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Proposition 6.3 Let \( A \) be a self-adjoint operator on a Hilbert space \( \mathcal{H} \), and let \( {\mathcal{D}}_{1} \) and \( {\mathcal{D}}_{2} \) be linear subspaces of \( \mathcal{H} \) such that \( {\mathcal{D}}_{2} \subseteq \mathcal{D}\left( A\right) \) and \( {\mathcal{D}}_{1} \) is dense in \( \mathcal{H} ... | Proof Suppose that \( \tau \in \{ 1, - 1\} \) and \( y \in \mathcal{N}\left( {{\left( A \mid {\mathcal{D}}_{2}\right) }^{ * } - \tau \mathrm{i}I}\right) \) . Let \( x \in {\mathcal{D}}_{1} \) . By assumption, \( {e}^{\mathrm{i}{tA}}x \in {\mathcal{D}}_{2} \subseteq \mathcal{D}\left( A\right) \) for \( \left| t\right| <... | Yes |
The translation group \( \left( {U\left( t\right) f}\right) \left( x\right) = f\left( {x + t}\right), t \in \mathbb{R} \), is a unitary group on \( \mathcal{H} = {L}^{2}\left( \mathbb{R}\right) \) with generator \( \mathrm{i}A \) given by \( \mathrm{i}{Af} = {f}^{\prime } \) for \( f \in \mathcal{D}\left( A\right) = {H... | Clearly, \( U\left( t\right) \) is unitary and axiom (i) of Definition 6.1 is satisfied.\n\nLet \( f \in {C}_{0}^{\infty }\left( \mathbb{R}\right) \) . For all \( h \in \mathbb{R},0 < \left| h\right| < 1 \), the functions \( U\left( h\right) f \) and \( {h}^{-1}\left( {\left( {U\left( h\right) - I}\right) f}\right) \le... | Yes |
For any \( {u}_{0} \in \mathcal{D}\left( A\right) \), the Cauchy problem of the Schrödinger equation (6.13) has a unique solution, which is given by \[ u\left( t\right) = {e}^{-\mathrm{i}{tA}}{u}_{0},\;t \in \mathbb{R}. \] | Proof That \( u\left( t\right) \) satisfies Eq. (6.13) follows at once from Eq. (6.4). Since obviously \( u\left( 0\right) = {u}_{0}, u\left( t\right) \) solves the Cauchy problem.\n\nTo prove the uniqueness, let \( {u}_{1} \) and \( {u}_{2} \) be two solutions and set \( v \mathrel{\text{:=}} {u}_{1} - {u}_{2} \) . Th... | Yes |
Proposition 6.6 The Cauchy problem for the heat equation has a unique solution | \[ u\left( t\right) = {e}^{-{tA}}{u}_{0},\;t > 0. \] Proof Fix \( t > 0 \) . Put \( {f}_{h}\left( \lambda \right) \mathrel{\text{:=}} {h}^{-1}\left( {{e}^{-\left( {t + h}\right) \lambda } - {e}^{-{t\lambda }}}\right) + \lambda {e}^{-{t\lambda }} \) for \( 0 < \left| h\right| < t/2 \) and \( {g}_{s}\left( \lambda \right... | Yes |
Proposition 6.7 Suppose that \( {u}_{0} \in \mathcal{D}\left( A\right) \) and \( {u}_{1} \in \mathcal{D}\left( {A}^{1/2}\right) \) . Then\n\n\[ u\left( t\right) \mathrel{\text{:=}} \left( {\cos {A}^{1/2}t}\right) {u}_{0} + \left( {{A}^{-1/2}\sin {A}^{1/2}t}\right) {u}_{1},\;t \in \mathbb{R}, \]\n\nis the unique solutio... | Proof First we note that \( u\left( t\right) \) is well defined, since \( \mathcal{N}\left( {A}^{1/2}\right) = \{ 0\} \) by the assumption \( \mathcal{N}\left( A\right) = \{ 0\} \) . Proceeding as in the proof of Proposition 6.6, one shows that \( u\left( t\right) \) is in \( {C}^{2}\left( {\mathbb{R},\mathcal{H}}\righ... | Yes |
Example 6.2 Suppose that \( A \) is a self-adjoint operator on a Hilbert space \( \mathcal{H} \) . Then, by Proposition 6.2, \( \left\{ {{e}^{\mathrm{i}{tA}} : t \geq 0}\right\} \) is a contraction semigroup. | Equations (6.3) and (6.4) say that the operator \( \mathrm{i}A \) is the generator of this contraction semigroup. | Yes |
Let \( \mathcal{J} \) be an interval, and \( E = {L}^{p}\left( \mathcal{J}\right) \) , \( p \in \lbrack 1, + \infty ) \) . For \( f \in E \) and \( t \in \lbrack 0, + \infty ) \), we define the right translation \( T\left( t\right) f \) by\n\n\[ \left( {T\left( t\right) f}\right) \left( x\right) = f\left( {x - t}\right... | It is not difficult to verify that \( \{ T\left( t\right) : t \geq 0\} \) is a contraction semigroup on \( E \) . \( \; \circ \) | No |
Lemma 6.9 \( \mathop{\lim }\limits_{{t \rightarrow + 0}}{t}^{-1}{x}_{a, t} = T\left( a\right) x \) . In particular, \( \mathop{\lim }\limits_{{t \rightarrow + 0}}{t}^{-1}{x}_{t} = x \) . | Proof By axiom (ii) in Definition 6.2, given \( \varepsilon > 0 \), there exists \( \delta > 0 \) such that \( \parallel \left( {T\left( s\right) - T\left( a\right) }\right) x\parallel \leq \varepsilon \) when \( s \in \left\lbrack {a, a + \delta }\right\rbrack \) . For \( 0 < t \leq \delta \), we then obtain\n\n\[ \be... | Yes |
Proposition 6.10 Let \( B \) be the generator of a contraction semigroup \( T \) on \( E \) . Each number \( \lambda \in \mathbb{C},\operatorname{Re}\lambda > 0 \), is in \( \rho \left( B\right) \) and \( \begin{Vmatrix}{\left( B - \lambda I\right) }^{-1}\end{Vmatrix} \leq {\left( \operatorname{Re}\lambda \right) }^{-1... | Proof Fix a number \( \lambda \in \mathbb{C},\operatorname{Re}\lambda > 0 \), and let \( {S}_{\lambda } \) denote the operator defined by the right-hand side of Eq. (6.18). Since \( \parallel T\left( s\right) \parallel \leq 1 \) and hence \[ \begin{Vmatrix}{{S}_{\lambda }x}\end{Vmatrix} \leq {\int }_{0}^{\infty }{e}^{-... | Yes |
Theorem 6.11 A linear operator B on a Banach space \( E \) is generator of a strongly continuous one-parameter contraction semigroup if and only if \( B \) is densely defined, closed, \( \left( {0,\infty }\right) \subseteq \rho \left( B\right) \), and\n\n\[ \begin{Vmatrix}{\left( B - \lambda I\right) }^{-1}\end{Vmatrix... | Proof The necessity of these conditions was already shown by Propositions 6.8 and 6.10. It remains to prove the sufficiency.\n\nLet \( n \in \mathbb{N} \) . Since \( n \in \rho \left( B\right) \), we can define a bounded operator \( {B}_{n} \) on \( E \) by\n\n\[ {B}_{n} \mathrel{\text{:=}} {nB}{\left( nI - B\right) }^... | Yes |
Theorem 6.12 A linear operator \( B \) on a Hilbert space \( \mathcal{H} \) is the generator of a strongly continuous one-parameter contraction semigroup if and only if \( B \) is \( m \) - dissipative, or equivalently, \( - B \) is \( m \) -accretive. | Proof Combine Theorem 6.11 and Proposition 3.22, applied to - B.\n\nIf \( B \) is the generator of a contraction semigroup \( T \), then in particular \( B \) is dissipative, and hence \( \operatorname{Re}\langle {Bx}, x\rangle \leq 0 \) for \( x \in \mathcal{D}\left( B\right) \) by Definition 3.6. The latter fact can ... | Yes |
Proposition 6.13 If \( T = \{ T\left( t\right) \} \) is a contraction semigroup on \( \mathcal{H} \) with generator \( B \) , then \( {T}^{ * } \mathrel{\text{:=}} \left\{ {T{\left( t\right) }^{ * }}\right\} \) is also a contraction semigroup whose generator is \( {B}^{ * } \) . | Proof Obviously, \( T{\left( t\right) }^{ * } \) is also a contraction, and axiom (i) of Definition 6.2 holds.\n\nTo prove axiom (ii), it suffices to show that \( \mathop{\lim }\limits_{{t \rightarrow + 0}}{T}^{ * }\left( t\right) x = x \) for \( x \in \mathcal{H} \) . For \( t > 0 \), we have\n\n\[ \n{\begin{Vmatrix}\... | Yes |
Proposition 6.14 If \( A \) is a positive self-adjoint operator on a Hilbert space \( \mathcal{H} \), then \( T \mathrel{\text{:=}} \left\{ {T\left( t\right) \mathrel{\text{:=}} {e}^{-{tA}} : t \geq 0}\right\} \) is a contraction semigroup of self-adjoint operators on \( \mathcal{H} \) whose generator is \( - A \) . Ea... | Proof That \( T \) is a contraction semigroup and \( \mathop{\lim }\limits_{{h \rightarrow + 0}}{h}^{-1}\left( {T\left( h\right) - I}\right) x = - {Ax} \) for \( x \in \mathcal{D}\left( A\right) \) can be shown in a similar manner as in the case of unitary groups (see the proof of Proposition 6.6). Thus, if \( B \) is ... | Yes |
Proposition 6.15 Two self-adjoint operators \( {A}_{1} \) and \( {A}_{2} \) acting on the Hilbert space \( \mathcal{H} \) commute strongly if and only if \( {e}^{\mathrm{i}t{A}_{1}}{e}^{\mathrm{i}s{A}_{2}} = {e}^{\mathrm{i}s{A}_{2}}{e}^{\mathrm{i}t{A}_{1}} \) for all \( t, s \in \mathbb{R} \) . | Proof First, suppose that \( {A}_{1} \) and \( {A}_{2} \) strongly commute. Let \( {E}_{A} \) be the spectral measure of the pair \( A = \left( {{A}_{1},{A}_{2}}\right) \) (see Theorem 5.23). Then \( {e}^{\mathrm{i}t{A}_{j}} \) is the corresponding spectral integral \( \mathbb{I}\left( {f}_{j, t}\right) \) of the funct... | Yes |
Lemma 7.1 \( \mathcal{D}\left( T\right) = \mathcal{D}\left( \left| T\right| \right) \), and \( \parallel {Tx}\parallel = \parallel \left| T\right| x\parallel \) for \( x \in \mathcal{D}\left( T\right) \) . | Proof If \( x \in \mathcal{D}\left( {{T}^{ * }T}\right) = \mathcal{D}\left( {\left| T\right| }^{2}\right) \), then\n\n\[ \parallel {Tx}{\parallel }^{2} = \langle {Tx},{Tx}\rangle = \left\langle {{T}^{ * }{Tx}, x}\right\rangle = \left\langle {{\left| T\right| }^{2}x, x}\right\rangle = \parallel \left| T\right| x{\parall... | Yes |
Theorem 7.2 Suppose that \( T \) is a densely defined closed operator of \( {\mathcal{H}}_{1} \) into \( {\mathcal{H}}_{2} \) . Then there is a partial isometry \( {U}_{T} \) with initial space \( \mathcal{N}{\left( T\right) }^{ \bot } = \overline{\mathcal{R}\left( {T}^{ * }\right) } = \overline{\mathcal{R}\left( \left... | Proof Set \( {U}_{T}\left( {\left| T\right| x}\right) = {Tx} \) for \( x \in \mathcal{D}\left( \left| T\right| \right) \) . By Lemma 7.1, \( {U}_{T} \) is a well-defined isometric linear map of \( \mathcal{R}\left( \left| T\right| \right) \) onto \( \mathcal{R}\left( T\right) \) . It extends by continuity to an isometr... | Yes |
Example 7.1 (Polar decomposition of a self-adjoint operator) Let \( T \) be a selfadjoint operator on a Hilbert space \( \mathcal{H} \), and let \( T = {U}_{T}\left| T\right| \) be its polar decomposition. Since \( {U}_{T} = {U}_{{T}^{ * }} = {U}_{T}^{ * },{U}_{T} \) is self-adjoint. Let \( {\mathcal{H}}_{0} = \mathcal... | \[ \mathcal{H} = {\mathcal{H}}_{ + } \oplus {\mathcal{H}}_{ - } \oplus {\mathcal{H}}_{0} \] (7.3) Since \( {U}_{T}T = T{U}_{T} \) and \( {U}_{T}\left| T\right| = T = \left| T\right| {U}_{T} \) by the above formulas, the decomposition (7.3) reduces the self-adjoint operators \( T \) and \( \left| T\right| \) . That is, ... | Yes |
Theorem 7.5 The mapping \( T \rightarrow {Z}_{T} \) is a bijection of \( \mathcal{C}\left( \mathcal{H}\right) \) onto \( \mathcal{Z}\left( \mathcal{H}\right) \) with inverse given by \( Z \rightarrow {T}_{Z} \) . Both mappings preserve adjoints, that is, \( {\left( {Z}_{T}\right) }^{ * } = {Z}_{{T}^{ * }} \) and \( {\l... | Proof From Lemma 5.8 we already know that the map \( T \rightarrow {Z}_{T} \) takes \( \mathcal{C}\left( \mathcal{H}\right) \) into \( \mathcal{Z}\left( \mathcal{H}\right) \) and that \( {Z}_{T}^{ * } = {Z}_{{T}^{ * }} \) and \( {\left( I + {T}^{ * }T\right) }^{-1} = I - {Z}_{T}^{ * }{Z}_{T} \) . Therefore, it suffices... | Yes |
Corollary 7.7 Let \( {\left( {m}_{n}\right) }_{n \in {\mathbb{N}}_{0}} \) be a positive sequence such that\n\n\[ \mathop{\sum }\limits_{{n = 1}}^{\infty }{m}_{n}^{-1/n} = \infty \]\n\n(7.22)\n\nSuppose that \( f \in {C}^{\infty }\left( \mathcal{J}\right) \) and there is a constant \( {K}_{f} > 0 \) such that (7.20) is ... | Proof Since obviously \( {m}_{n}^{1/n} \geq \mathop{\inf }\limits_{{k \geq n}}{m}_{k}^{1/k} \) ,(7.22) implies (7.21), so \( C\left\{ {m}_{n}\right\} \) is a quasi-analytic class by Proposition 7.6. This gives the assertion. | No |
Example 7.2 \( {\left( {m}_{n} = n!\right) }_{n \in \mathbb{N}} \) Since \( n! \leq {n}^{n} \), the sequence \( {\left( n!\right) }_{n \in {\mathbb{N}}_{0}} \) satisfies (7.21) and (7.22), so \( C\{ n!\} \) is a quasi-analytic class. But any function \( f \in C\{ n!\} \) has a holomorphic extension to some strip \( \{ ... | Using the latter fact, the assertion of Corollary 7.7 is well known in this special case. | No |
Example 7.3 (log convex sequences) Let \( {\left( {m}_{n}\right) }_{n \in {\mathbb{N}}_{0}} \) be a positive sequence such that \( {m}_{0} = 1 \) and\n\n\[ \n{m}_{n}^{2} \leq {m}_{n - 1}{m}_{n + 1}\;\text{ for }n \in \mathbb{N}.\n\]\n\n(7.23)\n\nCondition (7.23) means that \( {\left( \log {m}_{n}\right) }_{n \in {\math... | It is well known and easily checked that then \( {m}_{n}^{1/n} \leq {m}_{k}^{1/k} \) for \( n \leq k \), so \( {m}_{n}^{1/n} = \mathop{\inf }\limits_{{k \geq n}}{m}_{k}^{1/k} \) . Therefore, by Proposition 7.6, in this case \( C\left\{ {m}_{n}\right\} \) is a quasi-analytic class if and only if \( \mathop{\sum }\limits... | No |
Suppose that the operator \( T \) is symmetric and \( x \) is a unit vector of \( {\mathcal{D}}^{\infty }\left( T\right) \) . Set \( {m}_{n} \mathrel{\text{:=}} \begin{Vmatrix}{{T}^{n}x}\end{Vmatrix} \) . Then the assumptions of Example 7.3 are fulfilled, since \( {m}_{0} = 1 \) and | \[ {m}_{n}^{2} = {\begin{Vmatrix}{T}^{n}x\end{Vmatrix}}^{2} = \left\langle {{T}^{n}x,{T}^{n}x}\right\rangle = \left\langle {{T}^{n - 1}x,{T}^{n + 1}x}\right\rangle \leq {m}_{n - 1}{m}_{n + 1}\;\text{ for }n \in \mathbb{N}. \] Hence, as stated in Example 7.3, \( C\left\{ \begin{Vmatrix}{{T}^{n}x}\end{Vmatrix}\right\} \)... | Yes |
Corollary 7.9 Suppose that \( T \) is a self-adjoint operator on \( \mathcal{H} \). For any \( x \in \mathcal{H} \), the following statements are equivalent:\n\n(i) \( x \in {\mathcal{D}}^{a}\left( T\right) \).\n\n(ii) There is a \( c > 0 \) such that \( x \in \mathcal{D}\left( {e}^{zT}\right) \) for all \( z \in \math... | Proof Clearly, by formula (4.26), \( x \in {\mathcal{D}}^{\infty }\left( T\right) \) if (ii) or (iii) holds. The implication (i) \( \rightarrow \) (ii) was proved by Proposition 7.8. The equivalence of (ii) and (iii) follows at once from (4.26). Suppose that (iii) is satisfied and set \( u = {e}^{c\left| T\right| }x \)... | Yes |
Corollary 7.10 Let \( T \) be a self-adjoint operator on \( \mathcal{H}, y \in \mathcal{H} \), and \( x \in {\mathcal{D}}^{a}\left( T\right) \) . Then \( f\left( z\right) \mathrel{\text{:=}} \left\langle {{e}^{\mathrm{i}{zT}}x, y}\right\rangle \) is a holomorphic function on the strip \( \left\{ {z : \left| {\operatorn... | Proof Let \( s \in \mathbb{R} \) . By formula (7.30), applied with \( z \) replaced by \( \mathrm{i}\left( {z - s}\right) \), the series\n\n\[ f\left( z\right) = \left\langle {{e}^{\mathrm{i}{zT}}x, y}\right\rangle = \left\langle {{e}^{\left( {\mathrm{i}z - \mathrm{i}s}\right) T}x,{e}^{-\mathrm{i}{sT}}y}\right\rangle =... | Yes |
Lemma 7.11 Let \( {\left( {r}_{n}\right) }_{n \in \mathbb{N}} \) be a positive sequence and set \( {m}_{n} = \alpha {r}_{n} + \beta \) for \( n \in \mathbb{N} \) , where \( \alpha > 0 \) and \( \beta > 0 \) . If \( \mathop{\sum }\limits_{n}{r}_{n}^{-1/n} = \infty \), then \( \mathop{\sum }\limits_{n}{m}_{n}^{-1/n} = \i... | Proof Let \( M \) be the set \( \left\{ {n : \alpha {r}_{n} \leq \beta }\right\} \) . For \( n \in M \), we have \( {m}_{n} \leq {2\beta } \) and\n\n\[ \n{m}_{n}^{-1/n} \geq {\left( 2\beta \right) }^{-1/n} \geq {\left( 1 + 2\beta \right) }^{-1},\n\]\n\nso \( \mathop{\sum }\limits_{n}{m}_{n}^{-1/n} = \infty \) when the ... | Yes |
Lemma 7.12 If \( T \) if a symmetric operator, then \( {\mathcal{D}}^{s}\left( T\right) \subseteq {\mathcal{D}}^{s}\left( {T + {zI}}\right) \) for \( z \in \mathbb{C} \) . | Proof Let \( x \in {\mathcal{D}}^{s}\left( T\right) \) . Since \( {\lambda x} \in {\mathcal{D}}^{s}\left( T\right) \) for \( \lambda \in \mathbb{C} \), we can assume without loss of generality that \( {Tx} \neq 0 \) and \( \parallel x\parallel = 1 \) . As noted in Examples 7.3 and 7.4, we then have \( {\begin{Vmatrix}{... | Yes |
Lemma 7.13 If \( T \) is a self-adjoint operator on \( \mathcal{H} \), then \( {\mathcal{D}}^{b}\left( T\right) \) is dense in \( \mathcal{H} \) . | Proof Let \( E \) be the spectral measure of \( T \) . If \( a > 0 \) and \( x \in E\left( \left\lbrack {-a, a}\right\rbrack \right) \mathcal{H} \), then\n\n\[ \n{\begin{Vmatrix}{T}^{n}x\end{Vmatrix}}^{2} = {\int }_{-a}^{a}{\lambda }^{2n}d\langle E\left( \lambda \right) x, x\rangle \leq {a}^{2n}\parallel x{\parallel }^... | Yes |
Theorem 7.16 Let \( T \) be a symmetric operator on \( \mathcal{H} \). If the space \( {\mathcal{D}}^{a}\left( T\right) \) of analytic vectors for \( T \) is dense in \( \mathcal{H} \), then \( T \) is essentially self-adjoint. If \( T \) is closed, then \( T \) is self-adjoint if and only if \( {\mathcal{D}}^{a}\left(... | Proof Since \( {\mathcal{D}}^{a}\left( T\right) \) is a linear space and \( {\mathcal{D}}^{b}\left( T\right) \subseteq {\mathcal{D}}^{a}\left( T\right) \subseteq {\mathcal{D}}^{qa}\left( T\right) \) by (7.28), the assertions follow immediately from Theorem 7.14 and Lemma 7.13. | No |
Suppose that there exist two positive real constants \( \alpha \) and \( \beta \) such that \( \left| {a}_{k}\right| \leq {\alpha n} + \beta \) and \( \left| {b}_{k}\right| \leq {\alpha n} + \beta \) for all \( k \leq n, k, n \in \mathbb{N} \) . Then we have \( {\mathcal{D}}^{a}\left( T\right) = \) \( \mathcal{D}\left(... | The main part of the proof is to show that each vector \( {e}_{k} \) is analytic for \( T \) . From formula (7.33) we conclude \( {T}^{n}{e}_{k} \) is a sum of at most \( {3}^{n} \) summands of the form \( {\gamma }_{m}{e}_{m} \), where \( m \leq k + n \) and \( {\gamma }_{m} \) is a product of \( n \) factors \( {a}_{... | Yes |
Recall from (7.11) that the annihilation operator \( A \) and the creation operator \( {A}^{ + } \) act on the standard orthonormal basis \( \left\{ {{e}_{n} : n \in {\mathbb{N}}_{0}}\right\} \) of \( \mathcal{H} = {l}^{2}\left( {\mathbb{N}}_{0}\right) \) by\n\n\[ A{e}_{n} = \sqrt{n}{e}_{n - 1}\;\text{ and }\;{A}^{ + }... | Since \( {A}^{ * } = {A}^{ + } \), setting\n\n\[ {P}_{0} = \frac{1}{\sqrt{2}\mathrm{i}}\left( {A - {A}^{ + }}\right) \;\text{ and }\;{Q}_{0} = \frac{1}{\sqrt{2}}\left( {A + {A}^{ + }}\right) ,\]\n\n\( {P}_{0} \) and \( {Q}_{0} \) are symmetric operators on \( {l}^{2}\left( {\mathbb{N}}_{0}\right) \), and we have\n\n\[ ... | Yes |
Statement \( f \in {L}^{2}\left( \mathbb{R}\right) \) is an analytic vector for \( T \) if and only if \( f \) is the restriction to \( \mathbb{R} \) of a holomorphic function \( F \) on a strip \( \{ z : \left| {\operatorname{Im}z}\right| < c\} \) for some \( c > 0 \) satisfying\n\n\[\n\mathop{\sup }\limits_{{\left| y... | Proof Since \( T \) and \( {M}_{x} \) are unitarily equivalent, \( f \in {\mathcal{D}}^{a}\left( T\right) \) if and only if \( \widehat{f} = \) \( \mathcal{F}\left( f\right) \) is in \( {\mathcal{D}}^{a}\left( {M}_{x}\right) \) . Obviously, \( {\mathcal{D}}^{a}\left( {M}_{x}\right) = {\mathcal{D}}^{a}\left( \left| {M}_... | Yes |
Lemma 7.17 Let \( T \) be a symmetric operator with domain \( \mathcal{D} \) such that \( T\mathcal{D} \subseteq \mathcal{D} \) . For any operator \( S \in \{ T{\} }^{c} \), we have \( S{\mathcal{D}}^{qa}\left( T\right) \subseteq {\mathcal{D}}^{qa}\left( T\right) \) . In particular, \( {\mathcal{D}}^{qa}\left( T\right)... | Proof Let \( x \in \mathcal{D} \) . Using the properties of \( T \) and \( S \in \{ T{\} }^{c} \), we obtain\n\n\[ \n{\begin{Vmatrix}{T}^{n}Sx\end{Vmatrix}}^{2} = \left\langle {{T}^{2n}{Sx},{Sx}}\right\rangle = \left\langle {S{T}^{2n}x,{Sx}}\right\rangle = \left\langle {{T}^{2n}x,{S}^{ * }{Sx}}\right\rangle \leq \begin... | Yes |
Theorem 7.18 Let \( A \) and \( B \) be symmetric operators acting on the same dense domain \( \mathcal{D} \) of \( \mathcal{H} \) such that \( A\mathcal{D} \subseteq \mathcal{D}, B\mathcal{D} \subseteq \mathcal{D} \), and \( {ABx} = {BAx} \) for \( x \in \mathcal{D} \) . If the linear span \( {\mathcal{D}}_{\mathcal{Q... | Proof By the definition of the set \( \{ T{\} }^{c} \) we have \( \mathcal{Q} \subseteq \mathcal{D} \) . From Lemma 7.17 it follows that \( \mathcal{Q} \) is contained in \( {\mathcal{D}}^{qa}\left( A\right) \) and \( {\mathcal{D}}^{qa}\left( B\right) \) . Therefore, \( \bar{A} \) and \( \bar{B} \) are self-adjoint by ... | Yes |
Lemma 7.19 \( \langle \cdot , \cdot \rangle \) is a well-defined scalar product on \( {\mathcal{H}}_{1} \odot {\mathcal{H}}_{2} \) . | Proof First, we prove that \( \langle \cdot , \cdot \rangle \) is well defined, that is, the definition of \( \langle u, v\rangle \) does not depend on the particular representations of \( u \) and \( v \) as sums of elementary tensors. For this, it suffices to show that \( \langle u, v\rangle = 0 \) when \( u = 0 \) i... | Yes |
Example 7.9 (Tensor product of two \( {L}^{2} \) -spaces) Let \( \left( {{X}_{1},{\mu }_{1}}\right) \) and \( \left( {{X}_{2},{\mu }_{2}}\right) \) be two \( \sigma \) -finite measure spaces, and let \( \left( {{X}_{1} \times {X}_{2},\mu }\right) \) be the measure space with the product measure \( \mu = {\mu }_{1} \tim... | \[ J : \mathop{\sum }\limits_{k}{f}_{k} \otimes {g}_{k} \rightarrow \mathop{\sum }\limits_{k}{f}_{k}\left( {x}_{1}\right) {g}_{k}\left( {x}_{2}\right) ,\;{x}_{1} \in {X}_{1},{x}_{2} \in {X}_{2}, \] is a well-defined linear map of \( {\mathcal{H}}_{1} \odot {\mathcal{H}}_{2} \) onto a dense subspace of \( {L}^{2}\left( ... | Yes |
Lemma 7.21 Suppose that \( {T}_{1} \) and \( {T}_{2} \) are densely defined and closable. Then \( {T}_{1} \odot {T}_{2} \) is also densely defined and closable, and we have \( {\left( {T}_{1} \odot {T}_{2}\right) }^{ * } \supseteq {T}_{1}^{ * } \odot {T}_{2}^{ * } \) . | Proof Since \( {T}_{1} \) and \( {T}_{2} \) are densely defined and closable, \( {T}_{1}^{ * } \) and \( {T}_{2}^{ * } \) are densely defined by Theorem 1.8(i). Therefore, \( {T}_{1}^{ * } \odot {T}_{2}^{ * } \) is densely defined. A simple computation shows that \( {\left( {T}_{1} \odot {T}_{2}\right) }^{ * } \supsete... | Yes |
Lemma 7.22 \( \mathcal{D}\left( {{T}_{1} \odot {T}_{2}}\right) \) is a core for \( {\bar{T}}_{1} \odot {\bar{T}}_{2} \), and we have \( {\bar{T}}_{1} \otimes {\bar{T}}_{2} = {T}_{1} \otimes {T}_{2} \) . | Proof The first assertion follows easily by approximating all vectors \( {x}_{i} \in \mathcal{D}\left( {\bar{T}}_{1}\right) \) and \( {y}_{i} \in \mathcal{D}\left( {\bar{T}}_{2}\right) \) of \( u = \mathop{\sum }\limits_{i}{x}_{i} \otimes {y}_{i} \in \mathcal{D}\left( {{\bar{T}}_{1} \odot {\bar{T}}_{2}}\right) \) in th... | Yes |
Theorem 7.23 Suppose that \( {T}_{1} \) and \( {T}_{2} \) are self-adjoint. Then the tensor products \( P \mathrel{\text{:=}} {T}_{1} \otimes {T}_{2},{T}_{1} \otimes I \), and \( I \otimes {T}_{2} \) are self-adjoint and the sum \( S \mathrel{\text{:=}} {T}_{1} \otimes I + I \otimes {T}_{2} \) is essentially self-adjoi... | Proof From Proposition 7.20(iii) it follows that each of the operators \( T = {T}_{1} \otimes {T}_{2} \) , \( {T}_{1} \otimes I, I \otimes {T}_{2} \), and \( S \) is symmetric.\n\nLet \( x \in {\mathcal{D}}^{b}\left( {T}_{1}\right) \) and \( y \in {\mathcal{D}}^{b}\left( {T}_{2}\right) \) . Then there is a constant \( ... | Yes |
Lemma 7.24 If \( {T}_{1} \) and \( {T}_{2} \) are self-adjoint operators, then \( {T}_{1} \otimes I \) and \( I \otimes {T}_{2} \) are strongly commuting self-adjoint operators, and the joint spectrum \( \sigma \left( T\right) \) of the pair \( T = \left\{ {{T}_{1} \otimes I, I \otimes {T}_{2}}\right\} \) is equal to \... | Proof Let \( {\lambda }_{1} \in \rho \left( {T}_{1}\right) \) . Clearly, the bounded operator \( {R}_{{\lambda }_{1}}\left( {T}_{1}\right) \otimes I \) leaves \( \mathcal{D}\left( {{T}_{1} \odot I}\right) \) invariant, and its restriction to this domain is the inverse of \( {T}_{1} \odot I - {\lambda }_{1}I \) . Since ... | Yes |
Corollary 7.25 If \( {T}_{1} \) and \( {T}_{2} \) are self-adjoint operators on \( {\mathcal{H}}_{1} \) and \( {\mathcal{H}}_{2} \), then\n\n\[ \sigma \left( {{T}_{1} \otimes {T}_{2}}\right) = \overline{\sigma \left( {T}_{1}\right) \cdot \sigma \left( {T}_{2}\right) },\;\sigma \left( \overline{{T}_{1} \otimes I + I \ot... | In general none of the three bars in (7.45) can be omitted (see Example 7.10 below and Exercise 17). But \( \sigma \left( {T}_{1}\right) \cdot \sigma \left( {T}_{2}\right) \) is closed if one of the operators is bounded and 0 is not in the spectrum of the other one (Exercise 18.b). Further, \( {T}_{1} \otimes I + I \ot... | No |
For \( j = 1,2 \), let \( {T}_{j} \) be the multiplication operator by the variable \( {t}_{j} \) on \( {\mathcal{H}}_{j} \mathrel{\text{:=}} {L}^{2}\left( \mathbb{R}\right) \) . Then \( {T}_{1} \otimes I, I \otimes {T}_{2} \), and \( {T}_{1} \otimes {T}_{2} \) are the multiplication operators by \( {t}_{1},{t}_{2} \),... | For the function\n\n\[ f\left( {{t}_{1},{t}_{2}}\right) = {\chi }_{\left( 1, + \infty \right) }\left( {t}_{1}\right) {\chi }_{\left( -1,1\right) }\left( {{t}_{1} + {t}_{2}}\right) {t}_{1}^{-\alpha } \in {L}^{2}\left( {\mathbb{R}}^{2}\right) ,\;\text{ where }1/2 < \alpha \leq 3/2, \]\n\nwe have \( \left( {{t}_{1} + {t}_... | Yes |
Our aim is to prove that\n\n\[ {UT}{U}^{ * } = - \frac{1}{2M}{\Delta }_{{y}_{s}} + \left( {-\frac{1}{2m}{\Delta }_{{y}_{r}} + V\left( {y}_{r}\right) }\right) . \] | Now let us prove formula (7.49). Set \( g = {U}^{ * }f \) . Then \( g\left( x\right) = f\left( {\mathfrak{A}x}\right) \) . Using the substitution (7.48), we compute for \( j = 1,2,3 \) ,\n\n\[ \frac{{\partial }^{2}}{\partial {x}_{1j}^{2}}g\left( x\right) \equiv \frac{{\partial }^{2}}{\partial {x}_{1j}^{2}}f\left( {\mat... | Yes |
Proposition 8.1 \( {\left( p\left( D\right) \mathcal{F}\right) }^{ * } = {p}^{ + }{\left( D\right) }_{\mathcal{F}} \) and \( p{\left( D\right) }_{\mathcal{F}} = p{\left( D\right) }_{\min } = p{\left( D\right) }_{\max } \) . | Proof Since \( {\left( {M}_{p}\right) }^{ * } = {M}_{\bar{p}} = {M}_{{p}^{ + }} \) (by Example 1.3) and \( \mathcal{F} \) is unitary (by Theorem C.4), we have \( {\left( p\left( D\right) \mathcal{F}\right) }^{ * } = {\mathcal{F}}^{-1}{\left( {M}_{p}\right) }^{ * }\mathcal{F} = {\mathcal{F}}^{-1}{M}_{{p}^{ + }}\mathcal{... | Yes |
Proposition 8.2 If \( p\left( x\right) = \mathop{\sum }\limits_{\alpha }{a}_{\alpha }{x}^{\alpha } \) is a polynomial with real coefficients \( {a}_{\alpha } \), then:\n\n(i) \( p\left( D\right) \) is a self-adjoint operator on \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \).\n\n(ii) If \( g \) is a Borel function on \( \s... | Proof (i) and (ii) follow at once from the unitary equivalence (8.2).\n\n(iii): Let \( f \in {L}^{2}\left( {\mathbb{R}}^{d}\right) \). By (ii) we have\n\n\[ g\left( {p\left( D\right) }\right) f = {\mathcal{F}}^{-1}\left( {\left( {g \circ p}\right) \cdot \mathcal{F}\left( f\right) }\right) = {\mathcal{F}}^{-1}\left( {\m... | Yes |
Example 8.1 (Laplace operator on \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \) ) Let \( p\left( x\right) \mathrel{\text{:=}} \parallel x{\parallel }^{2} = {x}_{1}^{2} + \cdots + {x}_{d}^{2} \) and \( {p}_{k}\left( x\right) \mathrel{\text{:=}} {x}_{k} \) . Then \( p\left( D\right) = - \Delta \) and \( {p}_{k}\left( D\righ... | \[ \mathcal{D}\left( {-\Delta }\right) = \left\{ {f \in {L}^{2}\left( {\mathbb{R}}^{d}\right) : \parallel x{\parallel }^{2}\widehat{f}\left( x\right) \in {L}^{2}\left( {\mathbb{R}}^{d}\right) }\right\} ,\;\left( {-\Delta }\right) f = {\mathcal{F}}^{-1}\left( {\parallel x{\parallel }^{2}\widehat{f}}\right) ,\] with spec... | Yes |
Lemma 8.3 Each symmetric operator \( B \) is relatively \( {B}^{2} \) -bounded with \( {B}^{2} \) -bound equal to zero. | Proof Obviously, \( \mathcal{D}\left( B\right) \supseteq \mathcal{D}\left( {B}^{2}\right) \) . For \( \varepsilon > 0 \) and \( x \in \mathcal{D}\left( {B}^{2}\right) \), we have\n\n\[ \left\langle {\left( {{\varepsilon }^{2}{B}^{2} - I}\right) x,\left( {{\varepsilon }^{2}{B}^{2} - I}\right) x}\right\rangle = {\varepsi... | Yes |
Lemma 8.4 If \( A \) is a closed operator and \( B \) is a densely defined closable operator on a Hilbert space \( \mathcal{H} \) such that \( \mathcal{D}\left( B\right) \supseteq \mathcal{D}\left( A\right) \), then \( B \) is relatively \( A \) -bounded. | Proof Since \( A \) is closed, \( {\mathcal{D}}_{A} = \left( {\mathcal{D}\left( A\right) ,\parallel \cdot {\parallel }_{A}}\right) \) is a Hilbert space by Proposition 1.4.\n\nWe show that \( B \), considered as a mapping of \( {\mathcal{D}}_{A} \) into \( \mathcal{H} \), is closed. Suppose that \( {x}_{n} \rightarrow ... | Yes |
Theorem 8.5 Let \( A \) be a self-adjoint operator on \( \mathcal{H} \). Suppose that \( B \) is a relatively \( A \) -bounded symmetric operator on \( \mathcal{H} \) with \( A \) -bound \( {a}_{A}\left( B\right) < 1 \). Then:\n\n(i) The operator \( A + B \) on \( \mathcal{D}\left( {A + B}\right) = \mathcal{D}\left( A\... | Proof (i): By assumption, there are positive constants \( a, b \) such that \( a < 1 \) and (8.8) holds. Suppose that \( \beta \in \mathbb{R},\beta \neq 0 \). Since \( A \) is symmetric, by formula (3.2),\n\n\[ \parallel \left( {A - \mathrm{i}{\beta I}}\right) x{\parallel }^{2} = \parallel {Ax}{\parallel }^{2} + {\left... | Yes |
Theorem 8.8 Let \( V \) be a real-valued Kato-Rellich potential. Then \( - \Delta + V \) is selfadjoint on \( \mathcal{D}\left( {-\Delta }\right) \) and essentially self-adjoint on each core for \( - \Delta \) . | Proof We write \( V \) as \( V = {V}_{1} + {V}_{2} \) with \( {V}_{2} \in {L}^{\infty }\left( {\mathbb{R}}^{d}\right) \) and \( {V}_{1} \in {L}^{p}\left( {\mathbb{R}}^{d}\right) \), where \( p = 2 \) for \( d \leq 3 \) and \( p > d/2 \) for \( d \geq 4 \) .\n\nFirst, suppose that \( d \leq 3 \) . Applying (8.13) with \... | Yes |
Example 8.2 Let \( \alpha \in \lbrack 0,\infty ) \) . Suppose that \( \alpha < d/2 \) if \( d \leq 3 \) and \( \alpha < 2 \) if \( d \geq 4 \) . Then, for any \( c \in \mathbb{R} \), the function\n\n\[ V\left( x\right) = \frac{c}{\parallel x{\parallel }^{\alpha }} \]\n\non \( {\mathbb{R}}^{d} \) is a Kato-Rellich poten... | Indeed, let \( {\chi }_{n} \) be the characteristic function of \( \{ x : \parallel x\parallel < n\} \) for \( n \in \mathbb{N} \) and set \( {V}_{1, n} \mathrel{\text{:=}} V \cdot {\chi }_{n} \) . Then \( {V}_{1, n} \in {L}^{2}\left( {\mathbb{R}}^{d}\right) \) if \( d < 3 \) and \( {V}_{1, n} \in {L}^{p}\left( {\mathb... | Yes |
Statement Let \( V \in {L}^{2}\left( {\mathbb{R}}^{3}\right) + {L}^{\infty }\left( {\mathbb{R}}^{3}\right) \), and \( {a}_{1},{a}_{2},{a}_{3} \in {C}^{1}\left( {\mathbb{R}}^{3}\right) \) be real-valued functions such that \( {a}_{1},{a}_{2},{a}_{3},\operatorname{div}a \in {L}^{\infty }\left( {\mathbb{R}}^{3}\right) \) ... | The domain of \( \mathcal{D}\left( T\right) \) is \( \mathcal{D}\left( {-\Delta }\right) \) . Fix \( \varepsilon > 0 \) . For \( f \in \mathcal{D}\left( T\right) \), we derive\n\n\[{\begin{Vmatrix}{D}_{k}f\end{Vmatrix}}^{2} = \left\langle {{D}_{k}f,{D}_{k}f}\right\rangle = \left\langle {{D}_{k}^{2}f, f}\right\rangle \l... | Yes |
Lemma 8.9 If \( V \) is a Borel function on \( {\mathbb{R}}^{d} \) such that \( \mathcal{D}\left( V\right) \supseteq \mathcal{D}\left( {-\Delta }\right) \), then we have \( V \in {L}_{\text{loc }}^{2}\left( {\mathbb{R}}^{d}\right) \) and\n\n\[{\delta }_{V} \mathrel{\text{:=}} \mathop{\sup }\limits_{{c \in {\mathbb{R}}^... | Proof Since \( \mathcal{D}\left( V\right) \supseteq \mathcal{D}\left( {-\Delta }\right) \), it follows from Lemma 8.4 that \( V \) is \( \left( {-\Delta }\right) \) -bounded, that is, there are positive constants \( a \) and \( b \) such that\n\n\[ \parallel {Vf}\parallel \leq a\parallel - {\Delta f}\parallel + b\paral... | Yes |
Theorem 8.12 Let \( {A}_{1} \) and \( {A}_{2} \) be self-adjoint operators on a Hilbert space \( \mathcal{H} \) . Suppose that there exists a number \( \mu \in \rho \left( {A}_{1}\right) \cap \rho \left( {A}_{2}\right) \) such that\n\n\[ \n{C}_{\mu } \mathrel{\text{:=}} {\left( {A}_{2} - \mu I\right) }^{-1} - {\left( {... | Proof Let \( \lambda \in {\sigma }_{\text{ess }}\left( {A}_{1}\right) \) . We apply Weyl’s criterion (Proposition 8.11). Then there exists a singular sequence \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) for \( {A}_{1} \) at \( \lambda \) . In order to prove that \( \lambda \in {\sigma }_{\text{ess }}\left( {A}_{2... | Yes |
Corollary 8.13 Let \( T \) be a symmetric operator on \( \mathcal{H} \) such that \( {d}_{ + }\left( T\right) < \infty \) . If \( {A}_{1} \) and \( {A}_{2} \) are self-adjoint extensions of \( T \) on \( \mathcal{H} \), then \( {\sigma }_{\mathrm{{ess}}}\left( {A}_{1}\right) = {\sigma }_{\mathrm{{ess}}}\left( {A}_{2}\r... | Proof Since \( T \subseteq {A}_{2} \) and \( T \subseteq {A}_{1} \), the resolvents \( {\left( {A}_{2} - \mathrm{i}I\right) }^{-1} \) and \( {\left( {A}_{1} - \mathrm{i}I\right) }^{-1} \) coincide on \( \mathcal{R}\left( {T - \mathrm{i}I}\right) \) . Hence, the range of \( {C}_{\mathrm{i}} \mathrel{\text{:=}} {\left( {... | Yes |
Proposition 8.14 Let \( A \) be a closed operator, and \( C \) a linear operator on \( \mathcal{H} \) . (i) Suppose that \( \rho \left( A\right) \) is not empty. Then \( C \) is relatively \( A \) -compact if and only if \( \mathcal{D}\left( C\right) \supseteq \mathcal{D}\left( A\right) \) and \( C{\left( A - \mu I\rig... | Proof (i): If \( C \) is relatively \( A \) -compact, then \( C{\left( A - \mu I\right) }^{-1} \) is the composition of the continuous operator \( {\left( A - \mu I\right) }^{-1} : \mathcal{H} \rightarrow {\mathcal{D}}_{A} \) and the compact operator \( C : {\mathcal{D}}_{A} \rightarrow \mathcal{H} \) and hence compact... | Yes |
Theorem 8.15 Let \( A \) be a self-adjoint operator, and \( C \) a symmetric relatively \( A \) - compact operator on \( \mathcal{H} \) . Then \( A + C \) is self-adjoint, and \( {\sigma }_{\mathrm{{ess}}}\left( {A + C}\right) = {\sigma }_{\mathrm{{ess}}}\left( A\right) \) . | Proof Because \( C \) has the \( A \) -bound zero by Proposition 8.14(ii), the operator \( A + C \) is self-adjoint by the Kato-Rellich theorem (Theorem 8.5). Since \( \mathcal{D}\left( {A + C}\right) = \) \( \mathcal{D}\left( A\right) \), the first resolvent identity (2.4) applies with \( T = A + C, S = A \) and yield... | Yes |
Corollary 8.16 If \( A \) is a self-adjoint operator and \( C \) is a compact self-adjoint operator on a Hilbert space \( \mathcal{H} \), then \( {\sigma }_{\mathrm{{ess}}}\left( {A + C}\right) = {\sigma }_{\mathrm{{ess}}}\left( A\right) \) . | Let \( A \) be a self-adjoint operator, \( C \) a compact self-adjoint operator, and \( U \) a unitary operator on \( \mathcal{H} \) . Set \( B = U\left( {A + C}\right) {U}^{-1} \) . Since the essential spectrum is obviously preserved under unitary transformations, by Corollary 8.16 we have\n\n\[ \n{\sigma }_{\mathrm{{... | No |
Lemma 8.18 Let \( p \in \left\lbrack {2,\infty }\right\rbrack \) and \( V,\psi \in {L}^{p}\left( {\mathbb{R}}^{d}\right) \) . (i) The operator \( V\left( x\right) \psi \left( D\right) \) is bounded, and there is a constant \( c > 0 \) such that \[ \parallel V\left( x\right) \psi \left( D\right) f\parallel \leq c\parall... | Proof (i): Define \( q \in \left\lbrack {2,\infty }\right\rbrack \) and \( r \in \left\lbrack {1,2}\right\rbrack \) by \( {p}^{-1} + {q}^{-1} = {2}^{-1} \) and \( {q}^{-1} + {r}^{-1} = 1 \) . By the Hausdorff-Young Theorem C. 5 there exists a constant \( c > 0 \) such that \[ {\begin{Vmatrix}{\mathcal{F}}^{-1}g\end{Vma... | Yes |
Theorem 8.19 Let \( V \in {L}^{p}\left( {\mathbb{R}}^{d}\right) + {L}^{\infty }{\left( {\mathbb{R}}^{d}\right) }_{\varepsilon } \) be a real-valued function, where \( p = 2 \) if \( d \leq 3 \) and \( p > d/2 \) if \( d \geq 4 \) . Then \( V \) is relatively \( \left( {-\Delta }\right) \) -compact. The operator \( - \D... | Proof Since \( V \in {L}^{p}\left( {\mathbb{R}}^{d}\right) + {L}^{\infty }{\left( {\mathbb{R}}^{d}\right) }_{\varepsilon } \), we can choose a sequence of functions \( {V}_{n} \in \) \( {L}^{p}\left( {\mathbb{R}}^{d}\right) \) such that \( V - {V}_{n} \in {L}^{\infty }\left( {\mathbb{R}}^{d}\right) \) and \( \mathop{\l... | Yes |
For the decomposition of \( V = {V}_{1, n} + {V}_{2, n} \) given in Example 8.2, we have \( \mathop{\lim }\limits_{{n \rightarrow \infty }}{\begin{Vmatrix}{V}_{2, n}\end{Vmatrix}}_{\infty } = 0 \) | Hence, \( V \in {L}^{2}\left( {\mathbb{R}}^{d}\right) + {L}^{\infty }{\left( {\mathbb{R}}^{d}\right) }_{\varepsilon } \), so that \( {\sigma }_{\text{ess }}\left( {-\Delta + V}\right) = \lbrack 0, + \infty ) \) by Theorem 8.19. | No |
Corollary 9.4 Let \( p\left( x\right) \) be a nonconstant polynomial in \( d \) variables with real coefficients. Then the self-adjoint operator \( p\left( D\right) \) on \( {L}^{2}\left( {\mathbb{R}}^{d}\right) \) defined by (8.2) has a purely absolutely continuous spectrum. | Proof Since the multiplication operator \( {A}_{p\left( x\right) } \) has a purely absolutely continuous spectrum by Example 9.2, so has the operator \( p\left( D\right) = {\mathcal{F}}^{-1}{A}_{p\left( x\right) }\mathcal{F} \) . | Yes |
Proposition 9.5 A vector \( x \in \mathcal{H} \) belongs to the subspace \( {\mathcal{H}}_{\mathrm{c}}\left( A\right) \) if and only if\n\n\[ \mathop{\lim }\limits_{{T \rightarrow \infty }}{\left( 2T\right) }^{-1}{\int }_{T}^{T}{\left| \left\langle {e}^{-\mathrm{i}{tA}}x, x\right\rangle \right| }^{2}{dt} = 0. \] | Proof For a finite positive Borel measure \( \mu \) on \( \mathbb{R} \), we set \( {F}_{\mu }\left( t\right) = {\int }_{\mathbb{R}}{e}^{-\mathrm{i}{ts}}{d\mu }\left( s\right) \) . A classical theorem of N. Wiener ([Wr], see e.g. [RS3, Theorem XI.114]) states that\n\n\[ \mathop{\lim }\limits_{{T \rightarrow \infty }}{\l... | Yes |
Theorem 9.6 For any \( \alpha \in \mathbb{R} \), the absolutely continuous parts of \( {A}_{\alpha } \) and \( A \) are unitarily equivalent. In particular, \( {\sigma }_{\mathrm{{ac}}}\left( {A}_{\alpha }\right) = {\sigma }_{\mathrm{{ac}}}\left( A\right) \) . | First, we develop some preliminaries. Let \( \alpha \in \mathbb{R} \) . As usual, \( {E}_{{A}_{\alpha }} \) denotes the spectral measure of \( {A}_{\alpha } \) . We define the measure \( {\mu }_{\alpha } \) on \( \mathbb{R} \) and the function \( {F}_{\alpha } \) by\n\n\[ \n{\mu }_{\alpha }\left( \cdot \right) \mathrel... | No |
Lemma 9.7 For \( \alpha ,\beta \in \mathbb{R} \) and \( z \in \mathbb{C} \smallsetminus \mathbb{R} \), we have \( 1 + \left( {\alpha - \beta }\right) {F}_{\beta }\left( z\right) \neq 0 \) , \[ {F}_{\alpha }\left( z\right) = \frac{{F}_{\beta }\left( z\right) }{1 + \left( {\alpha - \beta }\right) {F}_{\beta }\left( z\rig... | Proof From the resolvent identity (2.4) and Eq. (9.7) we conclude that \[ {R}_{z}\left( {A}_{\alpha }\right) - {R}_{z}\left( {A}_{\beta }\right) = {R}_{z}\left( {A}_{\alpha }\right) \left( {{A}_{\beta } - {A}_{\alpha }}\right) {R}_{z}\left( {A}_{\beta }\right) \] \[ = \left( {\beta - \alpha }\right) \left\langle {{R}_{... | Yes |
Lemma 9.8 For any \( \alpha \in \mathbb{R},{\mathcal{H}}_{0} \) is the smallest reducing subspace for \( {A}_{\alpha } \) containing \( u \) . The vector \( u \) is generating for \( {A}_{\alpha } \) if and only if it is generating for \( A \) . | The subspace \( {\mathcal{H}}_{0} \) contains \( u \) and is reducing for \( {A}_{\alpha } \) and \( A \) . By (9.7) the parts of \( {A}_{\alpha } \) and \( A \) on the subspace \( {\left( {\mathcal{H}}_{0}\right) }^{ \bot } \) coincide. To study the spectrum of the operators \( {A}_{\alpha } \), we can therefore assum... | No |
Lemma 9.9 Let \( t \in \mathbb{R} \) . Suppose that \( G\left( t\right) < \infty \) . Then \( F\left( t\right) \in \mathbb{R} \) exists, and\n\n\[ F\left( t\right) = F\left( {t + \mathrm{i}0}\right) ,\;\mathrm{i}G\left( t\right) = \mathop{\lim }\limits_{{\varepsilon \rightarrow + 0}}{\varepsilon }^{-1}\left( {F\left( {... | Proof Put \( f\left( \lambda \right) \mathrel{\text{:=}} {\left( \lambda - t\right) }^{-1} \) . Since \( f \in {L}^{2}\left( {\mathbb{R},\mu }\right) \) (by \( G\left( t\right) < \infty \) ) and the measure \( \mu \) is finite, \( f \in {L}^{1}\left( {\mathbb{R},\mu }\right) \) by Hölder’s inequality, that is, \( F\lef... | Yes |
Theorem 9.10 Suppose that \( u \) is a generating vector for the self-adjoint operator \( A \) . Let \( \alpha ,\beta ,{\beta }_{1},{\beta }_{2} \in \mathbb{R} \) and suppose that \( \alpha \neq 0 \) .\n\n(i) A real number \( t \) is an eigenvalue of \( {A}_{\alpha } \) if and only if \( t \) belongs to the set\n\n\[ \... | Proof (i): Both descriptions of the set \( {P}_{\alpha } \) are equal, since \( F\left( t\right) = F\left( {t + \mathrm{i}0}\right) \) by (9.12) when \( G\left( t\right) < \infty \) .\n\nSince \( u \) is a generating vector, \( A \) has a simple spectrum, and by Proposition 5.18 we can assume that, up to unitary equiva... | Yes |
Let \( \mu \) be the sum of the Lebesgue measure on \( \left\lbrack {a, b}\right\rbrack \) and the delta measure \( {\delta }_{c} \), where \( a < c < b \) . As above, \( A \) is the multiplication operator \( \left( {Af}\right) \left( \lambda \right) = {\lambda f}\left( \lambda \right) \) on \( {L}^{2}\left( {\mathbb{... | \[ \operatorname{Im}F\left( {t + \mathrm{i}0}\right) = \left\{ \begin{array}{ll} \pi & \text{ if }a < t < b, t \neq c \\ 0 & \text{ if }t \notin \left\lbrack {a, b}\right\rbrack \\ \pi /2 & \text{ if }t = a, b \\ + \infty & \text{ if }t = c \end{array}\right\} \] (9.18) Thus, \( L = \left\lbrack {a, b}\right\rbrack \),... | Yes |
Example 9.5 (Singularly continuous spectrum is not invariant) We replace in Example 9.4 the measure \( {\delta }_{c} \) by a finite singular measure \( v \) on \( \left\lbrack {a, b}\right\rbrack \) . Then \( {\sigma }_{\mathrm{{sc}}}\left( A\right) \neq \varnothing \) . In fact, the singularly continuous part \( {A}_{... | We show that \( {\sigma }_{\mathrm{{sc}}}\left( {A}_{\alpha }\right) \) is empty for all \( \alpha \neq 0 \) . We have \( F = {H}_{1} + {H}_{2} \), where \( {H}_{1}\left( z\right) \mathrel{\text{:=}} {\int }_{a}^{b}{\left( \lambda - z\right) }^{-1}{d\lambda } \) and \( {H}_{2}\left( z\right) \mathrel{\text{:=}} {\int }... | Yes |
Let \( {\left( {a}_{n}\right) }_{n \in \mathbb{N}} \) and \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) be real sequences such that \( \left( {a}_{n}\right) \in {l}^{1}\left( \mathbb{N}\right) \) and \( {a}_{n} > 0 \) for all \( n \) . Then \( \mu \mathrel{\text{:=}} \mathop{\sum }\limits_{{n = 1}}^{\infty }{a}_{n}... | Let us assume for a moment that\n\n\[ G\left( t\right) \equiv {\int }_{\mathbb{R}}{\left( \lambda - t\right) }^{-2}{d\mu }\left( \lambda \right) = + \infty \;\text{ for all }t \in \mathbb{R}. \]\n\n(9.19)\n\nThen \( {P}_{\alpha } = \varnothing \), and so \( {\sigma }_{\mathrm{p}}\left( {A}_{\alpha }\right) = \varnothin... | Yes |
Lemma 9.11 For \( z \in \mathbb{C} \smallsetminus \mathbb{R} \), we have \( 1 + {\alpha F}\left( z\right) \neq 0 \) and\n\n\[ \n\operatorname{Tr}\left\lbrack {{R}_{z}\left( A\right) - {R}_{z}\left( B\right) }\right\rbrack = \frac{\alpha }{1 + {\alpha F}\left( z\right) }\left\langle {{R}_{z}{\left( A\right) }^{2}u, u}\r... | Proof The proof is based on identities (9.10) and (9.9) applied with \( \beta = 0 \) . Then \( A = {A}_{\beta } \) and \( B = {A}_{\alpha } \) . From (9.10) we obtain \( {R}_{z}\left( B\right) u = {\left( 1 + \alpha F\left( z\right) \right) }^{-1}{R}_{z}\left( A\right) u \) . Inserting this equation into (9.9), we deri... | Yes |
Theorem 9.12 There exists a function \( \xi \in {L}^{1}\left( \mathbb{R}\right) \) such that\n\n\[ \xi \left( \lambda \right) = {\pi }^{-1}\arg {\Delta }_{B/A}\left( {\lambda + \mathrm{i}0}\right) \mathrel{\text{:=}} {\pi }^{-1}\mathop{\lim }\limits_{{\varepsilon \rightarrow + 0}}\arg {\Delta }_{B/A}\left( {\lambda + \... | Proof Assume first that \( \alpha > 0 \) . Obviously, \( \operatorname{Im}{\Delta }_{B/A}\left( {\mathrm{i}y}\right) = \alpha \operatorname{Im}F\left( {\mathrm{i}y}\right) > 0 \) if \( y > 0 \) by (9.23). Hence, the function \( C\left( z\right) \mathrel{\text{:=}} \log {\Delta }_{B/A}\left( z\right) \) is holomorphic w... | Yes |
Lemma 9.13 Let \( {\left( {A}_{n}\right) }_{n \in \mathbb{N}} \) and \( {\left( {B}_{n}\right) }_{n \in \mathbb{N}} \) be sequences of operators \( {A}_{n},{B}_{n} \in \mathbf{B}\left( \mathcal{H}\right) \) such that \( \mathrm{s} - \mathop{\lim }\limits_{{n \rightarrow \infty }}{A}_{n} = I \) and \( \mathrm{s} - \math... | Proof The Banach-Steinhaus theorem yields \( c \mathrel{\text{:=}} \mathop{\sup }\limits_{{n \in \mathbb{N}}}\left( {\begin{Vmatrix}{A}_{n}\end{Vmatrix} + \begin{Vmatrix}{B}_{n}^{ * }\end{Vmatrix}}\right) < \infty \) . Let \( \varepsilon > 0 \) be given. The finite rank operators are dense in \( \left( {{\mathbf{B}}_{1... | Yes |
Corollary 9.14 Let \( \\left\\{ {{x}_{n} : n \\in \\mathbb{N}}\\right\\} \) be an orthonormal basis of \( \\mathcal{H} \) . Then\n\n\[ \n\\det \\left( {I + T}\\right) = \\mathop{\\lim }\\limits_{{n \\rightarrow \\infty }}\\det {\\left( {\\delta }_{kj} + \\left\\langle T{x}_{k},{x}_{j}\\right\\rangle \\right) }_{j, k = ... | Proof Let \( {P}_{n} \) be the projection on the linear span of \( {x}_{1},\\ldots ,{x}_{n} \) . Then we have \( {P}_{n}T{P}_{n} = \\mathop{\\sum }\\limits_{{k = 1}}^{n}\\left\\langle {\\cdot ,{x}_{k}}\\right\\rangle T{x}_{k} \) . Clearly, \( \\det {\\left( {\\delta }_{kj} + \\left\\langle T{x}_{k},{x}_{j}\\right\\rang... | Yes |
Lemma 9.15 Suppose that \( T \in {\mathbf{B}}_{1}\left( \mathcal{H}\right) \) . (i) \( I + T \) has an inverse in \( \mathbf{B}\left( \mathcal{H}\right) \) if and only if \( \det \left( {I + T}\right) \neq 0 \) . | Proof (i): If \( I + T \) is invertible in \( \mathbf{B}\left( \mathcal{H}\right) \), then \( S \mathrel{\text{:=}} - T{\left( I + T\right) }^{-1} \in {\mathbf{B}}_{1}\left( \mathcal{H}\right) \) and \( \left( {I + S}\right) \left( {I + T}\right) = I \), so \( \det \left( {I + T}\right) \det \left( {I + S}\right) = \de... | Yes |
Example 9.8 Let \( D \) be the rank one operator \( \alpha \langle \cdot, u\rangle u \), where \( u \in \mathcal{H} \) and \( \alpha \in \mathbb{C} \). Then \( D{R}_{z}\left( A\right) = \alpha \left\langle {{R}_{z}\left( A\right) \cdot, u}\right\rangle u \) has the eigenvalues \( {\lambda }_{1} = \alpha \left\langle {{... | Therefore, by (9.40) and (9.30), we obtain \[ {\Delta }_{B/A}\left( z\right) = 1 + \alpha \left\langle {{R}_{z}\left( A\right) u, u}\right\rangle = 1 + {\alpha F}\left( z\right) , \] that is, the expression in (9.23) is the perturbation determinant \( {\Delta }_{B/A}\left( z\right) \). | Yes |
Proposition 9.17 \( {\Delta }_{B/A}\left( z\right) \) is a holomorphic function on the resolvent set \( \rho \left( A\right) \) . | Proof We apply Corollary 9.14 with \( T \mathrel{\text{:=}} D{R}_{z}\left( A\right) \) . Since the determinant of the operator \( I + {P}_{n}D{R}_{z}\left( A\right) {P}_{n} \) is \( \det {\left( {\delta }_{kj} + \left\langle D{R}_{z}\left( A\right) {x}_{k},{x}_{j}\right\rangle \right) }_{j, k = 1}^{n} \), it is holomor... | Yes |
Proposition 9.19 Suppose that \( A \) and \( D \), hence \( B \), are self-adjoint operators. Then\n\n\[ \overline{{\Delta }_{B/A}\left( z\right) } = {\Delta }_{B/A}\left( \bar{z}\right) \;\text{ for }z \in \rho \left( A\right) ,\ ]\n\n(9.47)\n\n\[ {e}^{-\parallel D{\parallel }_{1}{\left| \operatorname{Im}z\right| }^{-... | Proof Using equalities (9.40),(9.35), and (9.37), we obtain for \( z \in \rho \left( A\right) \) ,\n\n\[ \overline{{\Delta }_{B/A}\left( z\right) } = \overline{\det \left( {I + D{R}_{z}\left( A\right) }\right) } = \det \left( {I + {R}_{\bar{z}}\left( A\right) D}\right) \]\n\n\[ = \det \left( {I + D{R}_{\bar{z}}\left( A... | Yes |
Corollary 9.23 If the operator \( D \) has precisely \( {n}_{ + } \) positive (resp. \( {n}_{ - } \) negative) eigenvalues (counted with multiplicities), then \( \xi \left( \lambda \right) \leq {n}_{ + } \) (resp. \( \xi \left( \lambda \right) \geq - {n}_{ - } \) ) a.e. on \( \mathbb{R} \) . In particular, if \( D \geq... | Proof Suppose that \( D \) has precisely \( {n}_{ + } \) positive eigenvalues. Then \( {n}_{ + } \) numbers \( {\alpha }_{k} \) are positive, and all others are negative or zero. Then, by (9.27), \( {n}_{ + } \) summands \( {\xi }_{k} \) have values in \( \left\lbrack {0,1}\right\rbrack \), while all other summands hav... | Yes |
Let \( B \) be the self-adjoint operator \( {A}_{\alpha } \) from Example 9.3. Assume that \( \alpha > 0 \) and \( {\lambda }_{n} < {\lambda }_{n + 1} \) for \( n \in \mathbb{N} \). Combining Corollary 9.25 and formula (9.17), in this case the spectral shift \( \xi \left( \cdot \right) \equiv \xi \left( {\cdot ;B, A}\r... | \[ \xi \left( \lambda \right) = 1\;\text{ if }{\lambda }_{k} < \lambda < v{\left( \alpha \right) }_{k},\;\xi \left( \lambda \right) = 0\;\text{ if }\lambda < {\lambda }_{1}\text{ or }v{\left( \alpha \right) }_{k} < \lambda < {\lambda }_{k + 1}.\;\text{ 。 } \] | Yes |
Theorem 9.27 Let \( f \in {W}_{1}\left( \mathbb{R}\right) \) . Then the operator \( f\left( B\right) - f\left( A\right) \) is of trace class and Krein’s trace formula (9.58) is satisfied for \( f \) . Moreover, if (9.62) holds, then\n\n\[ \parallel f\left( B\right) - f\left( A\right) {\parallel }_{1} \leq \left| \mu \r... | Proof We can assume without loss of generality that \( \mu \) is a finite positive measure. Then functional calculus and integration in (9.62) can be interchanged, so that\n\n\[ f\left( B\right) - f\left( A\right) = {\int }_{\mathbb{R}}\left( {{e}^{-\mathrm{i}{tB}} - {e}^{-\mathrm{i}{tA}}}\right) \mathrm{i}{t}^{-1}{d\m... | Yes |
Theorem 9.28 Let \( A \) and \( B \) be resolvent comparable self-adjoint operators on a Hilbert space \( \mathcal{H} \). Then there exists a real-valued function \( \xi \) on \( \mathbb{R} \) such that \( {\left( 1 + {\lambda }^{2}\right) }^{-1}\xi \left( \lambda \right) \in {L}^{1}\left( \mathbb{R}\right) \), and for... | \[ \xi \left( \lambda \right) = {\pi }^{-1}\arg {\widetilde{\Delta }}_{B/A}\left( {\lambda + \mathrm{i}0}\right) + c \mathrel{\text{:=}} {\pi }^{-1}\mathop{\lim }\limits_{{\varepsilon \rightarrow + 0}}{\widetilde{\Delta }}_{B/A}\left( {\lambda + \mathrm{i}\varepsilon }\right) + c\;\text{ a.e. on }\mathbb{R}. \] Here \(... | Yes |
Theorem 9.29 Let \( A \) and \( B \) be self-adjoint operators on \( \mathcal{H} \). Suppose that one of the following assumptions is fulfilled:\n\n(i) There is a trace class operator \( D \) on \( \mathcal{H} \) such that \( B = A + D \).\n\n(ii) \( {\left( B - zI\right) }^{-1} - {\left( A - zI\right) }^{-1} \) is of ... | Theorem 9.29 is usually proved in scattering theory, where the unitary equivalence of \( {A}_{\mathrm{{ac}}} \) and \( {B}_{\mathrm{{ac}}} \) is provided by means of the wave operators \( {\Omega }_{ \pm }\left( {A, B}\right) \). We do not carry out the proof and refer to [RS3, Theorems XI. 8 and XI.9]. | No |
Corollary 10.2 The sum of finitely many lower semibounded closed forms of \( \mathcal{H} \) is also closed. | Proof The sum of finitely many lower semicontinuous functions is obviously lower semicontinuous, so the assertion follows from Proposition 10.1,(i) \( \leftrightarrow \) (iii). | Yes |
Example 10.1 (Nonclosable forms) Let \( a, b \in \mathbb{R}, a < b, c \in \left\lbrack {a, b}\right\rbrack \), and \( \mathcal{H} = \) \( {L}^{2}\left( {a, b}\right) \) . Define positive forms by\n\n\[ \n{\mathrm{t}}_{1}\left\lbrack {f, g}\right\rbrack = f\left( c\right) \overline{g\left( c\right) }, \n\]\n\n\[ \n{\mat... | For any \( \alpha \in \mathbb{C} \), there is a sequence \( {\left( {f}_{n}\right) }_{n \in \mathbb{N}} \) from \( \mathcal{D}\left( {\mathfrak{t}}_{1}\right) = \mathcal{D}\left( {\mathfrak{t}}_{2}\right) \) such \( {f}_{n}\left( c\right) = \alpha \) for all \( n \in \mathbb{N} \) and \( \mathop{\lim }\limits_{n}{f}_{n... | Yes |
Proposition 10.5 Let \( A \geq {mI} \) be a lower semibounded self-adjoint operator.\n\n(i) For \( x, y \in \mathcal{D}\left\lbrack A\right\rbrack = \mathcal{D}\left( {\left( A - mI\right) }^{1/2}\right) \), we have\n\n\[ A\left\lbrack {x, y}\right\rbrack = \left\langle {{\left( A - mI\right) }^{1/2}x,{\left( A - mI\ri... | Proof (i): Clearly, \( \mathcal{D}\left\lbrack A\right\rbrack = \mathcal{D}\left( {\left| A\right| }^{1/2}\right) = \mathcal{D}\left( {\left| A - mI\right| }^{1/2}\right) \) by (4.26). Since \( A \geq m \cdot I \) , we have \( A = {\int }_{m}^{\infty }{\lambda d}{E}_{A}\left( \lambda \right) \) . Therefore, using (10.7... | Yes |
Proposition 10.6 Suppose that \( A \) is a lower semibounded self-adjoint operator and \( m < {m}_{A} \) . Then the following assertions are equivalent:\n\n(i) The embedding map \( {\mathcal{I}}_{{\mathfrak{t}}_{A}} : \left( {\mathcal{D}\left\lbrack A\right\rbrack ,\parallel \cdot {\parallel }_{{\mathfrak{t}}_{A}}}\rig... | Proof Note that \( \begin{Vmatrix}{{\left( A - mI\right) }^{1/2}x}\end{Vmatrix} \geq {\left( {m}_{A} - m\right) }^{1/2}\parallel x\parallel \) by the functional calculus. Hence, it follows from (10.11) that \( \parallel \cdot {\parallel }_{{\mathrm{t}}_{A}}^{\prime } \mathrel{\text{:=}} \parallel {\left( A - mI\right) ... | Yes |
Theorem 10.9 Let \( \mathfrak{t} \) be a densely defined lower semibounded closed form on \( \mathcal{H} \) , and let \( \lambda \in \mathbb{R},\lambda < {m}_{\mathrm{t}} \) . For \( u \in \mathcal{H} \), let \( {J}_{u} \) denote the functional on \( \mathcal{D}\left( \mathrm{t}\right) \) defined by\n\n\[ \n{J}_{u}\lef... | Proof Upon replacing \( t \) by \( t - \lambda \) and \( {A}_{t} \) by \( {A}_{t} - {\lambda I} \) we can assume without loss of generality that \( \lambda = 0 \) . Since then \( \mathrm{t} \geq {m}_{\mathrm{t}} > 0 \) and \( {A}_{\mathrm{t}} \geq {m}_{\mathrm{t}} \cdot I \) by Proposition 10.4(iii), we have \( {A}_{\m... | Yes |
Lemma 10.10 If \( A \) and \( B \) are lower semibounded self-adjoint operators, then:\n\n(i) \( A \geq B \) if and only if \( \mathcal{D}\left( A\right) \subseteq \mathcal{D}\left\lbrack B\right\rbrack \) and \( A\left\lbrack y\right\rbrack \geq B\left\lbrack y\right\rbrack \) for all \( y \in \mathcal{D}\left( A\righ... | Proof Upon adding a multiple of the identity we can assume that \( A \geq 0 \) and \( B \geq 0 \) . (i): Since \( \mathcal{D}\left( A\right) \subseteq \mathcal{D}\left\lbrack A\right\rbrack \), the only if implication is trivial.\n\nWe prove the converse direction. From the above condition it follows that\n\n\[ \begin{... | Yes |
Lemma 10.11 If \( T \) is a positive self-adjoint operator such that \( \mathcal{N}\left( T\right) = \{ 0\} \), then\n\n\[ \n{t}_{{T}^{-1}}^{\prime }\left\lbrack x\right\rbrack = \sup \left\{ {{\left| \langle x, y\rangle \right| }^{2}T{\left\lbrack y\right\rbrack }^{-1} : y \in \mathcal{D}\left\lbrack T\right\rbrack, y... | Proof First, let us note that \( {T}^{-1} \) is self-adjoint by Corollary 1.9, and \( \mathcal{D}\left\lbrack {T}^{-1}\right\rbrack = \) \( \mathcal{D}\left( {T}^{-1/2}\right) \), because \( {\left( {T}^{-1}\right) }^{1/2} = {T}^{-1/2} \) (see Exercise 5.9). Fix a nonzero \( x \in \mathcal{H} \).\n\nLet \( y \in \mathc... | Yes |
Corollary 10.12 Let \( A \) and \( B \) be self-adjoint operators on \( \mathcal{H} \) such that \( A \geq B \geq 0 \) and \( \mathcal{N}\left( B\right) = \{ 0\} \) . Then we have \( \mathcal{N}\left( A\right) = \{ 0\} \) and \( {B}^{-1} \geq {A}^{-1} \) . | Proof Let \( x \in \mathcal{N}\left( A\right) \) . Then, \( 0 = \langle {Ax}, x\rangle = {\begin{Vmatrix}{A}^{1/2}x\end{Vmatrix}}^{2} \geq {\begin{Vmatrix}{B}^{1/2}x\end{Vmatrix}}^{2} \), so that we have \( x \in \mathcal{N}\left( {B}^{1/2}\right) \subseteq \mathcal{N}\left( B\right) \), and therefore \( x = 0 \) . Thu... | Yes |
Corollary 10.13 Let \( A \) and \( B \) be lower semibounded self-adjoint operators on a Hilbert space \( \mathcal{H} \), and let \( \lambda \in \mathbb{R},\lambda < {m}_{A} \), and \( \lambda < {m}_{B} \). Then \( \lambda \in \rho \left( A\right) \cap \rho \left( B\right) \), and the relation \( A \geq B \) holds if a... | Proof By Proposition 10.4(iii), we have \( \lambda \in \rho \left( A\right) \cap \rho \left( B\right) \). Since \( {\mathfrak{t}}_{A - {\lambda I}} = {\mathfrak{t}}_{A} - \lambda \) and \( {\mathfrak{t}}_{B - {\lambda I}} = {\mathfrak{t}}_{B} - \lambda \), the relation \( A \geq B \) is equivalent to \( A - {\lambda I}... | Yes |
Proposition 10.14 Let \( A \) and \( B \) be self-adjoint operators on \( \mathcal{H} \) . If \( A \geq B \geq 0 \), then \( {A}^{\alpha } \geq {B}^{\alpha } \) for any \( \alpha \in \left( {0,1}\right) \) . | Proof Let \( T \) be a positive self-adjoint operator. By Proposition 5.16 we have\n\n\[ \n{\begin{Vmatrix}{T}^{\alpha /2}x\end{Vmatrix}}^{2} = {\pi }^{-1}\sin {\pi \alpha }{\int }_{0}^{\infty }{t}^{\alpha - 1}\left\langle {T{\left( T + tI\right) }^{-1}x, x}\right\rangle {dt} \n\]\n\n(10.22)\n\nfor \( x \in \mathcal{D}... | Yes |
Example 10.3 \( \left( {A \geq B \geq 0\text{does not imply}{A}^{2} \geq {B}^{2}}\right) \) For the operators \( A \) and \( B \) on the Hilbert space \( {\mathbb{C}}^{2} \) given by the matrices | \[ A = \left( \begin{array}{ll} 2 & 1 \\ 1 & 1 \end{array}\right) \;\text{ and }\;B = \left( \begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right) \] it is obvious that \( A \geq B \geq 0 \), but \( {A}^{2} \ngeqslant {B}^{2} \), since \[ {A}^{2} - {B}^{2} = \left( \begin{array}{ll} 4 & 3 \\ 3 & 2 \end{array}\right) \] | Yes |
Theorem 10.15 A function \( f \) on an open interval \( \mathcal{J} \) is operator monotone on \( \mathcal{J} \) if and only if there are numbers \( a \in \mathbb{R}, b \geq 0 \), and a finite positive regular Borel measure \( \mu \) on \( \mathbb{R} \) satisfying \( \mu \left( \mathcal{J}\right) = 0 \) such that \( f\... | Since \( \mu \left( \mathcal{J}\right) = 0 \), the holomorphic function \( f \) on \( {\mathbb{C}}_{ + } \) given by (10.23) admits then an analytic continuation across the interval \( \mathcal{J} \) into the lower half-plane. | No |
Lemma 10.16 If \( T \) is a densely defined lower semibounded symmetric operator, then the form \( {\mathfrak{s}}_{T} \) defined by \( {\mathfrak{s}}_{T}\left\lbrack {x, y}\right\rbrack = \langle {Tx}, y\rangle, x, y \in \mathcal{D}\left( {\mathfrak{s}}_{T}\right) \mathrel{\text{:=}} \mathcal{D}\left( T\right) \), is c... | Proof We write \( \mathfrak{s} \) for \( {\mathfrak{s}}_{T} \) . Let \( {\mathcal{H}}_{\mathfrak{s}} \) denote the completion of \( \left( {\mathcal{D}\left( \mathfrak{s}\right) ,\langle \cdot , \cdot {\rangle }_{\mathfrak{s}}}\right) \) . By Proposition 10.3, it suffices to show that the embedding \( {\mathcal{I}}_{\m... | Yes |
Theorem 10.17 Let \( T \) be a densely defined lower semibounded symmetric operator on a Hilbert space \( \mathcal{H} \). (i) \( {T}_{F} \) is a lower semibounded self-adjoint extension of \( T \) which has the same greatest lower bound as \( T \) . | Proof (i): If \( x \in \mathcal{D}\left( T\right) \), then \( {\mathrm{t}}_{{T}_{F}}\left\lbrack {x, y}\right\rbrack = \overline{{\mathfrak{s}}_{T}}\left\lbrack {x, y}\right\rbrack = \langle {Tx}, y\rangle \) for all \( y \in \mathcal{D}\left( T\right) \). From Proposition \( {10.5}\left( \mathrm{v}\right) \), applied ... | Yes |
Example 10.4 (Friedrichs extension of \( A = - \frac{{d}^{2}}{d{x}^{2}} \) on \( \mathcal{D}\left( A\right) = {H}_{0}^{2}\left( {a, b}\right), a, b \in \mathbb{R} \) ) From Example 1.4 we know that \( A \) is the square \( {T}^{2} \) of the symmetric operator \( T = - \mathrm{i}\frac{d}{dx} \) with domain \( \mathcal{D... | \[ {\mathfrak{s}}_{A}\left\lbrack f\right\rbrack = \langle {Af}, f\rangle = \parallel {Tf}{\parallel }^{2} = {\begin{Vmatrix}{f}^{\prime }\end{Vmatrix}}^{2}\;\text{ for }f \in \mathcal{D}\left( {\mathfrak{s}}_{A}\right) = \mathcal{D}\left( A\right) = {H}_{0}^{2}\left( {a, b}\right) . \] Hence, the form norm of \( {\mat... | Yes |
Statement If \( \mathcal{D}\left( T\right) \) is dense in \( {\mathcal{H}}_{1} \) and the operator \( T \) is closed, then \( {A}_{\mathrm{t}} = {T}^{ * }T \) . | Proof For \( x \in \mathcal{D}\left( {A}_{\mathrm{t}}\right) \) and \( y \in \mathcal{D}\left( T\right) \), we have \( \langle {Tx},{Ty}{\rangle }_{2} = \mathfrak{t}\left\lbrack {x, y}\right\rbrack = {\left\langle {A}_{\mathrm{t}}x, y\right\rangle }_{1} \) . This implies that \( {Tx} \in \mathcal{D}\left( {T}^{ * }\rig... | Yes |
Example 10.6 We apply the preceding example to the closed operators \( T = - \mathrm{i}\frac{d}{dx} \) with different domains in the Hilbert space \( {L}^{2}\left( {a, b}\right) \), where \( a, b \in \mathbb{R}, a < b \) . The form \( \mathrm{t} \) is always given by the same expression\n\n\[ \mathrm{t}\left\lbrack {f,... | We define the operators \( {T}_{0} = - \mathrm{i}\frac{d}{dx} \) and \( {T}_{\left( \alpha ,\beta \right) } = - \mathrm{i}\frac{d}{dx} \) for \( \left( {\alpha ,\beta }\right) \in {\mathbb{C}}^{2} \) with domains\n\n\[ \mathcal{D}\left( {T}_{0}\right) \mathrel{\text{:=}} {H}_{0}^{1}\left( {a, b}\right) \;\text{ and }\;... | Yes |
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