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Theorem 9.23. The Neumann problem (9.45) is uniquely solvable in \( D\left( {A}_{\max }\right) \) for \( f \in {L}_{2}\left( {\mathbb{R}}_{ + }^{n}\right) ,\varphi \in {H}^{\frac{1}{2}}\left( {\mathbb{R}}^{n - 1}\right) \) ; the solution belongs to \( {H}^{2}\left( {\mathbb{R}}_{ + }^{n}\right) \) and is defined by the... | \[ \left( \begin{array}{l} A \\ \nu \end{array}\right) : {H}^{2}\left( {\mathbb{R}}_{ + }^{n}\right) \rightarrow \mathop{\sum }\limits_{{{H}^{\frac{1}{2}}\left( {\mathbb{R}}^{n - 1}\right) }}^{{{L}_{2}\left( {\mathbb{R}}_{ + }^{n}\right) }}\text{ has inverse }\left( {{R}_{\nu }{K}_{\nu }}\right) : \mathop{\sum }\limits... | Yes |
The trace operator \( \mu \) satisfies\n\n\[ \n{\mu u} = \nu {A}_{\gamma }^{-1}{Au},\text{ for }u \in D\left( {A}_{\max }\right) ,\n\]\n\nhence maps \( D\left( {A}_{\max }\right) \) continuously into \( {H}^{\frac{1}{2}} \) . The following Green’s formula holds for all \( u, v \in D\left( {A}_{\max }\right) \ :\n\n\[ \... | Proof. When \( u \in D\left( {A}_{\max }\right) \), we decompose it in\n\n\[ \nu u = {u}_{\gamma } + {u}_{\zeta },\;{u}_{\gamma } = {A}_{\gamma }^{-1}{Au} \in D\left( {A}_{\gamma }\right) ,\n\]\n\nwhere \( {u}_{\zeta } \in Z\left( {A}_{\max }\right) \) (as in Lemma 13.1). Then since \( {\gamma }_{0}{u}_{\gamma } = 0 \)... | Yes |
Consider a closed realization \( \widetilde{A} \) of \( A \), corresponding to \( T \) : \( V \rightarrow W \) as in Theorem 13.7 (with \( {A}_{1} = {A}_{1}^{\prime } = {A}_{\max } \) ), and let \( L : X \rightarrow {Y}^{ * } \) be the corresponding operator introduced in Definition 9.28. Then \( D\left( \widetilde{A}\... | Proof. We have from Theorem 13.5 that the elements of \( D\left( \widetilde{A}\right) \) are characterized by the two conditions: \[ {u}_{\zeta } \in D\left( T\right) ,\;\left( {{Au}, w}\right) = \left( {T{u}_{\zeta }, w}\right) \text{ for all }w \in W. \] (9.77) We just have to translate this to boundary conditions. I... | Yes |
Example 9.30. For \( {A}_{\gamma } \) itself, \( V = W = \{ 0\} \), and \( T \) is trivial (zero) then. So \( X = Y = \{ 0\} \) and \( L = 0 \) . The boundary condition is | \[ {\gamma }_{0}u = 0 \] (9.79) as we know very well. | No |
The Neumann realization is defined by the boundary condition\n\n\\[ \n{\\nu u} = 0.\n\\]\n\n(9.82)\n\nWe know from Section 9.3 that the domain of the hereby defined realization \\( {A}_{\\nu } \\) equals \\( \\left\\{ {u \\in {H}^{2}\\left( {\\mathbb{R}}_{ + }^{n}\\right) \\mid {\\nu u} = 0}\\right\\} \\) . | Since \\( \\varrho = \\left\\{ {{\\gamma }_{0},\\nu }\\right\\} \\) is surjective from \\( {H}^{2}\\left( {\\mathbb{R}}_{ + }^{n}\\right) \\) to \\( {H}^{\\frac{3}{2}} \\times {H}^{\\frac{1}{2}} \\) (cf. Theorem 9.5), \\( {\\gamma }_{0}u \\) runs through \\( {H}^{\\frac{3}{2}} \\) when \\( u \\) runs through the domain... | Yes |
Theorem 9.33. Let \( \widetilde{A} \) be the realization defined by the boundary condition (9.84). If \( l\left( {\xi }^{\prime }\right) \) in (9.92) is elliptic, then\n\n\[ D\left( \widetilde{A}\right) = \left\{ {u \in {H}^{2}\left( {\mathbb{R}}_{ + }^{n}\right) \mid {\nu u} + B{\gamma }_{0}u = 0}\right\} . \]\n\nIn t... | The last statement follows since the adjoint symbol \( \bar{l}\left( {\xi }^{\prime }\right) \) is then likewise elliptic.\n\nEllipticity clearly holds if\n\n\[ c \neq 1,{b}_{1},\ldots ,{b}_{n - 1} \in \mathbb{R} \]\n\n(9.94)\n\nin particular when \( c = 0 \) and the \( {b}_{j} \) are real. | No |
Theorem 9.35. Let \( \widetilde{A} \) be the realization determined by the boundary condition (9.84).\n\n\( {1}^{ \circ } \) If \( \widetilde{A} \) is lower bounded, so is \( L \), with a similar sign of the lower bound.\n\n\( {2}^{ \circ } \) If \( L \) has positive or zero lower bound, so has \( \widetilde{A} \) . | Proof. We use the equivalence of (9.74) and (9.75). Note that we are in a case where we know beforehand that \( V = W = Z\left( {A}_{1}\right) \) . The first statement follows from Theorem 13.15. The second statement follows from Theorem 13.17. | No |
Corollary 9.36. When \( B \) is a differential operator with real coefficients, the realization defined by the boundary condition (9.84) is variational with positive lower bound. | Proof. In this case, \( \operatorname{Re}l\left( {\xi }^{\prime }\right) = \left\langle {\xi }^{\prime }\right\rangle \geq 1 \) and \( D\left( L\right) = {H}^{\frac{3}{2}} \), so \( L \) has lower bound 1, as an operator in \( {L}_{2} = {H}^{0} \) . The same holds for \( {L}^{ * } \) . Moreover, \[ \left| {\operatornam... | Yes |
A parametrix symbol \( q\left( {x,\xi }\right) \) for an elliptic differential operator \( P \) of order \( d \) certainly has the transmission property, since its symbol terms are rational functions of \( \xi \) . In fact, it has the stronger property\n\n\[ \n{q}_{-d - l}\left( {x, - \xi }\right) = {\left( -1\right) }... | Polyhomogeneous symbols having the property (10.3) are in some texts said to have even-even alternating parity (the even-order symbols are even), or just to be even-even, for short. The opposite parity\n\n\[ \n{q}_{-d - l}\left( {x, - \xi }\right) = {\left( -1\right) }^{d - l + 1}{q}_{-d - l}\left( {x,\xi }\right) \tex... | Yes |
The operator \( {K}_{\gamma } \) introduced in Theorem 9.3 is the Poisson operator with symbol-kernel \( \widetilde{k}\left( {{x}_{n},{\xi }^{\prime }}\right) = {e}^{-\left\langle {\xi }^{\prime }\right\rangle {x}_{n}} \) ; it is of order 0 . Its symbol is \( k\left( {{\xi }^{\prime },{\xi }_{n}}\right) = \frac{1}{\lef... | Inserting the expansion (7.11) of \( \left\langle {\xi }^{\prime }\right\rangle \) in (10.33) one can expand in homogeneous terms of falling degree (beginning with degree -1), showing that the symbol and symbol-kernel are polyhomogeneous of degree \( - 1 \) | No |
Example 10.9. As a simple example of a singular Green symbol-kernel, let us take \( \widetilde{g}\left( {{x}_{n},{y}_{n},{\xi }^{\prime }}\right) = {e}^{-\left\langle {\xi }^{\prime }\right\rangle \left( {{x}_{n} + {y}_{n}}\right) } \) . Its symbol is | \[ g\left( {{\xi }^{\prime },{\xi }_{n},{\eta }_{n}}\right) = \frac{1}{\left( {\left\langle {\xi }^{\prime }\right\rangle + i{\xi }_{n}}\right) \left( {\left\langle {\xi }^{\prime }\right\rangle - i{\eta }_{n}}\right) }, \] it is of degree -2 and order -1 . In view of the last remark in Example 10.7, \( - \frac{1}{2\le... | Yes |
Proposition 10.10. Consider a \( \psi \) do symbol \( p\left( {{x}^{\prime },0,{\xi }^{\prime },{\xi }_{n}}\right) \) and trace, Poisson and singular Green symbol-kernels \( \widetilde{t}\left( {{x}^{\prime },{x}_{n},{\xi }^{\prime }}\right) ,\widetilde{k}\left( {{x}^{\prime },{x}_{n},{\xi }^{\prime }}\right) ,\widetil... | (i) If \( p \) is \( O\left( {\left\langle {\xi }_{n}\right\rangle }^{-1}\right) \), then \( {\gamma }_{0}{\mathrm{{OP}}}_{n}{\left( p\right) }_{ + } = {\mathrm{{OPT}}}_{n}\left( {\widetilde{t}}^{\prime \prime }\right) \), where \( {\widetilde{t}}^{\prime \prime }\left( {{x}^{\prime },{x}_{n},{\xi }^{\prime }}\right) =... | Yes |
Proposition 10.11. Let \( \zeta \in {C}^{\infty }\left( {\overline{\mathbb{R}}}_{ + }^{n}\right) \) be such that \( \zeta \left( x\right) = 0 \) for \( {x}_{n} \leq \varepsilon \) , some \( \varepsilon > 0 \) (e.g., \( \zeta \left( x\right) = 1 - \chi \left( {{x}_{n}/\varepsilon }\right) \) on \( {\overline{\mathbb{R}}... | Proof. For any \( N \in {\mathbb{N}}_{0},{\zeta }_{N}\left( x\right) = \zeta \left( x\right) /{x}_{n}^{N} \) is in \( {C}^{\infty }\left( {\overline{\mathbb{R}}}_{ + }^{n}\right) \), supported in \( \left\{ {{x}_{n} \geq }\right. \) \( \varepsilon \} \) . Then\n\n\[{\zeta K} = {\zeta }_{N}{x}_{n}^{N}K,\;{\zeta G} = {\z... | Yes |
Lemma 10.13. Let \( \sigma > 0 \) and let \( d \in \mathbb{Z} \) . Let \( f\left( t\right) \in {C}^{\infty }\left( \mathbb{R}\right) \), and define\n\n\[ \tau = {t}^{-1}, k\left( \tau \right) = {\tau }^{d}f\left( {\tau }^{-1}\right) \text{ for }\tau \in \mathbb{R} \smallsetminus \{ 0\} ; \]\n\n\[ z = \frac{\sigma - {it... | Proof. Consider first the case \( d = 0 \) . Assume that \( f \) satisfies the conditions (10.45) (which then also hold with \( {\partial }_{t}^{l}{t}^{k} \) replaced by \( {t}^{k}{\partial }_{t}^{l} \) ). Since \( f\left( t\right) - {s}_{0} \) is \( O\left( {t}^{-1}\right) \) for \( t \rightarrow \pm \infty, k\left( \... | Yes |
Lemma 10.14. Let \( u \in {L}_{2}\left( {\mathbb{R}}_{ + }\right) \), expanded in the Laguerre system \( {\left( {\varphi }_{k}\right) }_{k \in {\mathbb{N}}_{0}} \) , by\n\n\[ u\left( x\right) = \mathop{\sum }\limits_{{k \in {\mathbb{N}}_{0}}}{b}_{k}{\varphi }_{k}\left( {x,\sigma }\right) . \]\n\nThen \( u \in \mathcal... | Proof. The identity (10.58) follows from the orthonormality and completeness of the system \( {\varphi }_{k} \) in \( {L}_{2}\left( {\mathbb{R}}_{ + }\right) \) . (10.59) then follows easily from the eigenvalue property of the \( {\varphi }_{k} \) :\n\n\[ {\begin{Vmatrix}{\left( {b}_{k}\right) }_{k \in {\mathbb{N}}_{0}... | Yes |
Lemma 10.18. When \( f = \mathcal{F}{e}^{ + }u\left( {u \in {\mathcal{S}}_{ + }}\right) \), with the expansion (10.80), then \[ \frac{1}{2\pi }{\int }^{ + }{t}^{k}f\left( t\right) {dt} = i{s}_{-1 - k} = {\gamma }_{k}u. \] | The coefficient \( {s}_{-1 - k} \) can be estimated by use of the standard trace estimates \[ {\left| {s}_{-1 - k}\right| }^{2} = {\left| {\gamma }_{k}u\right| }^{2} = - {\int }_{0}^{\infty }{\partial }_{x}\left\lbrack {{u}^{\left( k\right) }{\bar{u}}^{\left( k\right) }}\right\rbrack {dx} \] \[ \leq 2{\begin{Vmatrix}{D... | Yes |
Theorem 10.21. When \( p\left( {X,{x}_{n},{y}_{n},\xi }\right) \) satisfies Definition 10.2, then, with \( r = \max \{ d + 1,0\} \)\n\n10.3 The complex formulation\n\n\[ \n{h}^{ + }p\left( {X,0,0,\xi }\right) \in {S}_{1,0}^{d}\left( {\Xi ,{\mathbb{R}}^{n - 1},{\mathcal{H}}^{ + }}\right) \]\n\n\[ \n{h}^{ - }p\left( {X,0... | Proof. From Definition 10.2 follows that \( {h}_{-1}\left\lbrack {{D}_{X}^{\beta }{D}_{\xi }^{\alpha }\left( {{\xi }_{n}^{m}p\left( {X,0,0,\xi }\right) }\right) }\right\rbrack \) satisfies estimates\n\n\[ \n{\begin{Vmatrix}{h}_{-1}\left\lbrack {D}_{X}^{\beta }{D}_{\xi }^{\alpha }\left( {\xi }_{n}^{m}p\left( X,0,0,\xi \... | Yes |
With \( \sigma = \left\langle {\xi }^{\prime }\right\rangle \), the nonnormalized Fourier-transformed Laguerre functions | \[ {\left( 2\sigma \right) }^{-\frac{1}{2}}{\widehat{\varphi }}_{l}\left( {{\xi }_{n},\sigma }\right) = \frac{{\left( \sigma - i{\xi }_{n}\right) }^{l}}{{\left( \sigma + i{\xi }_{n}\right) }^{l + 1}} \] (cf. (10.55)) with \( l \geq 0 \) lie in \( {S}^{-1}\left( {{\mathbb{R}}^{n - 1},{\mathbb{R}}^{n - 1},{\mathcal{H}}^{... | Yes |
Theorem 10.24. Let \( d \) and \( {d}^{\prime } \in \mathbb{R} \), and let \( r \) and \( {r}^{\prime } \in {\mathbb{N}}_{0} \). Let (i) \( p\left( {X,\xi }\right) \in {S}_{1,0}^{d}\left( {\Xi ,{\mathbb{R}}^{n}}\right) \), with transm. cond., (ii) \( g\left( {X,{\xi }^{\prime },{\xi }_{n},{\eta }_{n}}\right) \in {S}_{1... | \[ {1}^{ \circ }\;{p}_{ + }{ \circ }_{n}{k}^{\prime } = {h}_{{\xi }_{n}}^{ + }\left\lbrack {p\left( {X,\xi }\right) {k}^{\prime }\left( {X,\xi }\right) }\right\rbrack \in {S}_{1,0}^{{d}^{\prime \prime } - 1}\left( {\mathcal{H}}^{ + }\right) ,\] \[ {2}^{ \circ }\;g{ \circ }_{n}{k}^{\prime } = {\int }^{ + }g\left( {X,\xi... | Yes |
Theorem 10.25. Let \( P = \operatorname{OP}\left( {p\left( {x, y,\xi }\right) }\right) \) be of order \( d \), and define \( K \) by (10.115). Then \( K \) is a Poisson operator of order \( d + 1 \) . The symbol-kernel \( \widetilde{k} \) and symbol \( k \) satisfy (with \( \widetilde{p} = {\mathcal{F}}_{{\xi }_{n} \ri... | Proof. The last statement is obvious since \( P\left( {v\left( {x}^{\prime }\right) \otimes \delta \left( {x}_{n}\right) }\right) \) is supported in \( \left\{ {{x}_{n} = 0}\right\} \) when \( P \) is a differential operator.\n\nTo show the formula, let first \( p \) be independent of \( {x}_{n} \) . Then\n\n\[ {r}^{ +... | Yes |
Corollary 10.27. Let \( p \) and \( {p}^{\prime } \) be as in Theorem 10.24 and write\n\n\[ \n{p}^{\prime } = \mathop{\sum }\limits_{{0 \leq j \leq {d}^{\prime }}}{s}_{j}^{\prime }\left( {X,{\xi }^{\prime }}\right) {\xi }_{n}^{j} + {h}_{-1}{p}^{\prime }\n\]\n\nThen \( L\left( {p,{p}^{\prime }}\right) = {\left( p\left( ... | More precisely, the symbol is defined by\n\n\[ \nL\left( {p,{p}^{\prime }}\right) \left( {X,\xi ,{\eta }_{n}}\right) = \mathop{\sum }\limits_{{0 \leq m < {d}^{\prime }}}{k}_{m}\left( {X,\xi }\right) {\eta }_{n}^{m} + {g}^{ + }\left( p\right) { \circ }_{n}{g}^{ - }\left( {p}^{\prime }\right) ,\n\]\n\n(10.128)\n\nwhere \... | Yes |
Theorem 10.28. Let symbols be given as in Theorem 10.24, with \( \Xi = {\mathbb{R}}^{n - 1} \) with points \( {x}^{\prime } \) . Let \( a\left( {{x}^{\prime },{\xi }^{\prime },{D}_{n}}\right) \) stand for the boundary symbol operator, and let \( A \) stand for the full operator \( {\mathrm{{OP}}}^{\prime }\left( {a\lef... | Proof. \( {1}^{ \circ } \) If \( {a}^{\prime } \) were in \( {y}^{\prime } \) -form, the resulting operator would simply have the boundary symbol operator in \( \left( {{x}^{\prime },{y}^{\prime }}\right) \) -form\n\n\[ \na\left( {{x}^{\prime },{\xi }^{\prime },{D}_{n}}\right) { \circ }_{n}{a}^{\prime }\left( {{y}^{\pr... | Yes |
Theorem 10.29. \( {1}^{ \circ } \) Let \( K \) be a Poisson operator of order \( d \in \mathbb{R} \), with symbol-kernel \( \widetilde{k}\left( {{x}^{\prime },{y}^{\prime },{x}_{n},{\xi }^{\prime }}\right) \in {S}_{1,0}^{d - 1}\left( {{\mathbb{R}}^{2\left( {n - 1}\right) },{\mathbb{R}}^{n - 1},{\mathcal{S}}_{ + }}\righ... | Proof. We have for \( v \in \mathcal{S}\left( {\mathbb{R}}^{n - 1}\right), u \in \mathcal{S}\left( {\overline{\mathbb{R}}}_{ + }^{n}\right) \) :\n\n\[ \n{\left( Kv, u\right) }_{{L}_{2}\left( {\mathbb{R}}_{ + }^{n}\right) } = \int {e}^{i\left( {{x}^{\prime } - {y}^{\prime }}\right) \cdot {\xi }^{\prime }}\widetilde{k}\l... | Yes |
Corollary 10.30. When \( T = \mathop{\sum }\limits_{{0 < j < r}}{S}_{j}{\gamma }_{j} + {T}^{\prime } \) is a trace operator of class \( r \) and order \( d \), with symbol compactly supported with respect to \( {x}^{\prime } \) and \( {y}^{\prime } \), and \( s > r - \frac{1}{2}, s \geq 0, t \in \mathbb{R} \), then \( ... | Proof. For the part \( {T}^{\prime } \) of class 0 this follows from Theorem 10.29. For the terms \( {S}_{j}{\gamma }_{j} \), it follows by a straightforward generalization of the estimate for \( {\gamma }_{0} \) shown in Remark 9.4: For \( u \in \mathcal{S}\left( {\mathbb{R}}^{n}\right), s > j + \frac{1}{2} \) ,\n\n\[... | Yes |
Theorem 10.37. When \( \mathcal{A} \) is elliptic, the inverse \( {\mathfrak{b}}^{0} \) of the principal boundary symbol operator \( {\mathfrak{a}}^{0} \) belongs to the calculus; it is a boundary symbol operator of order \( - d \) and class \( {r}^{\prime } = \max \{ r - d,0\} \) . | Proof (indications). The proof of this theorem in the differential operator case, where the \( \psi \) dbo is as in (10.19) (with \( M = 0 \) ), is not so hard, since one can find the inverse constructively by analysis of the solutions in \( {\mathcal{S}}_{ + }^{N} \) of the equation \( {p}^{0}\left( {{x}^{\prime },0,{... | Yes |
Theorem 10.38. When \( \mathcal{A} \) is elliptic, there exists a \( {\psi dbo}\mathcal{B} \) (with principal boundary symbol operator \( {\mathfrak{b}}^{0} \) as in Theorem 10.37), which is a parametrix of \( \mathcal{A} \), in the sense that\n\n\[ \mathcal{A}\mathcal{B} - I\text{and}\mathcal{B}\mathcal{A} - I\text{ar... | Proof (indications). A first approximation to \( \mathcal{B} \) is the operator \( {\mathcal{B}}^{\prime } \) with boundary symbol \( {\mathfrak{b}}^{0} \) and interior symbol \( q \), where \( q\left( {x,\xi }\right) \) is a parametrix symbol for \( p \) . Then \( {\mathcal{{AB}}}^{\prime } \) equals the identity plus... | No |
Let \( {a}^{0}\left( {{x}^{\prime },\xi }\right) \) be the principal symbol at \( {x}_{n} = 0 \) of an elliptic partial differential operator \( A \) on \( {\mathbb{R}}^{n} \) of order \( d \), possibly \( N \times N \) -matrix-formed. Along with \( A \) there is given a trace operator \( T = \left\{ {{T}_{0},{T}_{1},\... | The problem is easily reduced to a semihomogeneous problem by use of the inverse symbol \( {q}^{0}\left( {{x}^{\prime },{\xi }^{\prime },{\xi }_{n}}\right) = {\left( {a}^{0}\right) }^{-1} \), defined for \( {\xi }^{\prime } \neq 0 \) . In fact, \( {a}_{ + }^{0}{q}_{ + }^{0} = I \) on \( {\mathbb{R}}_{ + } \), since \( ... | Yes |
Proposition 11.3. Let \( A \) be a differential operator of order \( d \) from \( {\widetilde{E}}_{1} \) to \( {\widetilde{E}}_{2} \), written as\n\n\[ A = \mathop{\sum }\limits_{{l = 0}}^{d}{S}_{l}\left( {{x}^{\prime },{x}_{n},{D}^{\prime }}\right) {D}_{n}^{l} \]\n\non \( U \), with differential operators \( {S}_{l} \... | Proof. We show the formula for smooth \( u \) and \( v \) ; then it extends by continuity to \( {H}^{d} \) spaces. By definition of \( {A}^{ * },{\left( Au, v\right) }_{{X}_{ + }} - {\left( u,{A}^{ * }v\right) }_{{X}_{ + }} = 0 \) if \( u \) or \( v \) has compact support in \( {X}_{ + } \), so the only nontrivial cont... | Yes |
Theorem 11.4. The map \( {\varrho }^{ + } \) extends to a continuous map from \( {Z}_{ + }^{s} \) to \( {\mathcal{H}}^{s}\left( {E}_{1}^{\prime d}\right) \) for all \( s \in \mathbb{R} \) . | Proof (indications). One ingredient in the proof is the fact that for any \( r \in \mathbb{Z} \) , the operator \( {\left( {\Xi }_{ - }^{r}\right) }_{ + } = \mathrm{{OP}}{\left( {\left( \left\langle {\xi }^{\prime }\right\rangle - i{\xi }_{n}\right) }^{r}\right) }_{ + } \) maps \( {H}^{s, t}\left( {\mathbb{R}}_{ + }^{n... | No |
Proposition 11.7. \( {1}^{ \circ } \) The \( \psi \) do’s \( {C}^{ \pm } \) defined in Definition 11.6 are projections in \( {\mathcal{H}}^{s}\left( {E}_{1}^{\prime d}\right) \) for all \( s \in \mathbb{R} \) ,\n\n\[{\left( {C}^{ + }\right) }^{2} = {C}^{ + },\;{\left( {C}^{ - }\right) }^{2} = {C}^{ - }.\]\n\n\( {2}^{ \... | Proof. The projection property follows, since\n\n\[{\left( {C}^{ + }\right) }^{2} = {\varrho }^{ + }{K}^{ + }{\varrho }^{ + }{K}^{ + } = {\varrho }^{ + }{K}^{ + } = {C}^{ + }\n\]\nin view of (11.19); the identity \( {\left( {C}^{ + }\right) }^{2} = {C}^{ + } \) thus holds for smooth sections and extends by continuity t... | Yes |
Assume that \( A \) has the inverse \( Q \) on \( \widetilde{X} \) . Define the spaces \( {Z}_{ \pm }^{s} \) and \( {N}_{ \pm }^{s} \) by (11.14). Then the spaces \( {N}_{ \pm }^{s} \) are complementing closed subspaces of \( {\mathcal{H}}^{s}\left( {E}_{1}^{\prime d}\right) \); \[ {\mathcal{H}}^{s}\left( {E}_{1}^{\pri... | Proof. It remains to account for surjectiveness and homeomorphism properties. The surjectiveness of \( {K}^{ + } : {\mathcal{H}}^{s}\left( {E}_{1}^{\prime d}\right) \rightarrow {Z}_{ + }^{s} \) follows from the identity \( z = \) \( {K}^{ + }{\varrho }^{ + }z \) for \( z \in {Z}_{ + }^{s} \) shown in Proposition 11.5. ... | Yes |
Proposition 11.9. Let \( {\xi }^{\prime } \neq 0 \) . The spaces \( {Z}_{ \pm }\left( {{x}^{\prime },{\xi }^{\prime }}\right) \) and \( {N}_{ \pm }\left( {{x}^{\prime },{\xi }^{\prime }}\right) \) defined in (11.32)-(11.34) have dimension \( {m}_{ \pm }\left( {{x}^{\prime },{\xi }^{\prime }}\right) \) (cf. Example 10.3... | The principal boundary symbol operators for \( {K}^{ \pm } \) and \( {C}^{ \pm } \) in Theorem 11.8 are determined as\n\n\[ \n{k}^{\pm ,0}\left( {{x}^{\prime },{\xi }^{\prime },{D}_{n}}\right) = {K}^{ \pm }\left( {{x}^{\prime },{\xi }^{\prime }}\right) ,\;{c}^{\pm ,0}\left( {{x}^{\prime },{\xi }^{\prime }}\right) = {\v... | No |
Theorem 11.11. Consider the model operator (principal boundary symbol operator)\n\n\[ \left( \begin{matrix} {a}^{0}\left( {{x}^{\prime },0,{\xi }^{\prime },{D}_{{x}_{n}}}\right) \\ {s}^{0}\left( {{x}^{\prime },{\xi }^{\prime }}\right) \varrho \end{matrix}\right) : {\mathcal{S}}_{ + }^{N} \rightarrow \begin{matrix} {\ma... | In particular,\n\n\( \left( \begin{matrix} A \\ S\varrho \end{matrix}\right) \) is injectively elliptic \( \Leftrightarrow \left( \begin{matrix} S \\ {C}^{ - } \end{matrix}\right) \) is injectively elliptic;\n\n(11.46)\n\n\( \left( \begin{matrix} A \\ S\varrho \end{matrix}\right) \) is surjectively elliptic \( \Leftrig... | Yes |
The systems \( \left( \begin{matrix} A \\ \varrho \end{matrix}\right) \) and \( \left( \begin{matrix} A \\ {C}^{ + }\varrho \end{matrix}\right) \) are injectively elliptic; they both have the left inverse \( \left( {{Q}_{ + }{K}^{ + }}\right) \). | In fact, by (11.42),\n\n\[ \n{Q}_{ + }A + {K}^{ + }\varrho = I;\;{Q}_{ + }A + {K}^{ + }{C}^{ + }\varrho = I.\n\]\n\nThis left inverse is also found from (11.39), when we use that \( \left( \begin{matrix} I \\ {Li}{C}^{ - } \end{matrix}\right) \) and \( \left( \begin{matrix} {C}^{ + } \\ {C}^{ - } \end{matrix}\right) \)... | Yes |
Lemma 11.16. Each of the blocks \( {C}_{ij}^{ + } \) in (11.65) is elliptic. | The proof is developed in Exercises 11.7-11.10 (originally shown in [G71]). Now, note that in particular,\n\n\[ \n{C}_{10}^{ + }{\psi }_{0} = {P}_{\gamma ,\nu }{C}_{00}^{ + }{\psi }_{0} \n\]\n\n(11.67)\n\nfor \( {\psi }_{0} \in \mathop{\prod }\limits_{{0 \leq j < m}}{H}^{{2m} - j - \frac{1}{2}}\left( {X}^{\prime }\righ... | No |
Theorem 11.18. When (11.48) holds also for the Dirichlet realization on \( {X}_{ - } \) (so that \( {P}_{\gamma ,\nu }^{ - } \) is well-defined), then \( {P}_{\gamma ,\nu }^{ - } \) and \( {P}_{\gamma ,\nu }^{ + } - {P}_{\gamma ,\nu }^{ - } \) are elliptic \( \psi \) do’s. | In this case, one can moreover show that \( {P}_{\gamma ,\nu }^{ + } - {P}_{\gamma ,\nu }^{ - } \) is invertible. Then \( {C}^{ + } \) can be described by an explicit formula from \( {P}_{\gamma ,\nu }^{ + } \) and \( {P}_{\gamma ,\nu }^{ - } \), and vice versa (details are worked out in Exercise 11.25):\n\n\[ \left( \... | No |
Theorem 11.19. When \( A \) is strongly elliptic, invertible on \( \widetilde{X} \), and satisfies (11.48), then the solution operator \( \left( {{R}_{\gamma }{K}_{\gamma }}\right) \) of the Dirichlet problem satisfies:\n\n\( {K}_{\gamma } \) is a Poisson operator (mapping as in (11.61)), and \( {R}_{\gamma } \) is the... | One can moreover show that the continuity of \( {R}_{\gamma } \) from \( {H}^{s}\left( X\right) \) to \( {H}^{s + {2m}}\left( X\right) \) for \( s \geq 0 \) extends down to \( s > - m - \frac{1}{2} \), see e.g. the analysis in [G90]. | No |
Lemma 12.2. \( T : X \rightarrow Y \) is closed if and only if the following holds: When \( {\left( {x}_{n}\right) }_{n \in \mathbb{N}} \) is a sequence in \( D\left( T\right) \) with \( {x}_{n} \rightarrow x \) in \( X \) and \( T{x}_{n} \rightarrow y \) in \( Y \), then \( x \in D\left( T\right) \) with \( y = {Tx} \... | The closed graph theorem (recalled in Appendix B, Theorem B.16) implies that if \( T : X \rightarrow Y \) is closed and has \( D\left( T\right) = X \), then \( T \) is bounded. Thus for closed, densely defined operators, \( D\left( T\right) \neq X \) is equivalent with unboundedness. | No |
Lemma 12.4. Let \( T \) be densely defined. Then there is an adjoint operator \( {T}^{ * } : {Y}^{ * } \rightarrow {X}^{ * } \), uniquely defined by (12.7)-(12.9). Moreover, \( {T}^{ * } \) is closed. | Proof. The definition of \( {T}^{ * } \) is accounted for above; it remains to show the closedness.\n\nLet \( {y}_{n}^{ * } \in D\left( {T}^{ * }\right) \) for \( n \in \mathbb{N} \), with \( {y}_{n}^{ * } \rightarrow {y}^{ * } \) and \( {T}^{ * }{y}_{n}^{ * } \rightarrow {z}^{ * } \) for \( n \rightarrow \infty \) ; t... | Yes |
Theorem 12.5. Let \( T : X \rightarrow Y \) be a densely defined operator between two Hilbert spaces \( X \) and \( Y \). Then\n\n\[ X \oplus Y = \overline{G\left( T\right) } \oplus {UG}\left( {T}^{ * }\right) ,\]\n\n(12.14)\n\nwhere \( U \) is the operator from \( Y \oplus X \) to \( X \oplus Y \) given by \( U\{ v, w... | Proof. Let \( \{ v, w\} \in X \oplus Y \). The following statements are equivalent:\n\n\[ \{ v, w\} \in {UG}\left( {T}^{ * }\right) \Leftrightarrow \{ w, - v\} \in G\left( {T}^{ * }\right) \]\n\n\[ \Leftrightarrow {\left( Tx, w\right) }_{Y} = - {\left( x, v\right) }_{X}\;\forall x \in D\left( T\right) \]\n\n\[ \Leftrig... | Yes |
Corollary 12.6. Let \( T : X \rightarrow Y \) be densely defined. Then \( T \) has a closed extension if and only if \( {T}^{ * } \) is densely defined, and in the affirmative case, \[ {T}^{ * } = {\left( \bar{T}\right) }^{ * }\text{ and }{T}^{* * } = \bar{T}. \] | Proof. If \( T \) has a closure, then in particular \( \overline{G\left( T\right) } = G\left( \bar{T}\right) \) . Then \( {T}^{ * } = {\left( \bar{T}\right) }^{ * } \) by (12.14), and \( {\left( \bar{T}\right) }^{ * } \) is densely defined according to Theorem 12.5, with \( {T}^{* * } = {\left( \bar{T}\right) }^{* * } ... | Yes |
Theorem 12.7. Assume that \( T : X \rightarrow Y \) has the properties:\n\n(1) \( T \) is densely defined,\n\n(2) \( T \) is closed,\n\n(3) \( T \) is injective,\n\n(4) \( T \) has range dense in \( Y \) .\n\nThen \( {T}^{ * } \) and \( {T}^{-1} \) also have the properties (1)-(4), and\n\n\[{\left( {T}^{ * }\right) }^{... | Proof. \( {T}^{-1} \) is clearly injective, densely defined and closed (cf. Lemma 12.2) with dense range, and the same holds for \( {T}^{ * } \) by Theorem 12.5, Corollary 12.6 and (12.11) (applied to \( T \) and \( {T}^{ * } \) ). It then follows moreover that \( {\left( {T}^{ * }\right) }^{-1} \) and \( {\left( {T}^{... | Yes |
Lemma 12.8. Let \( T \) be an operator in the complex Hilbert space \( H \) . \( {1}^{ \circ }\;T \) is symmetric if and only if \( \left( {{Tx}, x}\right) \) is real for all \( x \) . | Proof. When \( T \) is symmetric, \[ \left( {{Tx}, x}\right) = \left( {x,{Tx}}\right) = \overline{\left( Tx, x\right) }\;\text{ for }x \in D\left( T\right) , \] whereby \( \left( {{Tx}, x}\right) \in \mathbb{R} \) . Conversely, when \( \left( {{Tx}, x}\right) \in \mathbb{R} \) for all \( x \in D\left( T\right) \), then... | Yes |
Theorem 12.9. \( {1}^{ \circ } \) If \( m\left( T\right) \geq \alpha > 0 \), then \( T \) is injective, and \( {T}^{-1} \) (with \( \left. {D\left( {T}^{-1}\right) = R\left( T\right) }\right) \) is a bounded operator in \( H \) with norm \( \begin{Vmatrix}{T}^{-1}\end{Vmatrix} \leq {\alpha }^{-1} \) . | Proof. The basic observation is that \( m\left( T\right) \geq \alpha \) implies the inequality\n\n\[ \parallel {Tx}\parallel \parallel x\parallel \geq \left| \left( {{Tx}, x}\right) \right| \geq \operatorname{Re}\left( {{Tx}, x}\right) \geq \alpha \parallel x{\parallel }^{2}\text{ for }x \in D\left( T\right) ,\]\n\n(12... | Yes |
Theorem 12.10. Let \( S \) be densely defined and symmetric. Then \( S \) is selfadjoint if and only if\n\n\[ R\left( {S + {iI}}\right) = R\left( {S - {iI}}\right) = H \]\n\n(12.25)\n\nand in the affirmative case, \( \mathbb{C} \smallsetminus \mathbb{R} \subset \varrho \left( S\right) \) . | Proof. Let \( S \) be selfadjoint. Then \( {iS} \) and \( - {iS} \) satisfy the hypotheses of Theorem \( {12.93}^{ \circ } \) with \( \beta = 0 \), and hence the half-spaces\n\n\[ {\mathbb{C}}_{ \pm } = \{ \lambda \in \mathbb{C} \mid \operatorname{Im}\lambda \gtrless 0\}\]\n\n(12.26)\n\nare contained in the resolvent s... | Yes |
Theorem 12.12. \( {1}^{ \circ } \) When \( S \) is symmetric \( \geq 0 \), one has the following version of the Cauchy-Schwarz inequality:\n\n\[ \n{\left| \left( Sx, y\right) \right| }^{2} \leq \left( {{Sx}, x}\right) \left( {{Sy}, y}\right) \text{ for }x, y \in D\left( S\right) .\n\] | Proof. \( {1}^{ \circ } \) . When \( t \in \mathbb{R} \), we have for \( x \) and \( y \in D\left( S\right) \), \n\n\[ \n0 \leq \left( {S\left( {x + {ty}}\right), x + {ty}}\right) = \left( {{Sx}, x}\right) + t\left( {{Sx}, y}\right) + t\left( {{Sy}, x}\right) + {t}^{2}\left( {{Sy}, y}\right) \n\] \n\n\[ \n= \left( {{Sx... | Yes |
Theorem 12.13. Let \( \Omega \) be an open subset of \( {\mathbb{R}}^{n} \), and let \( p : \Omega \rightarrow \mathbb{C} \) be a measurable function. The multiplication operator \( {M}_{p} \) in \( {L}_{2}\left( \Omega \right) \) defined by\n\n\[ D\left( {M}_{p}\right) = \left\{ {u \in {L}_{2}\left( \Omega \right) \mi... | Proof. Clearly, the operator is linear. Observe that a measurable function \( f \) on \( \Omega \) lies in \( D\left( {M}_{p}\right) \) if and only if \( \left( {1 + \left| p\right| }\right) f \in {L}_{2}\left( \Omega \right) \) . Hence\n\n\[ D\left( {M}_{p}\right) = \left\{ {\left. \frac{\varphi }{1 + \left| p\right| ... | Yes |
Lemma 12.15. Let a be a bounded everywhere defined sesquilinear form on \( V \), and let \( \mathcal{A} \) be the associated operator in \( V \). Then \( \mathcal{A} \in \mathbf{B}\left( V\right) \) with norm \( \leq C \) (cf. (12.37)), and its adjoint \( {\mathcal{A}}^{ * } \) in \( V \) is the operator in \( V \) ass... | Proof. The boundedness of \( \mathcal{A} \) was shown above. That the adjoint is the operator in \( V \) associated with \( {a}^{ * } \) follows from (12.10) and (12.35). Now (12.38) implies that \( m\left( \mathcal{A}\right) \geq {c}_{0} \) as well as \( m\left( {\mathcal{A}}^{ * }\right) \geq {c}_{0} \), where \( {c}... | Yes |
Lemma 12.16. There are continuous injections\n\n\[ \nV \hookrightarrow H \hookrightarrow {V}^{ * } \n\]\n\n(12.43)\n\nhere, when \( f \in H \) ,\n\n\[ \n\parallel f{\parallel }_{{V}^{ * }} \leq {c}^{-1}\parallel f{\parallel }_{H} \n\]\n\n(12.44) | Proof. The injections are accounted for above, and (12.44) follows from the calculation\n\n\[ \n\parallel f{\parallel }_{{V}^{ * }} = \sup \left\{ {\left. \frac{\left| {\ell }_{f}\left( v\right) \right| }{\parallel v{\parallel }_{V}}\right| \;v \in V\smallsetminus \{ 0\} }\right\} = \sup \left\{ {\left. \frac{{\left| {... | Yes |
Theorem 12.18. Consider a triple \( \left( {H, V, a}\right) \) where \( H \) and \( V \) are complex Hilbert spaces with \( V \subset H \) algebraically, topologically and densely (satisfying (12.36)), and where \( a \) is a bounded sesquilinear form on \( V \) with \( D\left( a\right) = V \) (satisfying (12.37)). Let ... | Proof. By Corollary 12.17, \( a \) gives rise to a bijection \( \widetilde{\mathcal{A}} \) from \( V \) to \( {V}^{ * } \), such that\n\n\[ a\left( {u, v}\right) = \left( {\widetilde{\mathcal{A}}u}\right) \left( v\right) \text{ for all }u, v \in V. \]\n\nBy the definition of \( A \), the elements of \( D\left( A\right)... | Yes |
Corollary 12.19. Hypotheses as in Theorem 12.18, except that V-ellipticity is replaced by \( V \) -coercivity (12.39). Then \( A \) is a closed operator with \( D\left( A\right) \) dense in \( H \) and in \( V \), and with \( m\left( A\right) \geq {c}_{0}{c}^{2} - k \) . Moreover, \n\n\[ \n\left\{ {\lambda \mid \operat... | Proof. Note that for \( \mu \in \mathbb{C}, A + {\mu I} \) (with \( D\left( {A + {\mu I}}\right) = D\left( A\right) \) ) is the operator in \( H \) associated with the sesquilinear form \( {a}_{\mu } \) defined in (12.40). When (12.39) holds, we replace \( a \) by \( {a}_{k} \) . Theorem 12.18 applies to this form and ... | Yes |
Let \( V \) be the closure of \( {C}^{1}\left( \bar{I}\right) \) in the norm \( \parallel u{\parallel }_{1} = \) \( {\left( \parallel u{\parallel }_{{L}_{2}}^{2} + {\begin{Vmatrix}{u}^{\prime }\end{Vmatrix}}_{{L}_{2}}^{2}\right) }^{\frac{1}{2}} \) (identified with \( {H}^{1}\left( I\right) \) in Section 4.3), and \( \f... | The typical information here is that \( A \) acts as a differential operator, namely, \( - \frac{{d}^{2}}{d{t}^{2}} + q \) (of order 2, while \( a\left( {u, v}\right) \) is of order 1), and the domain \( D\left( A\right) \) involves a boundary condition, namely, \( {u}^{\prime }\left( \beta \right) = {u}^{\prime }\left... | Yes |
Corollary 12.21. When \( A \) and \( {A}^{ * } \) are defined from the triple \( \left( {H, V, a}\right) \) as in Corollary 12.19, then the spectra \( \sigma \left( A\right) \) and \( \sigma \left( {A}^{ * }\right) \) and the numerical ranges \( \nu \left( A\right) \) and \( \nu \left( {A}^{ * }\right) \) are contained... | \[ M = \left\{ {\lambda \in \mathbb{C} \mid \operatorname{Re}\lambda \geq - k + {c}_{0}{c}^{2},\left| {\operatorname{Im}\lambda }\right| \leq C{c}_{0}^{-1}\left( {\operatorname{Re}\lambda + k}\right) }\right\} ,\] | Yes |
Theorem 12.24 (FRIEDRICHS). Let \( S \) be densely defined, symmetric and lower bounded in \( H \) . There exists a selfadjoint extension \( T \) with the same lower bound \( m\left( T\right) = m\left( S\right) \), and with \( D\left( T\right) \) contained in the completion of \( D\left( S\right) \) in the norm \( {\le... | Proof. Assume first that \( m\left( S\right) = c > 0 \) . The sesquilinear form\n\n\[ \n{s}_{0}\left( {u, v}\right) = \left( {{Su}, v}\right) \n\]\n\nis then a scalar product on \( D\left( S\right) \) (cf. Theorem 12.12), and we denote the completion of \( D\left( S\right) \) with respect to this scalar product by \( V... | Yes |
Lemma 12.27. When \( b \) and \( {b}^{\prime } \) are symmetric sesquilinear forms with the same domain \( D\left( b\right) \), satisfying\n\n\[ \left| {b\left( {u, u}\right) }\right| \leq {b}^{\prime }\left( {u, u}\right) ,\text{ all }u \in D\left( b\right) ,\]\n\n(12.62)\n\nthen\n\n\[ \left| {b\left( {u, v}\right) }\... | Proof. Note that \( {b}^{\prime } \) is a nonnegative sesquilinear form. Let \( u, v \in D\left( b\right) \) . If \( {b}^{\prime }\left( {u, u}\right) \) or \( {b}^{\prime }\left( {v, v}\right) \) is 0, so is \( b\left( {u, u}\right) \) resp. \( b\left( {v, v}\right) \) according to (12.62), so (12.63) is valid then. W... | Yes |
Lemma 13.1. There are decompositions into direct sums:\n\n\[ \nD\left( {A}_{1}\right) = D\left( {A}_{\beta }\right) \dot{ + }Z\left( {A}_{1}\right) ,\;D\left( {A}_{1}^{\prime }\right) = D\left( {A}_{\beta }^{ * }\right) \dot{ + }Z\left( {A}_{1}^{\prime }\right) ; \n\] | Proof. Clearly, \( D\left( {A}_{\beta }\right) + Z\left( {A}_{1}\right) \subset D\left( {A}_{1}\right) \) . Define \( {\operatorname{pr}}_{\beta } \) by (13.9); it sends \( D\left( {A}_{1}\right) \) into \( D\left( {A}_{\beta }\right) \) and is continuous with respect to the graph norm:\n\n\[ \n{\begin{Vmatrix}{\operat... | Yes |
Lemma 13.2. For \( u \in D\left( {A}_{1}\right), v \in D\left( {A}_{1}^{\prime }\right) \), one has\n\n\[ \left( {{Au}, v}\right) - \left( {u,{A}^{\prime }v}\right) = \left( {{Au},{v}_{{\zeta }^{\prime }}}\right) - \left( {{u}_{\zeta },{A}^{\prime }v}\right) \]\n\n(13.12)\n\n\[ = \left( {{\left( Au\right) }_{{Z}^{\prim... | Proof. Since \( {Au} = A{u}_{\beta },{A}^{\prime }v = {A}^{\prime }{v}_{{\beta }^{\prime }} \), we can write\n\n\[ \left( {{Au}, v}\right) - \left( {u,{A}^{\prime }v}\right) = \left( {A{u}_{\beta },{v}_{{\beta }^{\prime }} + {v}_{{\zeta }^{\prime }}}\right) - \left( {{u}_{\beta } + {u}_{\zeta },{A}^{\prime }{v}_{{\beta... | Yes |
Proposition 13.4. Let \( V \) and \( W \) be closed subspaces of \( Z \), resp. \( {Z}^{\prime } \), and let \( T : V \rightarrow W,{T}^{ * } : W \rightarrow V \) be a pair of adjoint operators (generally unbounded). Define the operators \( \widetilde{A} \subset {A}_{1},{\widetilde{A}}^{\prime } \subset {A}_{1}^{\prime... | Proof. Let \( \widetilde{A} \) and \( {\widetilde{A}}^{\prime } \) be given by (13.22),(13.23); they clearly extend \( {A}_{0} \) resp. \( {A}_{0}^{\prime } \) . It follows by use of Lemma 13.2 that for \( u \in D\left( \widetilde{A}\right), v \in D\left( {\widetilde{A}}^{\prime }\right) \),\n\n\[ \left( {{Au}, v}\righ... | Yes |
Theorem 13.6. When \( \widetilde{A} \) corresponds to \( T : V \rightarrow W \) as above, the mapping\n\n\[ \Psi : \{ z, f, v\} \mapsto u = z + {A}_{\beta }^{-1}\left( {{Tz} + f}\right) + v \]\n\n(13.29)\n\ndefines a bijection\n\n\[ \Psi : D\left( T\right) \times \left( {{Z}^{\prime } \ominus W}\right) \times D\left( {... | Proof. Let \( u = z + {A}_{\beta }^{-1}\left( {{Tz} + f}\right) + v \), with \( \{ z, f, v\} \in D\left( T\right) \times \left( {{Z}^{\prime } \ominus W}\right) \times D\left( {A}_{0}\right) \) . Clearly, \( u \) belongs to \( D\left( {A}_{1}\right) \), with \( {u}_{\zeta } = z,{u}_{\beta } = {A}_{\beta }^{-1}\left( {{... | Yes |
Theorem 13.8. Let \( \widetilde{A} \in \mathcal{M} \) correspond to \( T : V \rightarrow W \) as in Theorem 13.7. Then:\n\n\( {1}^{ \circ }Z\left( \widetilde{A}\right) = Z\left( T\right) \) . In particular, \( \dim Z\left( \widetilde{A}\right) = \dim Z\left( T\right) \), and \( \widetilde{A} \) is injective if and only... | Proof. \( {1}^{ \circ } \) . Let \( u \in Z\left( \widetilde{A}\right) \) . Then \( {u}_{\zeta } = u \), and \( T{u}_{\zeta } = {\left( Au\right) }_{W} = 0 \), so \( u \in Z\left( T\right) \).\n\nConversely, if \( u \in Z\left( T\right) \), then \( u \in D\left( \widetilde{A}\right) \) (take \( f \) and \( v \) equal t... | Yes |
Theorem 13.9. Let \( \widetilde{A} \in \mathcal{M} \) correspond to \( T : V \rightarrow W \) as in Theorem 13.7. Assume that \( \widetilde{A} \) is injective, then so is \( T \) . Define \( {T}^{\left( -1\right) } \) as the linear extension of\n\n\[ \n{T}^{\left( -1\right) }f = \left\{ \begin{array}{ll} {T}^{-1}f & \t... | Proof. Let \( f \in R\left( \widetilde{A}\right) \) . Let \( u = {\widetilde{A}}^{-1}f, v = {A}_{\beta }^{-1}f \) . Then \( u - v = z \), where \( z \in Z \) . By the definition of \( T, z \) belongs to \( D\left( T\right) \) and \( {Tz} = {\left( Au\right) }_{W} = {f}_{W} \) . Therefore \( {f}_{W} \in R\left( T\right)... | Yes |
A simple example of the choice of \( T \) is to take \( V = Z \) , \( W = {Z}^{\prime }, T = 0 \) on \( D\left( T\right) = Z \) . This corresponds to an operator in \( \mathcal{M} \) that we shall denote \( {A}_{M} \) ( \( M \) here indicates that it is maximal in a certain sense). | Since \( {Z}^{\prime } \ominus W = \{ 0\} \), we see from Theorem 13.6 that \( D\left( {A}_{M}\right) = D\left( {A}_{0}\right) + Z \) . The adjoint \( {A}_{M}^{ * } \) corresponds to \( {T}^{ * } \) equal to the zero operator from \( {Z}^{\prime } \) to \( Z \) , so it is completely analogous to \( {A}_{M} \), with \( ... | Yes |
Corollary 13.12. Assume that (13.5) holds. Let \( \widetilde{A} \) correspond to \( T : V \rightarrow W \) as in Theorem 13.7. Then \( \widetilde{A} \) is selfadjoint if and only if: \( V = W \) and \( T \) is selfadjoint. | In more detail: If \( V \) is a closed subspace of \( Z \) and \( T : V \rightarrow V \) is selfadjoint, then the operator \( \widetilde{A} \subset {A}_{1} \) with domain \[ D\left( \widetilde{A}\right) = \left\{ {u \in D\left( {A}_{1}\right) \mid {u}_{\zeta } \in D\left( T\right) ,{\left( Au\right) }_{V} = T{u}_{\zeta... | Yes |
Lemma 13.13. When (13.5) holds and \( V \subset W \), then\n\n\[ \left( {{Au}, v}\right) = \left( {A{u}_{\beta },{v}_{\beta }}\right) + \left( {T{u}_{\zeta },{v}_{\zeta }}\right) \]\n\n(13.42)\n\nfor all \( u, v \in D\left( \widetilde{A}\right) \) . | Proof. This follows from the calculation\n\n\[ \left( {{Au}, v}\right) = \left( {{Au},{v}_{\beta }}\right) + \left( {{Au},{v}_{\zeta }}\right) = \left( {A{u}_{\beta },{v}_{\beta }}\right) + \left( {{Au},{v}_{\zeta }}\right) \]\n\n\[ = \left( {A{u}_{\beta },{v}_{\beta }}\right) + \left( {{\left( Au\right) }_{W},{v}_{\ze... | Yes |
Theorem 13.15. Assume that (13.5) and (13.43) hold, and let \( \widetilde{A} \) correspond to \( T : V \rightarrow W \) as in Theorem 13.7.\n\nIf \( \nu \left( \widetilde{A}\right) \) is not all of \( \mathbb{C} \), then \( V \subset W \), and\n\n\[ \overline{\nu \left( T\right) } \subset \overline{\nu \left( \widetild... | Proof. Let \( \nu \left( \widetilde{A}\right) \neq \mathbb{C} \) ; then since \( \nu \left( \widetilde{A}\right) \) is convex (Exercise 12.34(e)), it is contained in a half-plane, and so is \( \overline{\nu \left( \widetilde{A}\right) } \) . By definition, a number \( \lambda \in \mathbb{C} \) has a positive distance t... | No |
Corollary 13.16. Assumptions as in Theorem 13.15.\n\n\( {1}^{ \circ } \) If \( \widetilde{A} \) and \( {\widetilde{A}}^{ * } \) both have the lower bound \( a \in \mathbb{R} \), then \( V = W \), and \( T \) and \( {T}^{ * } \) have the lower bound a, their spectra being contained in \( \{ \lambda \in \mathbb{C} \mid \... | Proof. \( {1}^{ \circ } \) . The preceding theorem applied to \( \widetilde{A} \) gives that \( V \subset W \) and \( m\left( T\right) \geq \) \( a \), and when it is applied to \( {\widetilde{A}}^{ * } \) it gives that \( W \subset V \) and \( m\left( {T}^{ * }\right) \geq a \) . The statement on the spectra follows f... | Yes |
Corollary 13.21. Assume (13.5) and (13.43). A variational operator \( \widetilde{A} \) belongs to \( \mathcal{M} \) if and only if the associated sesquilinear form \( \widetilde{a} \) satisfies (i)-(iii): (i) \( \;D\left( {a}_{\gamma }\right) \subset D\left( \widetilde{a}\right) \subset D\left( {a}_{M}\right) \), with ... | Proof. The necessity of (i) and (ii) is immediate from (13.57) and (13.58), and we get (iii) by applying (13.58) with \( u = w, v = z \), or \( u = z, v = w \) . For the converse direction, let \( \widetilde{A} \) be the variational operator associated with a form \( \widetilde{a} \) satisfying (i)-(iii). Then \( \wide... | Yes |
Corollary 13.22. Assume (13.5) and (13.43). A selfadjoint nonnegative operator \( \widetilde{A} \) belongs to \( \mathcal{M} \) if and only if the associated sesquilinear form \( \widetilde{a} \) satisfies (i)-(iii):\n\n(i) \( \;D\left( {a}_{\gamma }\right) \subset D\left( \widetilde{a}\right) \subset D\left( {a}_{M}\r... | Proof. Here we note that if \( \widetilde{A} \in \mathcal{M} \), then the corresponding form \( t \) is \( \geq 0 \), so (iii) follows from (13.58).\n\nIn the converse direction, we reduce to an application of Corollary 13.21 by observing that (ii) and (iii) imply, for \( u \in D\left( \widetilde{a}\right), u = w + z \... | Yes |
Theorem 13.23. Let \( \lambda \in \varrho \left( {A}_{\beta }\right) \) . There is a 1-1 correspondence between the closed operators \( \widetilde{A} - \lambda \) in \( {\mathcal{M}}_{\lambda } \) and the closed, densely defined operators \( {T}^{\lambda } : {V}_{\lambda } \rightarrow {W}_{\bar{\lambda }} \), where \( ... | Proof. The first part is covered by Theorem 13.7, and the formulas in (13.71) and (13.72) follow from Theorems 3.8 and 3.9; see also (13.36). | No |
Lemma 13.24. \( {E}^{\lambda } \) and \( {F}^{\lambda } \) are inverses of one another, and so are \( {E}^{\prime \bar{\lambda }} \) and \( {F}^{\prime \bar{\lambda }} \) . In particular, the operators restrict to homeomorphisms\n\n\[ \n{E}_{Z}^{\lambda } : Z\overset{ \sim }{ \rightarrow }{Z}_{\lambda },\;{F}_{Z}^{\lam... | Proof. To show that \( {E}^{\lambda } \) and \( {F}^{\lambda } \) are inverses of one another, note that for \n\n\[ \nv \in {A}_{\beta }^{-1}H = D\left( {A}_{\beta }\right) = D\left( {{A}_{\beta } - \lambda }\right) = {\left( {A}_{\beta } - \lambda \right) }^{-1}H, \n\] \n\none has that \( v = {A}_{\beta }^{-1}{A}_{1}v... | Yes |
Theorem 13.25. Let \( T : V \rightarrow W \) correspond to \( \widetilde{A} \) by Theorem 13.7, let \( \lambda \in \varrho \left( {A}_{\beta }\right) \) and let \( \widetilde{A} - \lambda \) correspond to \( {T}^{\lambda } : {V}_{\lambda } \rightarrow {W}_{\bar{\lambda }} \) by Theorem 13.23.\n\nFor \( \lambda \in \var... | Proof. The first line in (13.79) follows from (13.70) in view of (13.75). The second line is calculated as follows: For \( u \in D\left( \widetilde{A}\right), w \in W \) ,\n\n\[ \n\left( {T{u}_{\zeta }, w}\right) = \left( {{Au}, w}\right) = \left( {{Au},{F}^{\prime \bar{\lambda }}{E}^{\prime \bar{\lambda }}w}\right) = ... | Yes |
Corollary 13.26. When \( \lambda \in \varrho \left( \widetilde{A}\right) \cap \varrho \left( {A}_{\beta }\right) ,{T}^{\lambda } \) is bijective, and\n\n\[{\left( {T}^{\lambda }\right) }^{-1} = {E}_{V}^{\lambda }{\left( T + {G}_{V, W}^{\lambda }\right) }^{-1}{\left( {E}_{W}^{\prime \bar{\lambda }}\right) }^{ * }.\]\n\n... | Proof. (13.83) follows from (13.82) by inversion, and insertion in (13.72) shows (13.84). | No |
Lemma 13.28. \( R\left( {{\Gamma }_{1} - T{\Gamma }_{0}}\right) = R\left( {{\operatorname{pr}}_{{Z}^{\prime }}{A}_{1} - T{\operatorname{pr}}_{\zeta }}\right) \) is equal to \( {Z}^{\prime } \) . | In fact, any \( f \in {Z}^{\prime } \) can be written as\n\n\[ f = \left( {{\operatorname{pr}}_{{Z}^{\prime }}{A}_{1} - T{\operatorname{pr}}_{\zeta }}\right) v = {\operatorname{pr}}_{{Z}^{\prime }}{A}_{1}v,\text{ for }v = {A}_{\beta }^{-1}f. \]\n\n(13.93)\n\nProof. Let \( v \) run through \( D\left( {A}_{\beta }\right)... | Yes |
Proposition 13.29. For any \( \lambda \in \varrho \left( \widetilde{A}\right) ,{M}_{\widetilde{A}}\left( \lambda \right) \) is well-defined as a bounded map from \( {Z}^{\prime } \) to \( Z \) by (13.91) or (13.92), and ranges in \( D\left( T\right) \) . In fact,\n\n\[ \n{M}_{\widetilde{A}}\left( \lambda \right) = {\op... | Proof. The mapping \( \Phi = {\Gamma }_{1} - T{\Gamma }_{0} = {\operatorname{pr}}_{{Z}^{\prime }}{A}_{1} - T{\operatorname{pr}}_{\zeta } \) is defined for those \( u \in D\left( {A}_{1}\right) \) for which \( {\operatorname{pr}}_{\zeta }u \in D\left( T\right) \) . Let\n\n\[ \n{Z}_{\lambda, T} = \left\{ {{z}^{\lambda } ... | Yes |
Theorem 13.30. Let \( \widetilde{A} \) be defined by (13.90), where \( T : Z \rightarrow {Z}^{\prime } \) . When the boundary triplet is chosen as in (13.88) and \( \lambda \in \varrho \left( \widetilde{A}\right) \cap \varrho \left( {A}_{\beta }\right) , - {M}_{\widetilde{A}}\left( \lambda \right) \) equals the inverse... | \[ - {M}_{\widetilde{A}}{\left( \lambda \right) }^{-1} = T + {G}^{\lambda } = {\left( {E}_{{Z}^{\prime }}^{\prime \bar{\lambda }}\right) }^{ * }{T}^{\lambda }{E}_{Z}^{\lambda }. \] (13.99) In particular, \( {M}_{\widetilde{A}}\left( \lambda \right) \) has range \( D\left( T\right) \) . | Yes |
Corollary 13.31. For \( \lambda \in \varrho \left( \widetilde{A}\right) \cap \varrho \left( {A}_{\beta }\right) \) , | \[ {\left( \widetilde{A} - \lambda \right) }^{-1} = {\left( {A}_{\beta} - \lambda \right) }^{-1} - {\mathrm{i}}_{{Z}_{\lambda} \rightarrow H}{E}_{Z}^{\lambda }{M}_{\widetilde{A}}\left( \lambda \right) {\left( {E}_{{Z}^{\prime }}^{\prime \bar{\lambda }}\right) }^{ * }{\operatorname{pr}}_{{Z}_{\bar{\lambda }}^{\prime }}.... | Yes |
Corollary 13.32. For any \( \lambda \in \varrho \left( {A}_{\beta }\right) \) , | \[ Z\left( {\widetilde{A} - \lambda }\right) = {E}_{Z}^{\lambda }Z\left( {T + {G}^{\lambda }}\right) \] (13.101) \[ R\left( {\widetilde{A} - \lambda }\right) = {\left( {F}_{{Z}^{\prime }}^{\prime \bar{\lambda }}\right) }^{ * }R\left( {T + {G}^{\lambda }}\right) + R\left( {{A}_{0} - \lambda }\right) . \] | Yes |
Theorem 13.33. Let \( \widetilde{A} \) be an arbitrary closed densely defined operator between \( {A}_{0} \) and \( {A}_{1} \), and let \( T : V \rightarrow W \) be the corresponding operator according to Theorem 13.7. For any \( \lambda \in \varrho \left( \widetilde{A}\right) \) there is a bounded operator \( {M}_{\wi... | Proof. Following the lines of proofs of Lemma 13.28 and Proposition 13.29, we define \( {M}_{\widetilde{A}}\left( \lambda \right) \) satisfying (13.102) as follows: Let \( f \in W \) . Let \( v = {A}_{\beta }^{-1}f \) ; then \( {\operatorname{pr}}_{\zeta }v = 0 \in D\left( T\right) \), and\n\n\[ \n{\left( Av\right) }_{... | Yes |
Lemma 14.1. When \( G\left( t\right) \) satisfies (a)-(c), the map \( t \mapsto G\left( t\right) x \) is for any \( x \in X \) a continuous function from \( {\overline{\mathbb{R}}}_{ + } \) to \( X \) (resp. from \( \mathbb{R} \) to \( X \) in case of a group). | Proof. By (a) and (c) we have for \( {t}_{1} \leq {t}_{2} \) that\n\n\[ \begin{Vmatrix}{G\left( {t}_{2}\right) x - G\left( {t}_{1}\right) x}\end{Vmatrix} = \begin{Vmatrix}{G\left( {t}_{1}\right) \left( {G\left( {{t}_{2} - {t}_{1}}\right) x - x}\right) }\end{Vmatrix} \leq \begin{Vmatrix}{G\left( {{t}_{2} - {t}_{1}}\righ... | Yes |
For \( x \in D\left( B\right) \), the function \( G\left( t\right) x : {\overline{\mathbb{R}}}_{ + } \rightarrow X \) is differentiable, and takes its values in \( D\left( B\right) \) : | \[ \mathop{\lim }\limits_{{h \rightarrow 0}}\frac{1}{h}\left( {G\left( {t + h}\right) x - G\left( t\right) x}\right) = G\left( t\right) {Bx} = {BG}\left( t\right) x\text{ for all }t \geq 0. \] (14.17) Proof. When \( h > 0 \) , \[ \frac{1}{h}\left( {G\left( {t + h}\right) x - G\left( t\right) x}\right) = G\left( t\right... | Yes |
Corollary 14.3. For \( x \in D\left( B\right) \) ,\n\n\[ G\left( t\right) x - x = {\int }_{0}^{t}G\left( s\right) {Bxds}. \]\n\n(14.18) | This follows from the general property (14.11). | No |
For all \( x \in X, t > 0,{\int }_{0}^{t}G\left( s\right) {xds} \) belongs to \( D\left( B\right) \) and\n\n\[ G\left( t\right) x - x = B{\int }_{0}^{t}G\left( s\right) {xds}. \] | Proof. It follows from the continuity and the semigroup property (a) that for \( h > 0 \) :\n\n\[ \frac{G\left( h\right) - I}{h}{\int }_{0}^{t}G\left( s\right) {xdx} = \frac{1}{h}{\int }_{0}^{t}\left( {G\left( {s + h}\right) - G\left( s\right) }\right) {xds} \]\n\n\[ = \frac{1}{h}{\int }_{h}^{t + h}G\left( s\right) {xd... | Yes |
Lemma 14.5. B is closed and densely defined. | Proof. According to Lemma 14.4, \( \frac{1}{h}{\int }_{0}^{h}G\left( s\right) {xds} \in D\left( B\right) \) for all \( x \in X, h > 0 \) , so since this converges to \( x \) for \( h \rightarrow 0, D\left( B\right) \) is dense in \( X \) . Now if \( {x}_{n} \in D\left( B\right) \) with \( {x}_{n} \rightarrow x \) and \... | Yes |
Lemma 14.6. A contraction semigroup is uniquely determined from its infinitesimal generator. | Proof. Assume that \( {G}_{1}\left( t\right) \) and \( {G}_{2}\left( t\right) \) have the same infinitesimal generator \( B \) . For \( x \in D\left( B\right) ,{G}_{2}\left( t\right) x \in D\left( B\right) \), and we have by a generalization of the Leibniz formula:\n\n\[ \n\frac{d}{ds}{G}_{1}\left( {t - s}\right) {G}_{... | Yes |
Lemma 14.9. When \( G\\left( t\\right) \) is a semigroup in \( H \) satisfying (a),(b) and (c), then its infinitesimal generator \( B \) is upper semibounded, with upper bound \( \\leq 0 \) . | Proof. For \( x \\in X \) one has that\n\n\\[ \n\\operatorname{Re}\\frac{1}{h}\\left( {G\\left( h\\right) x - x, x}\\right) = \\frac{1}{h}\\left( {\\operatorname{Re}\\left( {G\\left( h\\right) x, x}\\right) - \\parallel x{\\parallel }^{2}}\\right)\n\\]\n\n\\[\n\\leq \\frac{1}{h}\\left( {\\begin{Vmatrix}{G\\left( h\\rig... | Yes |
Theorem 14.10. When \( G\left( t\right) \) is a semigroup in \( H \) satisfying (a),(b) and (c), then \( {G}^{ * }\left( t\right) \) is likewise a semigroup in \( H \) satisfying (a),(b) and (c); and when the generator for \( G\left( t\right) \) is \( B \), then the generator for \( {G}^{ * }\left( t\right) \) is preci... | Proof. It is seen immediately that \( {G}^{ * }\left( t\right) \) satisfies (a) and (c). For (b) we observe that one for \( x \) and \( y \in H \) has:\n\n\[ \left( {{G}^{ * }\left( t\right) x, y}\right) = \left( {x, G\left( t\right) y}\right) \rightarrow \left( {x, y}\right) \text{ for }t \rightarrow 0. \]\n\nThis imp... | Yes |
An operator \( B \) in a Hilbert space \( H \) is the infinitesimal generator of a strongly continuous contraction semigroup if and only if \( B \) is densely defined and closed, has \( u\left( B\right) \leq 0 \), and has \( {\mathbb{R}}_{ + } \) contained in its resolvent set. | The necessity of the conditions follows from what we have just shown, together with Lemma 14.5 and Theorem 14.7. The sufficiency is seen from the fact that (14.33) implies that \( - \left( {B - {\lambda I}}\right) = - B + {\lambda I} \) for \( \lambda > 0 \) has a bounded inverse with norm \( \leq {\lambda }^{-1} \), b... | Yes |
Corollary 14.12. Let \( B \) be a closed, densely defined operator in a Hilbert space \( H \) . Then the following properties are equivalent:\n\n(i) \( B \) is the infinitesimal generator of a strongly continuous contraction semigroup.\n\n(ii) \( B \) is dissipative (i.e., \( u\left( B\right) \leq 0 \) ) and \( {\mathb... | Proof. The equivalence of (i) and (ii) is shown above. Condition (i) implies (iii) by Theorem 14.10, and (iii) implies (ii) by Theorem \( {12.93}^{ \circ } \) . | No |
Let \( X = {\mathbb{C}}^{n} \equiv \left\{ {\left( {{z}_{1},{z}_{2},\ldots ,{z}_{n}}\right) : {z}_{j} \in \mathbb{C}}\right\} \) with\n\n\[ \begin{Vmatrix}\left( {{z}_{1},{z}_{2},\ldots ,{z}_{n}}\right) \end{Vmatrix} = {\left( \mathop{\sum }\limits_{{j = 1}}^{n}{\left| {z}_{j}\right| }^{2}\right) }^{\frac{1}{2}}; \] | this is called the Euclidean norm. The Euclidean space \( {\mathbb{R}}^{n} \) is similarly defined; in this case we restrict to real scalars. | No |
Let \( Y = \left\lbrack {0,1}\right\rbrack \), or more generally any compact Hausdorff space, and let \( C\left( Y\right) \) be the vector space of continuous, complex-valued functions on \( Y \), under pointwise addition and scalar multiplication. Define a norm on \( C\left( Y\right) \) by \( \parallel f\parallel = \)... | This (specifically \( C\left\lbrack {a, b}\right\rbrack \), endowed with the metric which defines the distance between functions \( f \) and \( g \) to be \( \mathop{\max }\limits_{{a \leq x \leq b}}\left| {f\left( x\right) - g\left( x\right) }\right| \) ), was one of the important examples that Fréchet put forth in hi... | No |
Example 1.6. We can generalize the last example as follows. Consider a positive measure space \( \left( {Y,\mathfrak{M},\mu }\right) \), where \( Y \) is a set, \( \mathfrak{M} \) is a \( \sigma \) -algebra of subsets of \( Y \), and \( \mu \) is a positive measure. Choose \( 1 \leq p < \infty \), and denote by \( {L}^... | Minkowski's inequality (for integrals) provides the proof that the norm satisfies the triangle inequality. | Yes |
Proposition 1.15. If \( \langle \cdot , \cdot \rangle \) is an inner product on a vector space \( X \), then\n\n\[ \parallel x\parallel \equiv \langle x, x{\rangle }^{\frac{1}{2}} \]\n\n\nis a norm on \( X \) . | Proof. We will check the triangle inequality, and leave the verification of the other norm properties to the reader. Using the linearity of the inner product we have\n\n\[ \parallel x + y{\parallel }^{2} = \langle x + y, x + y\rangle = \langle x, x\rangle + \langle y, x\rangle + \langle x, y\rangle + \langle y, y\rangl... | No |
Proposition 1.18. If \( f \) is a analytic function in some closed disk \( \overline{B\left( {a, R}\right) } \), then\n\n\[ f\left( a\right) = \frac{1}{\pi {R}^{2}}{\int }_{B\left( {a, R}\right) }{fdA}. \] | Proof. As a consequence of Cauchy's integral formula we have the mean value property\n\n\[ f\left( a\right) = \frac{1}{2\pi }{\int }_{0}^{2\pi }f\left( {a + r{e}^{i\theta }}\right) {d\theta }\n\]\n\nfor all \( 0 < r < R \) . Multiplying by \( r \) and integrating with respect to \( r \) we have\n\n\[ {\int }_{0}^{R}{rf... | Yes |
Corollary 1.19. Fix \( w \in \mathbb{D} \) . For every \( f \in {L}_{a}^{2}\left( \mathbb{D}\right) \) we have\n\n\[ \left| {f\left( w\right) }\right| \leq \frac{1}{1 - \left| w\right| }\parallel f{\parallel }_{{L}_{a}^{2}\left( \mathbb{D}\right) } \] | Proof. Let \( 0 < r < 1 - \left| w\right| \) so that the closed disk \( \overline{B\left( {w, r}\right) } \) is contained in \( \mathbb{D} \) . Using Proposition 1.18 and Hölder's inequality we have\n\n\[ \left| {f\left( w\right) }\right| = \left| {\frac{1}{\pi {r}^{2}}{\int }_{B\left( {w, r}\right) }{fdA}}\right| \]\n... | Yes |
Theorem 1.20. The Bergman space \( {L}_{a}^{2}\left( \mathbb{D}\right) \) is a Hilbert space. | Proof. As we have discussed, we need only show that \( {L}_{a}^{2}\left( \mathbb{D}\right) \) is a closed subspace of \( {L}^{2}\left( {\mathbb{D},{dA}/\pi }\right) \) . That \( {L}_{a}^{2}\left( \mathbb{D}\right) \) is a subspace is immediate. To see that it is closed, suppose we have a sequence \( \left\{ {f}_{n}\rig... | Yes |
Proposition 1.23 (Nearest Point Property). Every nonempty, closed convex set \( K \) in a Hilbert space \( \mathcal{H} \) contains a unique element of smallest norm. Moreover, given any \( h \in \mathcal{H} \), there is a unique \( {k}_{0} \) in \( K \) such that\n\n\[ \begin{Vmatrix}{h - {k}_{0}}\end{Vmatrix} = \opera... | Proof. We begin with a proof of the first statement. The parallelogram equality says that for any vectors \( x, y \) in \( \mathcal{H} \), \n\n\[ {\begin{Vmatrix}\frac{x - y}{2}\end{Vmatrix}}^{2} = \frac{1}{2}\left( {\parallel x{\parallel }^{2} + \parallel y{\parallel }^{2}}\right) - {\begin{Vmatrix}\frac{x + y}{2}\end... | Yes |
Theorem 1.24 (Projection Theorem). Let \( M \) be a closed subspace of a Hilbert space \( \mathcal{H} \). There is a unique pair of mappings \( P : \mathcal{H} \rightarrow M \) and \( Q : \mathcal{H} \rightarrow {M}^{ \bot } \) such that \( x = {Px} + {Qx} \) for all \( x \in \mathcal{H} \). Furthermore, \( P \) and \(... | Proof. First we define \( P \) as follows: For every \( x \in \mathcal{H} \), let \( {Px} \) be the unique closest point to \( x \) in the (closed convex) set \( M \) ; here we are using the nearest point property of Proposition 1.23. Uniqueness says that \( P \) is well-defined. Moreover, \( P : \mathcal{H} \rightarro... | Yes |
Proposition 1.28. If \( X \) is a normed linear space, and \( \Lambda : X \rightarrow \mathbb{C} \) is a linear functional, then the following are equivalent:\n\n(a) \( \Lambda \) is continuous.\n\n(b) \( \Lambda \) is continuous at 0 .\n\n(c) \( \Lambda \) is bounded. | Proof. The implication (a) \( \Rightarrow \) (b) is trivial, so we look first at (b) \( \Rightarrow \) (c). Since \( \Lambda \left( 0\right) = \) 0 (why?), continuity of \( \Lambda \) at 0 means that given \( \varepsilon > 0 \) we may find \( \delta > 0 \) such that if \( \parallel x\parallel \leq \delta \), then \( \l... | No |
Theorem 1.29. Every bounded linear functional \( \Lambda \) on a Hilbert space \( \mathcal{H} \) is given by inner product with a (unique) fixed vector \( {h}_{0} \) in \( \mathcal{H} : \Lambda \left( h\right) = \left\langle {h,{h}_{0}}\right\rangle \) . Moreover, the norm of the linear functional \( \Lambda \) is \( \... | Proof. Suppose \( \Lambda \) is a bounded linear functional on \( \mathcal{H} \) . If \( \Lambda \) is identically 0, choose \( {h}_{0} = 0 \) . Otherwise, set\n\n\[ M = \ker \Lambda \equiv \{ h \in \mathcal{H} : \Lambda \left( h\right) = 0\} . \]\n\nSince \( \Lambda \) is linear, \( M \) is a subspace of \( \mathcal{H... | Yes |
Lemma 1.30. Let \( P : \mathcal{H} \rightarrow M \) be the orthogonal projection of a Hilbert space \( \mathcal{H} \) onto a closed subspace \( M \) of \( \mathcal{H} \). We have \( \langle f,{Pg}\rangle = \langle {Pf}, g\rangle \) for all vectors \( f \) and \( g \) in \( \mathcal{H} \). | Proof. Let \( f \) and \( g \) be in \( \mathcal{H} \) and write, using the projection theorem, \( f = {m}_{1} + {n}_{1} \), \( g = {m}_{2} + {n}_{2} \), where \( {m}_{1},{m}_{2} \in M \) and \( {n}_{1},{n}_{2} \in {M}^{ \bot } \). We have\n\n\[ \langle f,{Pg}\rangle = \left\langle {{m}_{1} + {n}_{1},{m}_{2}}\right\ran... | Yes |
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