \pi \left( T\right) : {d}_{\lambda }\left( T\right) = 0}\right\} \) . | Proof (i) follows at once from Proposition 2.1,(i) and (iv). Since \( \pi \left( T\right) \) is open and \( {d}_{\lambda }\left( T\right) \) is locally constant on \( \pi \left( T\right) \) by Proposition 2.4, the assertion of (i) implies that \( \rho \left( T\right) \) is open. Hence, \( \sigma \left( T\right) = \math... | Yes |
Proposition 2.7 Let \( T \) be a closed operator on \( \mathcal{H} \). (i) \( \rho \left( T\right) \) is the set of all numbers \( \lambda \in \mathbb{C} \) such that \( T - {\lambda I} \) is a bijective mapping of \( \mathcal{D}\left( T\right) \) on \( \mathcal{H} \) (or equivalently, \( \mathcal{N}\left( {T - {\lambd... | Proof (i): Clearly, \( T - {\lambda I} \) is bijective if and only if the inverse \( {\left( T - \lambda I\right) }^{-1} \) exists and is everywhere defined on \( \mathcal{H} \). It remains to prove that \( {\left( T - \lambda I\right) }^{-1} \) is bounded if \( T - {\lambda I} \) is bijective. Since \( T \) is closed,... | Yes |
Proposition 2.8 Let \( T \) be a closed operator on \( \mathcal{H} \). Let \( \mathcal{U} \) be a connected open subset of \( \mathbb{C} \smallsetminus \overline{\Theta \left( T\right) } \). If there exists a number \( {\lambda }_{0} \in \mathcal{U} \) which is contained in \( \rho \left( T\right) \), then \( \mathcal{... | Proof By Lemma 2.5 we have \( \mathcal{U} \subseteq \pi \left( T\right) \). Therefore, since \( T \) is closed, it follows from Proposition 2.1(iv) that \( \mathcal{R}\left( {T - {\lambda I}}\right) \) is closed in \( \mathcal{H} \) for all \( \lambda \in \mathcal{U} \). By Proposition 2.4, the defect number \( {d}_{\l... | Yes |
Statement \( \sigma \left( {M}_{\varphi }\right) \) is the closure of the set \( \varphi \left( \mathcal{J}\right) \). | Proof Let \( \lambda \in \varphi \left( \mathcal{J}\right) \), say \( \lambda = \varphi \left( {t}_{0}\right) \) for \( {t}_{0} \in \mathcal{J} \). Given \( \varepsilon > 0 \), by the continuity of \( \varphi \) there exists an interval \( K \subseteq \mathcal{J} \) of positive length such that \( \left| {\varphi \left... | Yes |
Proposition 2.9 Suppose that \( {\lambda }_{0} \in \rho \left( T\right) ,\lambda \in \mathbb{C} \), and \( \left| {\lambda - {\lambda }_{0}}\right| < {\begin{Vmatrix}{R}_{{\lambda }_{0}}\left( T\right) \end{Vmatrix}}^{-1} \) . Then we have \( \lambda \in \rho \left( T\right) \) and\n\n\[ \n{R}_{\lambda }\left( T\right)... | Proof As stated in Proposition 2.1(i),(2.1) holds with \( {c}_{{\lambda }_{0}} = {\begin{Vmatrix}{R}_{{\lambda }_{0}}\left( T\right) \end{Vmatrix}}^{-1} \), so that \( \left| {\lambda - {\lambda }_{0}}\right| < {c}_{{\lambda }_{0}} \) by our assumption. Therefore, \( \lambda \in \pi \left( T\right) \) and \( {d}_{\lamb... | Yes |
Example 2.2 Suppose that \( M \) is a nonempty closed subset of \( \mathbb{C} \). Since \( \mathbb{C} \) is separable, so is \( M \), that is, there exists a countable subset \( \left\{ {{r}_{n} : n \in \mathbb{N}}\right\} \) of \( M \) which is dense in \( M \). Define the operator \( T \) on \( {l}^{2}\left( \mathbb{... | It is easily seen that \( \mathcal{D}\left( T\right) = \mathcal{D}\left( {T}^{ * }\right) \) and \( {T}^{ * }\left( {x}_{n}\right) = \left( {\overline{{r}_{n}}{x}_{n}}\right) \) for \( \left( {x}_{n}\right) \in \mathcal{D}\left( {T}^{ * }\right) \). Hence, \( T = {T}^{* * } \), so \( T \) is closed. Each number \( {r}_... | Yes |
Proposition 2.10 Let \( {\lambda }_{0} \) be a fixed number of \( \rho \left( T\right) \), and let \( \lambda \in \mathbb{C},\lambda \neq {\lambda }_{0} \). (i) \( \lambda \in \rho \left( T\right) \) if and only if \( {\left( \lambda - {\lambda }_{0}\right) }^{-1} \in \rho \left( {{R}_{{\lambda }_{0}}\left( T\right) }\... | Proof Both assertions are easy consequences of the following identity:\n\n\[ T - {\lambda I} = \left( {{R}_{{\lambda }_{0}}\left( T\right) - {\left( \lambda - {\lambda }_{0}\right) }^{-1}I}\right) \left( {T - {\lambda }_{0}I}\right) \left( {{\lambda }_{0} - \lambda }\right) .\n\]\n\n(2.9)\n\n(i): Since \( \left( {T - {... | Yes |
Proposition 2.11 Suppose that there exists a \( {\lambda }_{0} \in \rho \left( T\right) \) such that \( {R}_{{\lambda }_{0}}\left( T\right) \) is compact. Then \( {R}_{\lambda }\left( T\right) \) is compact for all \( \lambda \in \rho \left( T\right) \), and \( T \) has a purely discrete spectrum. | Proof The compactness of \( {R}_{\lambda }\left( T\right) \) follows at once from the resolvent identity (2.5). By Theorem A. 3 all nonzero numbers in the spectrum of the compact operator \( {R}_{{\lambda }_{0}}\left( T\right) \) are eigenvalues of finite multiplicities which have no nonzero accumulation point. By Prop... | Yes |
Example 2.3 (Example 1.4 continued: bounded interval \( \left( {a, b}\right) ) \) Recall that \( \mathcal{D}\left( {T}^{ * }\right) = \) \( {H}^{1}\left( {a, b}\right) \) and \( {T}^{ * }f = - \mathrm{i}{f}^{\prime } \) for \( f \in \mathcal{D}\left( {T}^{ * }\right) \) . For each \( \lambda \in \mathbb{C},{f}_{\lambda... | Since \( T = {\left( {T}^{ * }\right) }^{ * } \) , Proposition 2.7(ii) implies that \( \sigma \left( T\right) = \mathbb{C} \) . | Yes |
Statement \( \sigma \left( {S}_{z}\right) = \left\{ {\lambda \in \mathbb{C} : {e}^{\mathrm{i}\lambda \left( {a - b}\right) }z = 1}\right\} \) for \( z \in \mathbb{C} \) and \( \sigma \left( {S}_{\infty }\right) = \varnothing \) . | Proof Let \( \lambda \in \mathbb{C} \) and \( g \in {L}^{2}\left( {a, b}\right) \) . In order to \ | No |
Then the operator \( T = - \mathrm{i}\frac{d}{dx} \) on \( {H}^{1}\left( \mathbb{R}\right) \) is self-adjoint. We show that \( \sigma \left( T\right) = \mathbb{R} \) . | Suppose that \( \lambda \in \mathbb{R} \) . Let us choose a function \( \omega \in {C}_{0}^{\infty }\left( \mathbb{R}\right) ,\omega \neq 0 \), and put \( {h}_{\epsilon }\left( x\right) \mathrel{\text{:=}} {\epsilon }^{1/2}{e}^{\mathrm{i}{\lambda x}}\omega \left( {\epsilon x}\right) \) for \( \epsilon > 0 \) . Since \(... | Yes |
Lemma 3.1 \( T \) is symmetric if and only if \( \langle {Tx}, x\rangle \) is real for all \( x \in \mathcal{D}\left( T\right) \) . | Proof If \( T \) is symmetric, then \( \langle {Tx}, x\rangle = \langle x,{Tx}\rangle = \overline{\langle {Tx}, x\rangle } \), so \( \langle {Tx}, x\rangle \in \mathbb{R} \) . Conversely, if \( \langle {Tx}, x\rangle \) is real for all \( x \in \mathcal{D}\left( T\right) \), it follows immediately from the polarization... | Yes |
Example 3.1 (A nonclosable symmetric operator) Let \( S \) be a nonclosable operator on a Hilbert space \( {\mathcal{H}}_{1} \) (see Example 1.1). Then the operator \( T \) on the Hilbert space \( \mathcal{H} \mathrel{\text{:=}} {\mathcal{H}}_{1} \oplus {\mathcal{H}}_{1} \) defined by \( T\left( {{x}_{1},0}\right) = \l... | \n\[
\left\langle {T\left( {{x}_{1},0}\right) ,\left( {{y}_{1},0}\right) }\right\rangle = {\left\langle 0,{y}_{1}\right\rangle }_{1} + {\left\langle S{x}_{1},0\right\rangle }_{1} = 0 = \left\langle {\left( {{x}_{1},0}\right), T\left( {{y}_{1},0}\right) }\right\rangle .
\] | Yes |
Proposition 3.2 If \( T \) is a symmetric operator on \( \mathcal{H} \), then:\n\n(i) \( \mathbb{C} \smallsetminus \mathbb{R} \subseteq \pi \left( T\right) \) . | Proof (i): Let \( \lambda = \alpha + \mathrm{i}\beta \), where \( \alpha ,\beta \in \mathbb{R} \), and \( x \in \mathcal{D}\left( T\right) \) . Then\n\n\[ \parallel \left( {T - {\lambda I}}\right) x{\parallel }^{2} = \langle \left( {T - {\alpha I}}\right) x - \mathrm{i}{\beta x},\left( {T - {\alpha I}}\right) x - \math... | Yes |
Proposition 3.3 Suppose that \( T \) is a densely defined symmetric operator. If \( T \) is semibounded or if \( \pi \left( T\right) \) contains a real number, then \( {d}_{ + }\left( T\right) = {d}_{ - }\left( T\right) \) . | Proof In both cases, \( \pi \left( T\right) \) is connected by Proposition 3.2, so the assertion follows from Proposition 2.4. | No |
Recall that in all three examples we have \( {Tf} = - \mathrm{i}{f}^{\prime } \) for \( f \in \mathcal{D}\left( T\right) = {H}_{0}^{1}\left( \mathcal{J}\right) \) and \( {T}^{ * }g = - \mathrm{i}{g}^{\prime } \) for \( g \in \mathcal{D}\left( {T}^{ * }\right) = \) \( {H}^{1}\left( \mathcal{J}\right) \), where \( \mathc... | First suppose that \( \mathcal{J} \) is a bounded interval \( \left( {a, b}\right) \) . Then \( \mathcal{N}\left( {{T}^{ * } - {\lambda I}}\right) = \mathbb{C} \cdot {e}^{\mathrm{i}{\lambda x}} \) and \( {d}_{ + }\left( T\right) = {d}_{ - }\left( T\right) = 1 \) . Likewise we have \( \mathcal{N}\left( {{\left( {T}^{2}\... | Yes |
Example 3.3 Let \( S = - \frac{{d}^{2}}{d{x}^{2}} \) with domain \( \mathcal{D}\left( S\right) = \left\{ {f \in {H}^{2}\left( \mathbb{R}\right) : f\left( 0\right) = {f}^{\prime }\left( 0\right) = 0}\right\} \) on \( {L}^{2}\left( \mathbb{R}\right) \) . Then \( {S}^{ * }f = - {f}^{\prime \prime } \) for \( f \in \mathca... | Fix \( \lambda \in \mathbb{C} \smallsetminus \lbrack 0,\infty ) \) . Let \( \sqrt{\lambda } \) denote the square root of \( \lambda \) with \( \operatorname{Im}\sqrt{\lambda } > 0 \), and let \( {\chi }_{ + } \) and \( {\chi }_{ - } \) be the characteristic functions of \( \lbrack 0,\infty ) \) and \( \left( {-\infty ,... | Yes |
Proposition 3.6 Let \( T \) be a densely defined closed symmetric operator, and let \( \mathcal{E} \) be a finite-dimensional linear subspace of \( \mathcal{D}\left( {T}^{ * }\right) \) . Define\n\n\[ \n\mathcal{D}\left( {T}_{\mathcal{E}}\right) \mathrel{\text{:=}} \mathcal{D}\left( T\right) + \mathcal{E}\;\text{ and }... | Proof Since \( \mathcal{G}\left( {T}_{\mathcal{E}}\right) \) is the sum of the closed subspace \( \mathcal{G}\left( T\right) \) and the finite-dimensional vector space \( \left\{ {\left( {x,{T}^{ * }x}\right) : x \in \mathcal{E}}\right\} ,\mathcal{G}\left( {T}_{\mathcal{E}}\right) \) is closed, and so is the operator \... | Yes |
Proposition 3.7 Let \( T \) be a densely defined symmetric operator. Then\n\n\[ \mathcal{D}\left( {T}^{ * }\right) = \mathcal{D}\left( \bar{T}\right) \dot{ + }\mathcal{N}\left( {{T}^{ * } - {\lambda I}}\right) \dot{ + }\mathcal{N}\left( {{T}^{ * } - \bar{\lambda }I}\right) \;\text{ for }\lambda \in \mathbb{C} \smallset... | Proof Let us abbreviate \( {\mathcal{N}}_{\lambda } \mathrel{\text{:=}} \mathcal{N}\left( {{T}^{ * } - {\lambda I}}\right) \) and \( {\mathcal{N}}_{\bar{\lambda }} \mathrel{\text{:=}} \mathcal{N}\left( {{T}^{ * } - \bar{\lambda }I}\right) \) . The inclusion \( \mathcal{D}\left( \bar{T}\right) + {\mathcal{N}}_{\lambda }... | Yes |
Proposition 3.9 Let \( T \) be a densely defined symmetric operator such that there is a real number in \( \pi \left( T\right) \) ; in particular, if \( T \) is lower semibounded, the latter is fulfilled, and \( \mathbb{C} \smallsetminus \left\lbrack {{m}_{T},\infty }\right) \subseteq \pi \left( T\right) \) . Then \( T... | Proof Since \( {d}_{\lambda }\left( T\right) = {d}_{ \pm }\left( T\right) \) by Proposition 3.3 and \( \mathbb{C} \smallsetminus \left\lbrack {{m}_{T},\infty }\right) \subseteq \pi \left( T\right) \) by Proposition 3.2, the assertions follow at once from Proposition 3.8. | Yes |
Proposition 3.10 Let \( T \) be a self-adjoint operator on a Hilbert space \( \mathcal{H} \). For any complex number \( \lambda \), the following conditions are equivalent:\n\n(i) \( \lambda \in \rho \left( T\right) \).\n\n(ii) \( \lambda \in \pi \left( T\right) \), that is, there exists a constant \( {c}_{\lambda } > ... | Proof Since \( T \) is self-adjoint, by Corollary 2.2 we have\n\n\[ \mathcal{R}{\left( T - \lambda I\right) }^{ \bot } = \mathcal{N}\left( {T - \bar{\lambda }I}\right) \]\n\n(3.13)\n\nFirst suppose that \( \lambda \in \mathbb{C} \smallsetminus \mathbb{R} \). Then \( \lambda \in \pi \left( T\right) \), and hence \( \mat... | Yes |
Proposition 3.11 Let \( T \) be a symmetric operator on \( \mathcal{H} \). If there exists a complex number \( \lambda \) such that \( \mathcal{R}\left( {T - {\lambda I}}\right) = \mathcal{H} \) and \( \mathcal{R}\left( {T - \bar{\lambda }I}\right) \) is dense in \( \mathcal{H} \), then \( T \) is selfadjoint, and \( \... | Proof We first show that \( \mathcal{D}\left( T\right) \) is dense in \( \mathcal{H} \). Let \( y \in \mathcal{D}{\left( T\right) }^{ \bot } \). Since \( \mathcal{R}\left( {T - {\lambda I}}\right) = \mathcal{H} \), there exists a vector \( u \in \mathcal{D}\left( T\right) \) such that \( y = \left( {T - {\lambda I}}\ri... | Yes |
Proposition 3.13 Let \( T \) be a closed symmetric operator on \( \mathcal{H} \), and let \( {\lambda }_{ + },{\lambda }_{ - } \in \mathbb{C} \) , where \( \operatorname{Im}{\lambda }_{ + } > 0 \) and \( \operatorname{Im}{\lambda }_{ - } < 0 \) . The operator \( T \) is self-adjoint if and only if \( {d}_{ + }\left( T\... | Proof From Proposition 3.10, a self-adjoint operator satisfies all these conditions.\n\nTo prove the converse, we first note that \( \mathcal{R}\left( {T - {\lambda I}}\right) \) is closed for any \( \lambda \in \mathbb{C} \smallsetminus \mathbb{R} \) by Proposition 2.1(iv), because \( \lambda \in \pi \left( T\right) \... | Yes |
Let \( a, b \in \mathbb{R}, a < b \) . Recall from Examples 1.4 and 3.2 that \( T = - \mathrm{i}\frac{d}{dx} \) on \( \mathcal{D}\left( T\right) = {H}_{0}^{1}\left( {a, b}\right) \) is a closed symmetric operator with deficiency indices \( \left( {1,1}\right) \) . In Example 1.5 it was shown that for any \( z = {e}^{\m... | We rederive this result by applying Proposition 3.6 to the subspace \( {\mathcal{E}}_{z} \mathrel{\text{:=}} \mathbb{C} \cdot {u}_{\varphi } \) , where \( {u}_{\varphi }\left( x\right) \mathrel{\text{:=}} \exp \left( {\mathrm{i}\varphi {\left( b - a\right) }^{-1}x}\right) \) . Using formula (1.13), one verifies that \(... | Yes |
Statement \( {T}_{B} \) is a self-adjoint extension of \( T \) . | Proof Using the integration-by-parts formula (3.8), one easily verifies that \( {T}_{B} \) is symmetric. From Example 3.3 we know that the symmetric operator \( T \) has deficiency indices \( \left( {1,1}\right) \) . Clearly, \( \mathcal{D}\left( T\right) \subseteq \mathcal{D}\left( {T}_{B}\right) \), and \( {T}_{B} \)... | Yes |
Proposition 3.16 If \( T \) is a densely defined closed symmetric operator on \( \mathcal{H} \) and \( \mu \) is a real number in \( \pi \left( T\right) \), then there exists a self-adjoint extension \( A \) of \( T \) on \( \mathcal{H} \) such that \( \mu \in \rho \left( A\right) \) . | Proof Upon replacing \( T \) by \( T + {\mu I} \), we can assume without loss of generality that \( \mu = 0 \) . Let \( P \) denote the projection of \( \mathcal{H} \) onto \( \mathcal{N}\left( {T}^{ * }\right) \) . Since \( \mu = 0 \in \pi \left( T\right) \) , we have \( \mathcal{R}\left( {T}^{ * }\right) = \mathcal{H... | Yes |
Proposition 3.17 Let \( T \) be a densely defined symmetric operator on a Hilbert space \( \mathcal{H} \). Then there exists a self-adjoint operator \( A \) on a Hilbert space \( \mathcal{G} \) which contains \( \mathcal{H} \) as a subspace such that \( T \subseteq A \) and \( \mathcal{D}\left( \bar{T}\right) = \mathca... | Proof Let \( {T}_{d} \) be the operator on the \ | No |
Proposition 3.18 Let \( T \) be a densely defined closed linear operator from a Hilbert space \( {\mathcal{H}}_{1} \) into a Hilbert space \( {\mathcal{H}}_{2} \). Then:\n\n(i) \( I + {T}^{ * }T \) is a bijective mapping of \( {\mathcal{H}}_{1} \). Its inverse \( C \mathrel{\text{:=}} {\left( I + {T}^{ * }T\right) }^{-... | Proof (i): Recall that \( \mathcal{G}\left( {T}^{ * }\right) = V{\left( \mathcal{G}\left( T\right) \right) }^{ \bot } \) by Lemma 1.10, where \( V\left( {x, y}\right) = \left( {-y, x}\right), x \in {\mathcal{H}}_{1}, y \in {\mathcal{H}}_{2} \). Hence, \( {\mathcal{H}}_{2} \oplus {\mathcal{H}}_{1} = \mathcal{G}\left( {T... | Yes |
Proposition 3.19 Let \( T \) be a closed operator on \( \mathcal{H} \), and \( \Theta \left( T\right) \subseteq {S}_{c,\theta } \), where \( c \in \mathbb{R} \) and \( \theta \in \lbrack 0,\pi /2) \) . Then the following assertions are equivalent:\n\n(i) \( T \) is \( m \) -sectorial.\n\n(ii) \( \lambda \in \rho \left(... | Proof Upon replacing \( T \) by \( T + {cI} \), we can assume throughout this proof that \( c = 0 \) .\n\n(i) \( \rightarrow \) (ii): Let \( {\lambda }_{0} \) be as in Definition 3.7. Then, by Lemma \( {2.5},{\lambda }_{0} \in \pi \left( T\right) \) . Therefore, since \( T \) is closed, \( \mathcal{R}\left( {T - {\lamb... | Yes |
Proposition 3.20 If \( T \) is a closed accretive operator, the following are equivalent:\n\n(i) \( T \) is \( m \) -accretive.\n\n(ii) \( \lambda \in \rho \left( T\right) \) for some, hence all, \( \lambda \in \mathbb{C},\operatorname{Re}\lambda < 0 \) .\n\n(iii) \( \mathcal{D}\left( T\right) \) is dense in \( \mathca... | If \( T \) is \( m \) -accretive, then \( \sigma \left( T\right) \subseteq \{ \lambda \in \mathbb{C} : \operatorname{Re}\lambda \geq 0\} \), and\n\n\[ \begin{Vmatrix}{\left( T - \lambda I\right) }^{-1}\end{Vmatrix} \leq {\left| \operatorname{Re}\lambda \right| }^{-1}\;\text{ for }\operatorname{Re}\lambda < 0. \] | No |
Lemma 3.21 A linear operator \( T \) is accretive if and only if \( \parallel \left( {T + {\lambda I}}\right) x\parallel \geq \lambda \parallel x\parallel \) for all \( \lambda > 0 \) and \( x \in \mathcal{D}\left( T\right) \) . | Proof Clearly, \( \parallel \left( {T + {\lambda I}}\right) x\parallel \geq \lambda \parallel x\parallel \) means that \( \parallel {Tx}{\parallel }^{2} + {2\lambda }\operatorname{Re}\langle {Tx}, x\rangle \geq 0 \) . Since \( \lambda > 0 \) is arbitrary, the latter is obviously equivalent to \( \operatorname{Re}\langl... | Yes |
Proposition 3.22 A linear operator \( T \) on \( \mathcal{H} \) is \( m \) -accretive if and only if \( \mathcal{D}\left( T\right) \) is dense in \( \mathcal{H}, T \) is closed, \( \left( {-\infty ,0}\right) \subseteq \rho \left( T\right) \), and \( \begin{Vmatrix}{\left( T + \lambda I\right) }^{-1}\end{Vmatrix} \leq {... | Proof The necessity of these conditions follows at once from Proposition 3.20. The sufficiency will follow from Proposition 3.20,(ii) \( \rightarrow \) (i), once we have shown that \( T \) is accretive. Indeed, if \( \lambda > 0 \), then \( \begin{Vmatrix}{{\left( T + \lambda I\right) }^{-1}y}\end{Vmatrix} \leq {\lambd... | Yes |
Proposition 3.24 An m-sectorial (resp. m-accretive) operator \( T \) is maximal sectorial (resp. maximal accretive), that is, it has no proper sectorial (resp. accretive) extension acting on the same Hilbert space. | Proof Let us prove this for sectorial operators. Suppose that \( {T}_{1} \) is a sectorial extension of \( T \), say \( \Theta \left( T\right) \subseteq {S}_{c,\theta } \) and \( \Theta \left( {T}_{1}\right) \subseteq {S}_{{c}_{1},{\theta }_{1}} \) . Take \( \lambda \) from the complement of both sectors. Then \( T - {... | Yes |
Proposition 3.25 \( T \) is normal if and only if \( T \) is closed and \( {T}^{ * }T = T{T}^{ * } \) . | Proof First suppose that \( T \) is normal. By (3.17), the graph norms \( \parallel \cdot {\parallel }_{T} \) and \( \parallel \cdot {\parallel }_{{T}^{ * }} \) coincide on \( \mathcal{D}\left( T\right) = \mathcal{D}\left( {T}^{ * }\right) \) . Since \( {T}^{ * } \) is closed, \( \left( {\mathcal{D}\left( {T}^{ * }\rig... | Yes |
For \( B \in \mathbf{B}\left( \mathcal{H}\right) \), there is an operator \( {e}^{B} \in \mathbf{B}\left( \mathcal{H}\right) \) defined by the series\n\n\[ \n{e}^{B} \mathrel{\text{:=}} \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{{B}^{n}}{n!} \]\n\nwhich converges in the operator norm. If \( B, C \in \mathbf{B}\left... | Proof From the convergence of the series \( {e}^{\parallel B\parallel } = \mathop{\sum }\limits_{{n = 0}}^{\infty }\frac{1}{n!}\parallel B{\parallel }^{n} \) it follows that \( {\left( {S}_{k} \mathrel{\text{:=}} \mathop{\sum }\limits_{{n = 0}}^{k}\frac{1}{n!}{B}^{n}\right) }_{k \in \mathbb{N}} \) is a Cauchy sequence ... | Yes |
Proposition 3.28 (Fuglede’s theorem) Let \( T, S \in \mathbf{B}\left( \mathcal{H}\right) \) . If the operator \( T \) is normal and \( {TS} = {ST} \), then \( {T}^{ * }S = S{T}^{ * } \) . | Proof Fix \( x, y \in \mathcal{H} \) . We define the function \( f \) on the complex plane by\n\n\[ f\left( z\right) = \left\langle {{e}^{z{T}^{ * }}S{e}^{-z{T}^{ * }}x, y}\right\rangle ,\;z \in \mathbb{C}. \]\n\n(3.19)\n\nInserting the power series expansions of \( {e}^{z{T}^{ * }} \) and \( {e}^{-z{T}^{ * }} \) (by (... | Yes |
Let \( \left( {X,\mathfrak{A},\mu }\right) \) be a \( \sigma \) -finite measure space. Suppose that \( \varphi : X \rightarrow \mathbb{C} \cup \{ \infty \} \) is an \( \mathcal{A} \) -measurable function on \( X \) which is \( \mu \) -a.e. finite (i.e., \( {K}_{\infty } \mathrel{\text{:=}} \{ t \in X : \varphi \left( t... | First we prove that the domain \( \mathcal{D}\left( {M}_{\varphi }\right) \) is dense in \( {L}^{2}\left( {X,\mu }\right) \) . Let \( f \in {L}^{2}\left( {X,\mu }\right) \) . Let \( {\chi }_{n} \) denote the characteristic function of \( {K}_{n} \mathrel{\text{:=}} \{ t : \left| {\varphi \left( t\right) }\right| \leq n... | Yes |
The operator \( T \) in Example 2.2 is normal. Hence, any nonempty (!) closed subset of \( \mathbb{C} \) is spectrum of some normal operator. | From the spectral theorem proved in Sect. 5.5 it follows that the spectrum of a normal operator is nonempty. | No |
Let \( {\left( {\lambda }_{n}\right) }_{n \in \mathbb{N}} \) be a real sequence, and let \( {\left( {P}_{n}\right) }_{n \in \mathbb{N}} \) be a sequence of orthogonal projections on \( \mathcal{H} \) such that \( {P}_{k}{P}_{n} = 0 \) for \( k \neq n \) and \( \mathop{\sum }\limits_{{n = 1}}^{\infty }{P}_{n} = I \) . (... | Since the projections \( {P}_{n} \) are pairwise orthogonal, \( E\left( \lambda \right) \) is indeed a projection. Let \( x \in \mathcal{H} \) and \( \varepsilon > 0 \) . Since \( \mathop{\sum }\limits_{k}{P}_{k} = I \) and hence \( \mathop{\sum }\limits_{k}{\begin{Vmatrix}{P}_{k}x\end{Vmatrix}}^{2} = \parallel x{\para... | No |
Example 4.2 Let \( \mu \) be a positive regular Borel measure on an interval \( \mathcal{J} \) and \( \mathcal{H} = \) \( {L}^{2}\left( {\mathcal{J},\mu }\right) \) . Define \[ \left( {E\left( \lambda \right) f}\right) \left( t\right) = {\chi }_{( - \infty ,\lambda \rbrack }\left( t\right) \cdot f\left( t\right) \;\tex... | We verify axiom (ii). Let \( f \in \mathcal{H} \), and let \( {\left( {\lambda }_{n}\right) }_{n \in \mathbb{N}} \) be a real sequence such that \( {\lambda }_{n} > {\lambda }_{0} \) and \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{\lambda }_{n} = {\lambda }_{0} \) . Put \( {f}_{n} = {\chi }_{\left( {\lambda }_{0... | Yes |
Example 4.3 Let \( \left( {\Omega ,\mathfrak{A},\mu }\right) \) be a measure space, \( \mathcal{H} = {L}^{2}\left( {\Omega ,\mu }\right) \), and \( h : \Omega \rightarrow \mathbb{R} \) an \( \mathfrak{A} \) -measurable function. Define \( \Omega \left( \lambda \right) = \{ t \in \Omega : h\left( t\right) \leq \lambda \... | Axioms (ii) and (iii) are proved as in Example 4.2 using Lebesgue's theorem. | No |
Proposition 4.2 For \( f \in C\left( \mathbb{R}\right) \), there exists a linear operator \( {\int }_{\mathbb{R}}{fdE} \) such that\n\n\[ \mathcal{D} \mathrel{\text{:=}} \mathcal{D}\left( {{\int }_{\mathbb{R}}{fdE}}\right) = \left\{ {x \in \mathcal{H} : {\int }_{\mathbb{R}}{\left| f\left( \lambda \right) \right| }^{2}d... | Proof For \( x \in \mathcal{H} \) and \( a, b, c, d \in \mathbb{R}, a < b \), we abbreviate \( {x}_{a, b} \mathrel{\text{:=}} \left( {{\int }_{\left\lbrack a, b\right\rbrack }{fdE}}\right) x \) , \( \{ c, d\} = \left\lbrack {c, d}\right\rbrack \) if \( c \leq d \) and \( \{ c, d\} = \left\lbrack {d, c}\right\rbrack \) ... | Yes |
Example 4.4 Let \( \left( {\Omega ,\mathfrak{A},\mu }\right) \) be a measure space, and \( \mathcal{H} = {L}^{2}\left( {\Omega ,\mu }\right) \) . For \( M \in \mathfrak{A} \) , let \( E\left( M\right) \) be the multiplication operator by the characteristic function \( {\chi }_{M} \), that is,\n\n\[ \left( {E\left( M\ri... | Since \( {\chi }_{M}^{2} = {\chi }_{M} = {\bar{\chi }}_{M} \), we have \( E{\left( M\right) }^{2} = E\left( M\right) = E{\left( M\right) }^{ * } \), so \( E\left( M\right) \) is an orthogonal projection. Obviously, \( E\left( \Omega \right) = I \) .\n\nWe verify axiom (ii). Let \( {\left( {M}_{n}\right) }_{n \in \mathb... | Yes |
Lemma 4.3 If \( E \) is a finitely additive map of an algebra \( \mathfrak{A} \) into the orthogonal projections on a Hilbert space \( \mathcal{H} \), then we have\n\n\[ E\left( M\right) E\left( N\right) = E\left( {M \cap N}\right) \;\text{ for }M, N \in \mathfrak{A}. \]\n\n(4.16)\n\nIn particular, \( E\left( M\right) ... | Proof First we note that \( E\left( {M}_{1}\right) E\left( {M}_{2}\right) = 0 \) if \( {M}_{1},{M}_{2} \in \mathfrak{A} \) are disjoint. Indeed, by the finite additivity of \( E \), the sum of the two projections \( E\left( {M}_{1}\right) \) and \( E\left( {M}_{2}\right) \) is again a projection. Therefore, \( E\left( ... | Yes |
Lemma 4.4 A map \( E \) of an algebra (resp. \( \sigma \) -algebra) \( \mathfrak{A} \) on a set \( \Omega \) into the orthogonal projections on \( \mathcal{H} \) is a spectral premeasure (resp. spectral measure) if and only if \( E\left( \Omega \right) = I \) and for each vector \( x \in \mathcal{H} \), the set functio... | Proof The only if assertion follows at once from Definition 4.2.\n\nWe prove the if direction. Let \( {\left( {M}_{n}\right) }_{n \in \mathbb{N}} \) be a sequence of disjoint sets in \( \mathfrak{A} \) such that \( M \mathrel{\text{:=}} \mathop{\bigcup }\limits_{n}{M}_{n} \) is also in \( \mathfrak{A} \) . Since \( {E}... | Yes |
Proposition 4.7 For \( j = 1,2 \), let \( {E}_{j} \) be a spectral measure on \( \mathfrak{B}\left( \mathbb{R}\right) \) in a Hilbert space \( {\mathcal{H}}_{j} \) . For any operator \( S \in \mathbf{B}\left( {{\mathcal{H}}_{1},{\mathcal{H}}_{2}}\right) \), we have \( S{E}_{1}\left( \lambda \right) = {E}_{2}\left( \lam... | Proof The if part follows at once from (4.18). We prove the only if direction.\n\nLet \( \mathfrak{A} \) denote the family of sets \( M \in \mathfrak{B}\left( \mathbb{R}\right) \) for which \( S{E}_{1}\left( M\right) = {E}_{2}\left( M\right) S \) . It is easy to check that \( \mathfrak{A} \) is a \( \sigma \) -algebra.... | Yes |
Lemma 4.8 Let \( E \) be a spectral measure on \( \left( {\Omega ,\mathfrak{A}}\right) \) in a Hilbert space \( \mathcal{H} \) . (i) \( \left| {E}_{x, y}\right| \left( M\right) \leq {E}_{x}{\left( M\right) }^{1/2}{E}_{y}{\left( M\right) }^{1/2} \) for \( x, y \in \mathcal{H} \) and \( M \in \mathfrak{A} \) . | Proof (i): Let \( M \) be a disjoint union of sets \( {M}_{1},\ldots ,{M}_{n} \in \mathfrak{A}, n \in \mathbb{N} \) . Since \[ \left| {{E}_{x, y}\left( {M}_{k}\right) }\right| = \left| \left\langle {E\left( {M}_{k}\right) x, E\left( {M}_{k}\right) y}\right\rangle \right| \] \[ \leq \begin{Vmatrix}{E\left( {M}_{k}\right... | Yes |
Theorem 4.10 For \( j = 1,\ldots, k \), let \( {\Omega }_{j} \) be a locally compact Hausdorff space which has a countable base of open sets, and let \( {E}_{j} \) be a spectral measure on the Borel algebra \( \mathfrak{B}\left( {\Omega }_{j}\right) \) . Suppose that these spectral measures act on the same Hilbert spac... | \[ E\left( {{M}_{1} \times \cdots \times {M}_{k}}\right) = {E}_{1}\left( {M}_{1}\right) \cdots E\left( {M}_{k}\right) \;\text{ for }{M}_{j} \in \mathfrak{B}\left( {\Omega }_{j}\right), j = 1,\ldots, k. \] (4.21) The unique spectral measure \( E \) satisfying Eq. (4.21) is called the product of the spectral measures \( ... | Yes |
Lemma 4.11 \( \parallel \mathbb{I}\left( f\right) \parallel \leq \parallel f{\parallel }_{\Omega } \) for \( f \in {\mathcal{B}}_{\mathrm{s}} \) . | Proof Since the sets \( {M}_{1},\ldots ,{M}_{n} \in \mathfrak{A} \) in (4.23) are disjoint, \( E\left( {M}_{k}\right) \mathcal{H} \) and \( E\left( {M}_{l}\right) \mathcal{H} \) are orthogonal for \( k \neq l \) by Lemma 4.3. Using this fact, we get for \( x \in \mathcal{H} \) ,\n\n\[ \parallel \mathbb{I}\left( f\right... | Yes |
Proposition 4.12 For \( f, g \in \mathcal{B}\left( {\Omega ,\mathfrak{A}}\right) ,\alpha ,\beta \in \mathbb{C} \), and \( x, y \in \mathcal{H} \), we have:\n\n(i) \( \mathbb{I}\left( \bar{f}\right) = \mathbb{I}{\left( f\right) }^{ * },\mathbb{I}\left( {{\alpha f} + {\beta g}}\right) = \alpha \mathbb{I}\left( f\right) +... | Proof It suffices to prove assertions (i)-(iv) for simple functions \( f \) and \( g \), because by continuity all relations extend then to arbitrary functions from \( \mathcal{B} \) . For simple functions, the proofs of (i)-(iv) are straightforward verifications.\n\nWe carry out the proof of the equality \( \mathbb{I}... | Yes |
Theorem 4.13 Suppose that \( f \in \mathcal{S} \) and define\n\n\[ \mathcal{D}\left( {\mathbb{I}\left( f\right) }\right) = \left\{ {x \in \mathcal{H} : {\int }_{\Omega }{\left| f\left( t\right) \right| }^{2}d\langle E\left( t\right) x, x\rangle < \infty }\right\} .\n\]\n\nLet \( {\left( {M}_{n}\right) }_{n \in \mathbb{... | Proof (i): First suppose that \( x \in \mathcal{D}\left( {\mathbb{I}\left( f\right) }\right) \) . Since \( f \) is bounded on \( {M}_{n} \) and hence \( f{\chi }_{{M}_{n}} \in \mathcal{B} \), the bounded operator \( \mathbb{I}\left( {f{\chi }_{{M}_{n}}}\right) \) is defined by the preceding subsection. Using Propositio... | Yes |
Corollary 4.14 Let \( n \in \mathbb{N} \) and \( {f}_{1},\ldots ,{f}_{n} \in \mathcal{S} \) . Then \( \mathop{\bigcap }\limits_{{k = 1}}^{n}\mathcal{D}\left( {\mathbb{I}\left( {f}_{k}\right) }\right) \) is a core for each operator \( \mathbb{I}\left( {f}_{k}\right) \) . | Proof Choose a bounding sequence for all \( {f}_{j} \) and apply Theorem 4.13(iii). | No |
Proposition 4.15 Let \( f, g \in \mathcal{S} \) and \( x \in \mathcal{D}\left( {\mathbb{I}\left( f\right) }\right), y \in \mathcal{D}\left( {\mathbb{I}\left( g\right) }\right) \) . Then we have\n\n\[ \langle \mathbb{I}\left( f\right) x,\mathbb{I}\left( g\right) y\rangle = {\int }_{\Omega }f\left( t\right) \overline{g\l... | Proof Equation (4.33) follows from (4.32) by setting \( f = g, x = y \) .\n\nWe prove formula (4.32). Applying Proposition 4.12(ii) to the bounded function \( f\bar{g}{\chi }_{{M}_{n}} \) and Proposition 4.12(i), we obtain\n\n\[ {\int }_{\Omega }f\bar{g}{\chi }_{{M}_{n}}d{E}_{x, y} = \left\langle {\mathbb{I}\left( {f\b... | Yes |
Theorem 4.16 For \( f, g \in \mathcal{S}\left( {\Omega ,\mathfrak{A}, E}\right) \) and \( \alpha ,\beta \in \mathbb{C} \), we have:\n\n(i) \( \mathbb{I}\left( \bar{f}\right) = \mathbb{I}{\left( f\right) }^{ * } \) | Proof We choose a bounding sequence \( {\left( {M}_{n}\right) }_{n \in \mathbb{N}} \) for \( f \) and \( g \) and abbreviate \( {E}_{n} \mathrel{\text{:=}} E\left( {M}_{n}\right) \) and \( {h}_{n} \mathrel{\text{:=}} h{\chi }_{{M}_{n}} \) for \( h \in \mathcal{S} \) . Recall that \( \mathop{\lim }\limits_{n}{E}_{n}x = ... | Yes |
Proposition 4.18 The operator \( \mathbb{I}\left( f\right) \) is bounded if and only if \( f \in {L}^{\infty }\left( {\Omega, E}\right) \) . In this case, \( \parallel \mathbb{I}\left( f\right) \parallel = \parallel f{\parallel }_{\infty } \) . | Proof Obviously, if \( f \in {L}^{\infty }\left( {\Omega, E}\right) \), then \( \mathbb{I}\left( f\right) \) is bounded and \( \parallel \mathbb{I}\left( f\right) \parallel \leq \parallel f{\parallel }_{\infty } \) . Conversely, suppose \( \mathbb{I}\left( f\right) \) is bounded. Set \( {M}_{n} \mathrel{\text{:=}} \lef... | Yes |
Proposition 4.19 The operator \( \mathbb{I}\left( f\right) \) is invertible if and only if \( f\left( t\right) \neq 0 \) E-a.e. on \( \Omega \) . In this case we have \( \mathbb{I}{\left( f\right) }^{-1} = \mathbb{I}\left( {f}^{-1}\right) \) . Here \( {f}^{-1} \) denotes the function from \( \mathcal{S} \) which is def... | Proof Set \( \mathcal{N}\left( f\right) \mathrel{\text{:=}} \{ t \in \Omega : f\left( t\right) = 0\} \) . Then \( \mathbb{I}\left( f\right) E\left( {\mathcal{N}\left( f\right) }\right) = \mathbb{I}\left( {f{\chi }_{\mathcal{N}\left( f\right) }}\right) = \) \( \mathbb{I}\left( 0\right) = 0 \) by Theorem 4.16. Hence, \( ... | Yes |
Example 4.8 Let \( \lambda \in \rho \left( {\mathbb{I}\left( f\right) }\right) \) . By Theorem 4.16(v) and Proposition 4.40(i) we then have \( \mathbb{I}\left( {f{\left( f - \lambda \right) }^{-1}}\right) = \mathbb{I}\left( f\right) \mathbb{I}\left( {\left( f - \lambda \right) }^{-1}\right) = \mathbb{I}\left( f\right) ... | Also, \( \mathbb{I}\left( {f{\left( f - \lambda \right) }^{-1}}\right) \supseteq \mathbb{I}\left( {\left( f - \lambda \right) }^{-1}\right) \mathbb{I}\left( f\right) \), but the domain of \( \mathbb{I}\left( {\left( f - \lambda \right) }^{-1}\right) \mathbb{I}\left( f\right) \) is only \( \mathcal{D}\left( {\mathbb{I}\... | Yes |
An operator \( T \in \mathbf{B}\left( \mathcal{H}\right) \) commutes with a spectral measure \( E \) on \( \mathcal{H} \) (that is, \( {TE}\left( M\right) = E\left( M\right) T \) for all \( M \in \mathfrak{A} \) ) if and only if \( T\mathbb{I}\left( f\right) \subseteq \mathbb{I}\left( f\right) T \) for all \( f \in \ma... | The if part is clear, because \( \mathbb{I}\left( {\chi }_{M}\right) = E\left( M\right) \) . We prove the only if assertion. For \( x \in \mathcal{H} \) and \( M \in \mathfrak{A} \), we have\n\n\[ \n{E}_{Tx}\left( M\right) = \parallel E\left( M\right) {Tx}{\parallel }^{2} = \parallel {TE}\left( M\right) x{\parallel }^{... | Yes |
Proposition 4.24 If \( h \in \mathcal{S}\left( {{\Omega }_{0},{\mathfrak{A}}_{0}, F}\right) \), then \( h \circ \varphi \in \mathcal{S}\left( {\Omega ,\mathfrak{A}, E}\right) \), and\n\n\[ \n{\int }_{{\Omega }_{0}}h\left( s\right) {dF}\left( s\right) = {\int }_{\Omega }h\left( {\varphi \left( t\right) }\right) {dE}\lef... | Proof If follows at once from definition (4.43) that \( h \circ \varphi \) is in \( \mathcal{S}\left( {\Omega ,\mathfrak{A}, E}\right) \) . From the transformation formula for scalar measures (see (B.6)) we derive\n\n\[ \n{\int }_{{\Omega }_{0}}{\left| h\left( s\right) \right| }^{2}d\langle F\left( s\right) x, x\rangle... | Yes |
\[ \mathcal{D}\left( {\mathbb{I}\left( f\right) }\right) = \left\{ {x \in \mathcal{H} : \mathop{\sum }\limits_{n}{\left| f\left( {\lambda }_{n}\right) \right| }^{2}{\begin{Vmatrix}{P}_{n}x\end{Vmatrix}}^{2} < \infty }\right\} , \] | \[ \mathbb{I}\left( f\right) x = \mathop{\sum }\limits_{n}f\left( {\lambda }_{n}\right) {P}_{n}x\;\text{ for }x \in \mathcal{D}\left( {\mathbb{I}\left( f\right) }\right) . \] | No |
Example 4.11 (Example 4.7 continued)\n\n\[ \mathcal{D}\left( {\mathbb{I}\left( f\right) }\right) = \left\{ {x \in {L}^{2}\left( {\Omega ,\mu }\right) : {\int }_{\Omega }{\left| f\left( h\left( t\right) \right) \right| }^{2}{\left| x\left( t\right) \right| }^{2}{d\mu }\left( t\right) < \infty }\right\} ,\n\]\n\n\[ \left... | For characteristic functions \( f = {\chi }_{M} \), these formulas are just the definitions of the spectral measures given in Examples 4.5, 4.6, and 4.7. Hence, the formulas hold for simple functions by linearity and for arbitrary functions \( f \in \mathcal{S} \) by taking limits. | No |
Lemma 5.2 \( \sigma \left( {p\left( A\right) }\right) \subseteq p\left( {\sigma \left( A\right) }\right) \) for \( p \in \mathbb{C}\left\lbrack t\right\rbrack \) . | Proof Let \( \gamma \in \sigma \left( {p\left( A\right) }\right) \) . Clearly, we can assume that \( n \mathrel{\text{:=}} \deg p > 0 \) . By the fundamental theorem of algebra, there are complex numbers \( {\alpha }_{1},\ldots ,{\alpha }_{n} \) such that \( p\left( t\right) - \gamma = {a}_{n}\left( {t - {\alpha }_{1}}... | Yes |
Lemma 5.3 For \( p \in \mathbb{C}\left\lbrack t\right\rbrack \), we have \( \parallel p\left( A\right) \parallel \leq \parallel p{\parallel }_{\mathcal{J}} \) . | Proof Using Lemma 5.2, applied to \( \bar{p}p \), and the relation \( \sigma \left( A\right) \subseteq \mathcal{J} \), we obtain\n\n\[ \parallel p\left( A\right) {\parallel }^{2} = \begin{Vmatrix}{p{\left( A\right) }^{ * }p\left( A\right) }\end{Vmatrix} = \begin{Vmatrix}{\left( {\overline{p}p}\right) \left( A\right) }\... | Yes |
For each continuous linear functional \( F \) on the normed linear space \( \left( {\mathbb{C}\left\lbrack t\right\rbrack ,\parallel \cdot {\parallel }_{\mathcal{J}}}\right) \), there exists a unique complex regular Borel measure \( \mu \) on \( \mathcal{J} \) such that\n\n\[ F\left( p\right) = {\int }_{\mathcal{J}}p\l... | Proof By the Weierstrass theorem, the polynomials are dense in \( \left( {C\left( \mathcal{J}\right) ,\parallel \cdot {\parallel }_{\mathcal{J}}}\right) \) , so each continuous linear functional on \( \left( {\mathbb{C}\left\lbrack t\right\rbrack ,\parallel \cdot {\parallel }_{\mathcal{J}}}\right) \) has a unique exten... | Yes |
Corollary 5.5 Let \( A \) be a positive bounded self-adjoint operator on \( \mathcal{H} \). There exists a unique positive self-adjoint operator \( B \) on \( \mathcal{H} \), denoted by \( {A}^{1/2} \), such that \( {B}^{2} = A \). If \( {\left( {p}_{n}\right) }_{n \in \mathbb{N}} \) is a sequence of polynomials such t... | Proof Since \( A \) is positive and bounded, \( \sigma \left( A\right) \subseteq \left\lbrack {0, b}\right\rbrack \) for some \( b > 0 \). By Theorem 5.1, there is a spectral measure \( E \) such that \( A = {\int }_{\left\lbrack 0, b\right\rbrack }{\lambda dE}\left( \lambda \right) \). Setting \( B \mathrel{\text{:=}}... | Yes |
Theorem 5.7 Let \( A \) be a self-adjoint operator on a Hilbert space \( \mathcal{H} \) . Then there exists a unique spectral measure \( E = {E}_{A} \) on the Borel \( \sigma \) -algebra \( \mathfrak{B}\left( \mathbb{R}\right) \) such that\n\n\[ A = {\int }_{\mathbb{R}}{\lambda d}{E}_{A}\left( \lambda \right) \] | First, we explain the idea of our proof of Theorem 5.7. Clearly, the mapping \( t \rightarrow {z}_{t} \mathrel{\text{:=}} t{\left( 1 + {t}^{2}\right) }^{-1/2} \) is a homeomorphism of \( \mathbb{R} \) on the interval \( \left( {-1,1}\right) \) . We define an operator analog of this mapping to transform the unbounded se... | Yes |
Let \( \mu \) be a positive regular Borel measure on an interval \( \mathcal{J} \) . For the self-adjoint operator \( A \) on \( \mathcal{H} = {L}^{2}\left( {\mathcal{J},\mu }\right) \) defined by\n\n\[ \left( {Ag}\right) \left( t\right) = {tg}\left( t\right) ,\;\mathcal{D}\left( A\right) = \left\{ {g \in {L}^{2}\left(... | the spectral measure and functions are given by\n\n\[ \left( {{E}_{A}\left( M\right) g}\right) \left( t\right) = {\chi }_{M}\left( t\right) \cdot g\left( t\right) ,\;g \in \mathcal{H}, M \in \mathfrak{B}\left( \mathcal{J}\right) ,\]\n\n\[ \left( {f\left( A\right) g}\right) \left( t\right) = f\left( t\right) \cdot g\lef... | Yes |
Proposition 5.12 For a self-adjoint operator \( A \) on an infinite dimensional Hilbert space \( \mathcal{H} \), the following are equivalent:\n\n(i) There exist a real sequence \( {\left( {\lambda }_{n}\right) }_{n \in \mathbb{N}} \) and an orthonormal basis \( \left\{ {{e}_{n} : n \in \mathbb{N}}\right\} \) of \( \ma... | Proof (i) \( \rightarrow \) (ii): From the properties stated in (i) one easily verifies that \( A \) has a purely discrete spectrum \( \sigma \left( A\right) = \left\{ {{\lambda }_{n} : n \in \mathbb{N}}\right\} \. | No |
Proposition 5.13 If \( A \) is a positive self-adjoint operator on \( \mathcal{H} \), then there is unique positive self-adjoint operator \( B \) on \( \mathcal{H} \) such that \( {B}^{2} = A \) . | Proof Since \( A \geq 0 \), we have \( \operatorname{supp}{E}_{A} = \sigma \left( A\right) \subseteq \lbrack 0, + \infty ) \) by Proposition 5.10. Hence, the function \( f\left( \lambda \right) \mathrel{\text{:=}} \sqrt{\lambda } \) of \( \mathcal{S} \) is nonnegative \( {E}_{A} \) -a.e. on \( \mathbb{R} \) . Therefore... | Yes |
Proposition 5.14 (Stone’s formulas) Suppose that \( a, b \in \mathbb{R} \cup \{ - \infty \} \cup \{ + \infty \} \), \( a < b \), and \( c \in \mathbb{R} \). Then we have\n\n\[ \n{E}_{A}\left( \left\lbrack {a, b}\right\rbrack \right) + {E}_{A}\left( \left( {a, b}\right) \right)\n\]\n\n\[ \n= \mathop{\operatorname{s-lim}... | Proof For \( a, b \in \overline{\mathbb{R}} \mathrel{\text{:=}} \mathbb{R} \cup \{ - \infty \} \cup \{ + \infty \} \) and \( \varepsilon > 0 \), we define the functions\n\n\[ \n{f}_{\varepsilon }\left( {\lambda, t}\right) = \frac{1}{\pi \mathrm{i}}\left( {{\left( \lambda - \left( t + \mathrm{i}\varepsilon \right) \righ... | Yes |
Proposition 5.15 Let \( {A}_{1} \) and \( {A}_{2} \) be self-adjoint operators on Hilbert spaces \( {\mathcal{H}}_{1} \) and \( {\mathcal{H}}_{2} \), respectively, and let \( S \in \mathbf{B}\left( {{\mathcal{H}}_{1},{\mathcal{H}}_{2}}\right) \) . Then the following are equivalent:\n\n(i) \( S{A}_{1} \subseteq {A}_{2}S... | Proof (i) \( \leftrightarrow \) (ii): Fix \( \lambda \in \rho \left( {A}_{1}\right) \cap \rho \left( {A}_{2}\right) \) . Clearly,(i) is equivalent to \( S\left( {{A}_{1} - {\lambda I}}\right) \subseteq \) \( \left( {{A}_{2} - {\lambda I}}\right) S \) and so to \( {R}_{\lambda }\left( {A}_{2}\right) S = S{R}_{\lambda }\... | Yes |
Proposition 5.16 Let \( A \) be a positive self-adjoint operator on \( \mathcal{H} \) and \( \alpha \in \left( {0,1}\right) \) .\n\n(i) A vector \( x \in \mathcal{H} \) belongs to the domain \( \mathcal{D}\left( {A}^{\alpha /2}\right) \) if and only if the improper integral \( {\int }_{0}^{\infty }{t}^{\alpha - 1}\left... | Proof All formulas in Proposition 5.16 remain unchanged when \( x \) is replaced by \( x - {E}_{A}\left( {\{ 0\} }\right) x \) . Hence, we can assume without loss of generality that \( {E}_{A}\left( {\{ 0\} }\right) x = 0 \) . Further, we abbreviate \( {b}_{\alpha } \mathrel{\text{:=}} {\pi }^{-1}\sin {\pi \alpha },{P}... | Yes |
Lemma 5.17 Let \( \mathcal{N} \) be a subset of \( \mathcal{H} \), and let \( {\mathcal{H}}_{\mathcal{N}} \) be the closed linear span of vectors \( {E}_{A}\left( M\right) x \), where \( x \in \mathcal{N} \) and \( M \in \mathfrak{B}\left( \mathbb{R}\right) \) . Then \( {\mathcal{H}}_{\mathcal{N}} \) is smallest reduci... | Proof Let \( {\mathcal{H}}_{0} \) be a closed subspace of \( \mathcal{H} \), and let \( {P}_{0} \) be the projection onto \( {\mathcal{H}}_{0} \) . By Proposition 1.15, \( {\mathcal{H}}_{0} \) is reducing for \( A \) if and only if \( {P}_{0}A \subseteq A{P}_{0} \), or equivalently by Proposition 5.15, if \( {P}_{0}{E}... | Yes |
Statement \( {1\lambda } \in \mathbb{R} \) is an eigenvalue of \( {A}_{t} \) if and only if \( \mu \left( {\{ \lambda \} }\right) \neq 0 \) . Each eigenvalue of \( {A}_{t} \) has multiplicity one. | Proof Both assertions follow immediately from that fact that \( \mathcal{N}\left( {{A}_{t} - {\lambda I}}\right) \) consists of complex multiples of the characteristic function of the point \( \lambda \) . | No |
Proposition 5.18 Let \( x \) be a generating vector for the self-adjoint operator \( A \) on \( \mathcal{H} \) . Set \( \mu \left( \cdot \right) \mathrel{\text{:=}} \left\langle {{E}_{A}\left( \cdot \right) x, x}\right\rangle \) . Then the map \( \left( {U\left( {f\left( A\right) x}\right) }\right) \left( t\right) = f\... | Proof For \( f \in {\mathcal{F}}_{x} \) it follows from Theorem 5.9,2) that\n\n\[ \parallel f\left( A\right) x{\parallel }^{2} = {\int }_{\mathbb{R}}{\left| f\left( t\right) \right| }^{2}d\left\langle {{E}_{A}\left( t\right) x, x}\right\rangle = \parallel f{\parallel }_{{L}^{2}\left( {\mathbb{R},\mu }\right) }^{2}. \]\... | Yes |
Corollary 5.19 If the self-adjoint operator \( A \) has a simple spectrum, each eigenvalue of \( A \) has multiplicity one. | Proof Combine Proposition 5.18 with Statement 1 of Example 5.4. | No |
Proposition 5.20 A self-adjoint operator A on \( \mathcal{H} \) has a simple spectrum if and only if there exists a vector \( x \in \mathop{\bigcap }\limits_{{n = 0}}^{\infty }\mathcal{D}\left( {A}^{n}\right) \) such that \( \operatorname{Lin}\left\{ {{A}^{n}x : n \in {\mathbb{N}}_{0}}\right\} \) is dense in \( \mathca... | Proof Suppose that the condition is fulfilled. By the definition of spectral integrals, \( {A}^{n}x \) is a limit of linear combinations of vectors \( {E}_{A}\left( M\right) x \) . Therefore, the density of \( \operatorname{Lin}\left\{ {{A}^{n}x : n \in {\mathbb{N}}_{0}}\right\} \) in \( \mathcal{H} \) implies that the... | Yes |
Lemma 5.22 Let \( E = {E}_{1} \times \cdots \times {E}_{m} \) be the product measure (by Theorem 4.10) of spectral measures \( {E}_{1},\ldots ,{E}_{m} \) on \( \mathfrak{B}\left( \mathbb{R}\right) \) . If \( f \) is a Borel function on \( \mathbb{R} \), then\n\n\[ \n{\int }_{\mathbb{R}}f\left( {\lambda }_{k}\right) d{E... | Proof Formula (5.27) holds for any characteristic function \( {\chi }_{M}, M \in \mathfrak{B}\left( \mathbb{R}\right) \), since\n\n\[ \n{\int }_{\mathbb{R}}{\chi }_{M}\left( {\lambda }_{k}\right) d{E}_{k}\left( {\lambda }_{k}\right) = {E}_{k}\left( M\right) = E\left( {\mathbb{R} \times \cdots \times M \times \cdots \ti... | Yes |
Proposition 5.24 Let \( s = \left( {{s}_{1},\ldots ,{s}_{n}}\right) \in {\mathbb{C}}^{n} \). Then:\n\n(i) \( s \) belongs to \( \sigma \left( T\right) \) if and only if there exists a sequence \( {\left( {x}_{k}\right) }_{k \in \mathbb{N}} \) of unit vectors \( {x}_{k} \in \mathcal{D}\left( {T}_{j}\right) \) such that ... | Proof Using properties of spectral integrals, we compute for \( M \in \mathfrak{B}\left( {\mathbb{C}}^{n}\right) \),\n\n\[ \mathop{\sum }\limits_{{j = 1}}^{n}{\begin{Vmatrix}\left( {T}_{j} - {s}_{j}I\right) {E}_{T}\left( M\right) x\end{Vmatrix}}^{2} = \mathop{\sum }\limits_{{j = 1}}^{n}{\int }_{M}{\left| {t}_{j} - {s}_... | Yes |
If \( f : \sigma \left( T\right) \rightarrow \mathbb{C} \) is a continuous function, then \( \sigma \left( {f\left( T\right) }\right) \) is the closure of the set \( f\left( {\sigma \left( T\right) }\right) \), that is,\n\n\[ \sigma \left( {f\left( T\right) }\right) = \overline{f\left( {\sigma \left( T\right) }\right) ... | Proof Let \( {\lambda }_{0} \) be in the closure of \( f\left( {\sigma \left( T\right) }\right) \) . Let \( \varepsilon > 0 \) . Then there exists \( {t}_{0} \in \sigma \left( T\right) \) such that \( \left| {{\lambda }_{0} - f\left( {t}_{0}\right) }\right| < \varepsilon /2 \) . By the continuity of \( f \) there is a ... | Yes |
Proposition 5.26 Let \( T = \left\{ {{T}_{1},\ldots ,{T}_{n}}\right\} \) and \( E = {E}_{T} \) be as in Theorem 5.21. Recall that \( {Z}_{k} = {Z}_{{T}_{k}} \) is the bounded transform of \( {T}_{k}, k = 1,\ldots, n \), defined by (5.6). For each bounded operator \( S \in \mathbf{B}\left( \mathcal{H}\right) \), the fol... | Proof (i) \( \rightarrow \) (iv): First we prove this in the case where all operators \( {T}_{1},\ldots ,{T}_{n} \) are bounded. Then supp \( E \) is compact, since \( \operatorname{supp}E = \sigma \left( T\right) \subseteq \sigma \left( {T}_{1}\right) \times \cdots \times \sigma \left( {T}_{n}\right) \) by Proposition... | Yes |
Proposition 5.27 Let \( {T}_{1} \) and \( {T}_{2} \) be normal operators on a Hilbert space \( \mathcal{H} \), and let \( {E}_{{T}_{1}} \) and \( {E}_{{T}_{2}} \) be their spectral measures. Consider the following statements:\n\n(i) \( {T}_{1} \) and \( {T}_{2} \) strongly commute, that is, \( {Z}_{{T}_{1}}{Z}_{{T}_{2}... | Proof All assertions are easily derived from Proposition 5.26. As a sample, we verify the equivalence of (i) and (iv). Indeed, \( {Z}_{{T}_{1}}{Z}_{{T}_{2}} = {Z}_{{T}_{2}}{Z}_{{T}_{1}} \) is equivalent to \( {Z}_{{T}_{1}}{\left( {T}_{2} - {s}_{2}I\right) }^{-1} = {\left( {T}_{2} - {s}_{2}I\right) }^{-1}{Z}_{{T}_{1}} \... | Yes |
Corollary 5.28 Suppose that \( {T}_{1} \) and \( {T}_{2} \) are strongly commuting normal operators on \( \mathcal{H} \) . Then there exists a dense linear subspace \( \mathcal{D} \) of \( \mathcal{H} \) such that\n\n(i) \( {T}_{k}\mathcal{D} \subseteq \mathcal{D} \) and \( {T}_{k}^{ * }\mathcal{D} \subseteq \mathcal{D... | Proof Let \( {E}_{T} \) be the spectral measure of the strongly commuting pair \( T = \left( {{T}_{1},{T}_{2}}\right) \) and put \( \mathcal{D} \mathrel{\text{:=}} \mathop{\bigcup }\limits_{{n = 1}}^{\infty }{E}_{T}\left( {M}_{n}\right) \mathcal{H} \), where \( {M}_{n} = \left\{ {\left( {{t}_{1},{t}_{2}}\right) \in {\m... | Yes |
Proposition 5.29 Let \( E \) be a spectral measure on \( \left( {\Omega ,\mathfrak{A}}\right) \), and \( f, g \in \mathcal{S}\left( {\Omega ,\mathfrak{A}, E}\right) \). (i) The spectral measure \( {E}_{\mathbb{I}\left( f\right) } \) of the normal operator \( \mathbb{I}\left( f\right) \) is given by \[ {E}_{\mathbb{I}\l... | Proof (i): Let \( F \) denote the spectral measure on \( \mathfrak{B}\left( \mathbb{C}\right) \) defined by (5.36). From Proposition 4.24, applied with \( {\Omega }_{0} = f\left( \Omega \right) \smallsetminus \{ \infty \} ,\varphi \left( t\right) = f\left( t\right), h\left( s\right) = s \), we obtain \[ {\int }_{\mathb... | Yes |
Proposition 5.30 If \( A \) and \( B \) are strongly commuting self-adjoint operators on a Hilbert space \( \mathcal{H} \), then \( N = A + \mathrm{i}B \) is a normal operator, and \( {N}^{ * } = A - \mathrm{i}B \) . | Proof Let \( {E}_{T} \) be the spectral measure of the pair \( T = \left( {A, B}\right) \). Let \( f \) be the function on \( {\mathbb{R}}^{2} \) defined by \( f = {f}_{1} + \mathrm{i}{f}_{2} \), where \( {f}_{j}\left( {{\lambda }_{1},{\lambda }_{2}}\right) = {\lambda }_{j} \). The spectral integral \( N \mathrel{\text... | Yes |
Lemma 5.31 Let \( X \) and \( Y \) be bounded self-adjoint operators on a Hilbert space \( \mathcal{H} \) such that \( \mathcal{N}\left( X\right) = \mathcal{N}\left( Y\right) = \{ 0\} \), and let \( Q \) be the projection onto the closure of \( \left\lbrack {X, Y}\right\rbrack \mathcal{H} \) . Set \( \mathcal{D} = {XY}... | Proof (i): Let \( y \in \mathcal{H} \) . For arbitrary \( u \in \mathcal{H} \), we have by the definition of \( Q \) ,\n\n\[ \langle \left( {{YX} - {XY}}\right) \left( {I - Q}\right) y, u\rangle = \langle \left( {I - Q}\right) y,\left\lbrack {X, Y}\right\rbrack u\rangle = 0. \]\n\nHence, \( x \mathrel{\text{:=}} {XY}\l... | Yes |
Statement \( \mathbf{1}A \mathrel{\text{:=}} {X}^{-1} \) and \( B \mathrel{\text{:=}} {Y}^{-1} \) are self-adjoint operators on \( \mathcal{H} \) . They commute pointwise on the common core \( \mathcal{D} \mathrel{\text{:=}} {XY}\left( {I - Q}\right) \mathcal{H} \) for \( A \) and \( B \), but they do not commute stron... | Proof Since \( \mathcal{N}\left( X\right) = \mathcal{N}\left( Y\right) = \{ 0\}, A \) and \( B \) are well defined. They are self-adjoint by Corollary 1.9 and commute on the common core \( \mathcal{D} \) by Lemma 5.31. Since their resolvents \( {R}_{0}\left( A\right) = X \) and \( {R}_{0}\left( B\right) = Y \) do not c... | Yes |
Proposition 6.1 Let \( A \) be a self-adjoint operator on a Hilbert space \( \mathcal{H} \). Then \( U = \left\{ {U\left( t\right) \mathrel{\text{:=}} {e}^{\mathrm{i}{tA}} : t \in \mathbb{R}}\right\} \) is a strongly continuous one-parameter unitary group on \( \mathcal{H} \). The operator \( A \) is uniquely determine... | Proof From the functional calculus of \( A \) (Theorem 5.9) it follows at once that axiom (i) of Definition 6.1 holds and that \( U\left( t\right) \) is unitary.\n\nLet \( x \in \mathcal{H} \). Put \( {f}_{h}\left( \lambda \right) \mathrel{\text{:=}} {e}^{\mathrm{i}{h\lambda }} - 1 \). Clearly, \( {f}_{h}\left( \lambda... | Yes |
Theorem 6.2 (Stone's theorem) If \( U \) is a strongly continuous one-parameter unitary group on \( \mathcal{H} \), then there is a unique self-adjoint operator \( A \) on \( \mathcal{H} \) such that \( U\left( t\right) = {e}^{\mathrm{i}{tA}} \) for \( t \in \mathbb{R} \) . | The uniqueness assertion is already contained in Proposition 6.1. It remains to prove the existence.\n\nFor \( x \in \mathcal{H} \) and \( f \in {C}_{0}^{\infty }\left( \mathbb{R}\right) \), we define a \ | No |
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