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Theorem 4. Let \( \varphi \in {C}_{c}^{\infty }\left( {N \smallsetminus G}\right) \) and assume \( \widehat{\varphi }\left\lbrack 0\right\rbrack = 0 \) . Then\n\n\[ Z\left( {{T}^{0}{T\varphi }, y,2 - {2s}}\right) \]\n\n is meromorphic, and holomorphic for \( \operatorname{Re}s \geq \frac{1}{2} \) . This function and\n\... | Proof. The Riemann zeta function \( \zeta \left( {2s}\right) \) has no zero for \( \operatorname{Re}{2s} \geq 1 \), or in other words, for \( \operatorname{Re}s \geq \frac{1}{2} \) . The denominator \( \zeta \left( {2s}\right) \) in Theorem 3 therefore introduces no singularity in our stated region. The rapid decrease ... | Yes |
Theorem 1. In the above region, \( \varphi \) is analytic in \( s,{C}^{\infty } \) in \( u \), and:\n\ni) \( {l\varphi } = s\left( {1 - s}\right) \varphi \)\n\nii) \( \varphi \left( {u, s}\right) = - \frac{1}{4\pi }\log u + O\left( 1\right) \; \) for fixed \( s, u \rightarrow 0 \) ;\n\niii) \( {\varphi }_{s}^{\prime }\... | Proof. Property (iv) is clear. To prove that \( \varphi \) satisfies the desired differential equation, we may differentiate under the integral sign, and it turns out that applying \( {l}_{u} \) to this integrand turns it into an exact differential, with zero boundary value. More precisely, a trivial direct computation... | Yes |
Theorem 2. Let \( f \in B{C}^{\infty }\left( \mathfrak{H}\right) \) . For \( \operatorname{Re}\left( s\right) > 1 \), let\n\n\[ \n{R}_{\mathfrak{G}}\left( s\right) f\left( z\right) = {\int }_{\mathfrak{H}}{\varphi }_{s}\left( {u\left( {z,{z}^{\prime }}\right) }\right) f\left( {z}^{\prime }\right) d{z}^{\prime }\n\]\n\n... | The proof will be carried out in three steps.\n\nFirst, we show that \( {R}_{\mathfrak{H}}\left( s\right) f \) is bounded.\n\nSecond, we prove the special case of the theorem when \( f \) has compact support, by potential theory.\n\nThird, we prove the general case by applying the fundamental theorem on elliptic operat... | No |
Lemma 1. If \( f \in B{C}^{\infty }\left( \mathfrak{H}\right) \), then \( {R}_{\mathfrak{H}}\left( s\right) f \) is bounded, if \( \sigma > 1 \) . | Proof. Let\n\n\[ h\left( z\right) = {\int }_{\mathfrak{H}}{\varphi }_{s}\left( {u\left( {z,{z}^{\prime }}\right) }\right) f\left( {z}^{\prime }\right) d{z}^{\prime }.\]\n\nThen\n\n\[ \left| {h\left( z\right) }\right| \leq \parallel f\parallel {\int }_{\mathfrak{H}}\left| {{\varphi }_{s}\left( {u\left( {z,{z}^{\prime }}... | Yes |
Lemma 2. The integral\n\n\[ \n{\int }_{\mathfrak{H}}\frac{1}{{\left( 1 + u\left( z,{z}^{\prime }\right) \right) }^{\sigma }}d{z}^{\prime } \]\n\nconverges for \( \sigma > 1 \) . | Proof. We may assume \( z = i \) . The integral is then equal to\n\n\[ \n{\int }_{-\infty }^{\infty }{\int }_{0}^{\infty }\frac{{y}^{\sigma - 2}}{{\left\lbrack {x}^{2} + {\left( y - 1\right) }^{2}\right\rbrack }^{\sigma }}{dydx}. \]\n\nWe write\n\n\[ \n{\int }_{0}^{\infty } = {\int }_{0}^{\delta } + {\int }_{\delta }^{... | No |
Let \( \sigma > 1 \) and \( f \in {C}_{c}^{\infty }\left( \mathfrak{H}\right) \). Let\n\n\[ h\left( z\right) = {\int }_{\mathfrak{H}}{\varphi }_{s}\left( {u\left( {z,{z}^{\prime }}\right) }\right) f\left( {z}^{\prime }\right) d{z}^{\prime }.\]\n\nthen \( h \) is \( {C}^{\infty } \). Let \( {M}_{s} = L - s\left( {1 - s}... | Proof. We omit the subscript \( s \) which is fixed. For \( z \) varying in a small open set, let\n\n\[ z \mapsto \gamma \left( z\right) \in G \]\n\nbe a \( {C}^{\infty } \) map such that \( \gamma \left( z\right) i = z \) (so \( z \mapsto \gamma \left( z\right) \) is a \( {C}^{\infty } \) section). Abbreviate \( \varp... | Yes |
Lemma 1. If \( f,{Lf} \) are in \( B{C}^{\infty }\left( {\Gamma \smallsetminus \mathfrak{H}}\right) \) and real, then the integral\n\n\[ \left\lbrack {f, f}\right\rbrack = {\iint }_{F}\left( {{\left( \frac{\partial f}{\partial x}\right) }^{2} + {\left( \frac{\partial f}{\partial y}\right) }^{2}}\right) {dxdy} \]\n\nove... | Proof. Let \( Y \) be a large positive number, and \( \zeta = {\zeta }_{Y} \) a cutoff function as above, between \( Y \) and \( {2Y} \) . Our first goal is to prove formula (*) below. Let\n\n\[ \omega \left( {x, y}\right) = \frac{\partial f}{\partial x}{f\zeta dy} - \frac{\partial f}{\partial y}{f\zeta dx} = \frac{\pa... | Yes |
Theorem 3. Let \( f, g \in B{C}^{\infty }\left( {\Gamma \smallsetminus \mathfrak{H}}\right) \) be real functions such that \( {Lf} \) and \( {Lg} \) are also in \( B{C}^{\infty }\left( {\Gamma \smallsetminus \mathfrak{S}}\right) \) . Then \( L \) is symmetric, i.e.\n\n\[ \langle {Lf}, g\rangle = \langle {Lg}, f\rangle ... | Proof. Let \( a \) be a large positive number, and \( {F}_{a} \) a cutoff fundamental domain as shown in Fig. 5. We may assume that \( f, g \) are real. Let\n\n\[ I = \langle {Lf}, g\rangle - \langle {Lg}, f\rangle . \]\n\nThen by Stokes, we have a truncated integral\n\n\[ {I}_{a} = {\iint }_{{F}_{a}}\left\lbrack {\lef... | Yes |
Lemma 1. If \( \sigma > 1 \), then the series\n\n\[\n\mathop{\sum }\limits_{{\gamma \in \Gamma }}\frac{1}{{\left\lbrack 1 + u\left( z,\gamma {z}^{\prime }\right) \right\rbrack }^{\sigma }}\n\]\n\nis convergent uniformly for \( z,{z}^{\prime } \) in compact domains. | Proof. Let \( {M}_{R} \) be the number of elements \( \gamma \in \Gamma \) such that \( u\left( {z,\gamma {z}^{\prime }}\right) \leq R \) . We contend that \( {M}_{R} \ll R \) . Indeed, let \( D \) be a disc of fixed small radius around \( {z}^{\prime } \), such that \( D \cap {\gamma D} \) is empty if \( \gamma {z}^{\... | Yes |
Theorem 4. Let \( \sigma > 1 \) . The kernel \( r\left( {z,{z}^{\prime };s}\right) \) defines an operator\n\n\[ R\left( s\right) \;B{C}^{\infty }\left( {\Gamma \smallsetminus \mathfrak{H}}\right) \rightarrow B{C}^{\infty }\left( {\Gamma \smallsetminus \mathfrak{H}}\right) \]\n\nsatisfying\n\n\[ \left( {L - s\left( {1 -... | From Theorem 4, we see that\n\n\[ {D}_{L} \supset R\left( s\right) B{C}^{\infty }\left( {\Gamma \smallsetminus \mathfrak{H}}\right) ,\]\n\ni.e. the domain of \( L \) contains the image of \( R\left( s\right) \), and also that\n\n\[ \left( {L - s\left( {1 - s}\right) }\right) {D}_{L} \]\n\nis dense in \( \mathfrak{H} \)... | Yes |
Theorem 5. The operator \( L \) with domain \( {D}_{L} \) has a closure, denoted by \( A \) , with domain \( {D}_{A} \) . The operator \( \left( {A,{D}_{A}}\right) \) is self adjoint. | Using appropriate estimates, we shall prove in \( §7 \) : | No |
Theorem 6. For \( \sigma > 3/2, R\left( s\right) \) is a bounded operator on \( H = {L}^{2}\left( {\Gamma \smallsetminus \mathfrak{H}}\right) \) . | This will involve decomposing the kernel \( r\left( {z,{z}^{\prime };s}\right) \) into various components. Thus \( R\left( s\right) \) is what is usually called the resolvant of \( A \) .\n\nIt follows from Theorem 5 that\n\n\[ R\left( s\right) H \subset {D}_{A}, \]\n\n\ni.e. the image of \( R\left( s\right) \) is cont... | Yes |
On \( \left( {0,\infty }\right) \) let\n\n\[ \n{M}_{y} = - {\left( \frac{d}{dy}\right) }^{2} - \frac{s\left( {1 - s}\right) }{{y}^{2}}.\n\]\n\nThe associated homogeneous equation is\n\n\[ \n{\psi }^{\prime \prime }\left( y\right) = - \frac{s\left( {1 - s}\right) }{{y}^{2}}\psi \left( y\right) .\n\] | For \( s \neq \frac{1}{2} \) we take the two linearly independent solutions\n\n\[ \n{y}^{1 - s}\;\text{and}\;{y}^{s}\text{.}\n\]\n\nTheir Wronskian is \( {2s} - 1 \), and therefore a Green’s function is given by\n\n\[ \nt\left( {y,{y}^{\prime };s}\right) = \frac{1}{{2s} - 1}\left\{ \begin{array}{lll} {y}^{\prime s}{y}^... | Yes |
We shall prove that there exist two solutions\n\n\[ J = {J}_{s, c}\;\text{ and }\;K = {K}_{s, c} \]\n\nhaving the following asymptotic behavior, uniformly under the stated conditions.\n\nFor \( y \rightarrow \infty, c \geq 1, s \) in a compact set, \( \operatorname{Re}\left( s\right) > 0 \), except for \( {J}^{\prime }... | Proofs. There remains to give the proof of the existence of \( {J}_{s, c} \) and \( {K}_{s, c} \) having the desired properties. This is done by recalling some classical results.\n\nLet\n\n\[ {W}_{s}\left( y\right) = \frac{1}{\Gamma \left( s\right) }{y}^{s}{e}^{-y/2}{\int }_{0}^{\infty }{e}^{-{ty}}{\left\lbrack t\left(... | No |
Lemma 0. Let \( k\left( {z,{z}^{\prime }}\right) \) be a kernel of type \( {\mathfrak{B}}_{-\mu } \) . Then the associated operator \( K \) is defined on \( {\mathcal{B}}_{1 + \mu - \epsilon } \) and maps it continuously into \( {\mathcal{B}}_{-\mu } \) . | The proof is trivial, by freshman integration applied to the integral, say on \( {F}_{1} \) :\n\n\[ \n{\int }_{a}^{\infty }{\left( y{y}^{\prime }\right) }^{-\mu }{y}^{\prime 1 + \mu - \epsilon }\frac{d{y}^{\prime }}{{y}^{\prime 2}} \ll {y}^{-\mu }.\n\] | No |
Lemma 2. For \( \operatorname{Re}\left( s\right) > 1 \) we have\n\n\[ \n{m}_{0}\left( {y,{y}^{\prime };s}\right) = t\left( {y,{y}^{\prime };s}\right) = \frac{1}{{2s} - 1}\left\{ \begin{array}{lll} {y}^{\prime s}{y}^{1 - s} & \text{ if } & {y}^{\prime } < y, \\ {y}^{\prime 1 - s}{y}^{s} & \text{ if } & {y}^{\prime } > y... | Proof. We have\n\n\[ \n{m}_{0}\left( {y,{y}^{\prime };s}\right) = {\int }_{-\frac{1}{2}}^{\frac{1}{2}}{r}^{0}\left( {z,{z}^{\prime };s}\right) {dx}.\n\]\n\nFix \( {y}^{\prime } \) . Let \( y > {y}^{\prime } \) . Since \( {m}_{0} \) satisfies the homogeneous differential equation away from the diagonal, we must have\n\n... | Yes |
Lemma 3. For \( \sigma > 1 \), we have an estimate\n\n\[ \left| {{m}_{k}\left( {y,{y}^{\prime };s}\right) }\right| \leq C\frac{{e}^{-{2\pi k}\left| {y - {y}^{\prime }}\right| }}{k} \]\n\nwhere \( C \) is a constant, uniform for \( s \) in a compact set, and\n\n\[ 0 < a \leq y,{y}^{\prime } < \infty \text{.} \] | Therefore \( m\left( {z,{z}^{\prime };s}\right) \) lies in \( {\mathcal{L}}^{2}\left( {{F}_{1} \times {F}_{1}}\right) \), and the corresponding operator\n\n\[ {\mathcal{L}}^{2}\left( {F}_{1}\right) \rightarrow {\mathcal{L}}^{2}\left( {F}_{1}\right) \]\n\nis compact. | No |
Lemma 4. Assume \( \sigma > \frac{1}{2} \) and let \( 1 - \sigma < \mu < \sigma \) . Then for \( y \geq a \) ,\n\n\[{\int }_{a}^{\infty }\left| {t\left( {y,{y}^{\prime };s}\right) }\right| {y}^{\prime \mu }\frac{d{y}^{\prime }}{{y}^{\prime 2}} \ll {y}^{\mu }.\]\n\nThus the associated operator\n\n\[T\left( s\right) : {\... | Proof. Write the integral on the left as\n\nIt is estimated by\n\[{\int }_{a}^{\infty } = {\int }_{a}^{y} + {\int }_{y}^{\infty }\]\n\n\[{\int }_{a}^{y}{y}^{1 - \sigma }{y}^{\prime \sigma }{y}^{\prime \mu - 2}d{y}^{\prime } + {\int }_{y}^{\infty }{y}^{\sigma }{y}^{\prime 1 - \sigma }{y}^{\prime \mu - 2}d{y}^{\prime }\]... | No |
Lemma 5. The operator \( T\left( s\right) \) having kernel \( t\left( {y,{y}^{\prime };s}\right) \) is bounded on \( {L}^{2}\left( {F}_{1}\right) \) for \( \sigma > \frac{1}{2} \) . | Proof. Select \( \mu \) as in Lemma 4. Without loss of generality, we may take a function \( f \) in \( {\mathcal{L}}^{2}\left( {\lbrack a,\infty }\right) ) \) and see what \( T\left( s\right) \) does to \( f \) . Apply the Schwarz inequality to the functions \[ {\left| t\left( y,{y}^{\prime }\right) {y}^{\prime \mu }\... | Yes |
Lemma 6. Let \( c \geq 1 \) . We have for any real number \( \mu \) :\n\n\[ \n{\int }_{a}^{\infty }{e}^{-c\left| {y - {y}^{\prime }}\right| }{y}^{\prime \mu }d{y}^{\prime } \leq {C}_{1}\frac{1}{c}{y}^{\mu }\n\]\n\nwhere \( {C}_{1} \) is a constant independent of \( c \) and \( y \) . | Proof. Integrate by parts, and split the integral from \( a \) to \( y \) and \( y \) to \( \infty \) . For instance, the integral from \( a \) to \( y \) yields\n\n\[ \n{\left. \frac{{y}^{\prime \mu }{e}^{-c\left( {y - {y}^{\prime }}\right) }}{-c}\right| }_{a}^{y} + \frac{1}{c}{\int }_{a}^{y}{e}^{-c\left( {y - {y}^{\p... | Yes |
Lemma 7. For \( \sigma > 1 \), the operator \( M\left( s\right) \) whose kernel is \( m\left( {z,{z}^{\prime };s}\right) \) maps \( {L}^{2}\left( {F}_{1}\right) \) continuously into \( {\mathcal{B}}_{-1}\left( {F}_{1}\right) \) . | Proof. The assertion reduces to the following estimates, for a function \( f \in {\mathcal{L}}^{2}\left( {\lbrack a,\infty }\right) ) \) . \n\n\[ \n\mathop{\sum }\limits_{{k = 1}}^{\infty }{\int }_{a}^{\infty }\frac{{e}^{-{2\pi k}\left| {y - {y}^{\prime }}\right| }}{k}\left| {f\left( {y}^{\prime }\right) }\right| \frac... | Yes |
For \( \sigma > 1 \) and any real number \( \mu \), the operator \( M\left( s\right) \) whose kernel is \( m\left( {z,{z}^{\prime };s}\right) \) gives a bounded linear map \[ M\left( s\right) : {\mathcal{B}}_{\mu }\left( {F}_{1}\right) \rightarrow {\mathcal{B}}_{\mu - 2}\left( {F}_{1}\right) \] and the induced linear m... | Proof. Again we may consider a function \( f \in {\mathcal{B}}_{\mu }\left( {\lbrack a,\infty }\right) ) \), and for the first assertion, we estimate the sum \[ \mathop{\sum }\limits_{{k = 1}}^{\infty }{\int }_{a}^{\infty }{m}_{k}\left( {y,{y}^{\prime };s}\right) {y}^{\prime \mu - 2}d{y}^{\prime } \] using \[ \left| {{... | Yes |
Lemma 9. For \( \sigma > 3/2 \) each kernel \( {n}_{ij}\left( {z,{z}^{\prime };s}\right) \) lies in \( {\ell }^{2}\left( {{F}_{i} \times {F}_{j}}\right) \), and \( N\left( s\right) \) is therefore a compact operator on \( {L}^{2}\left( F\right) \) . In fact, for \( \left( {i, j}\right) \neq \left( {0,0}\right) \) , for... | Proof. It is clear that the estimates imply that the kernels are in \( {\ell }^{2} \) . For the estimates, we need another lemma. | No |
Lemma 10. Let \( {y}_{0} > 0 \) . For \( \sigma > 1 \), uniformly in \( - \frac{1}{2} \leq x \leq \frac{1}{2} \) and \( y,{y}^{\prime } \) \( \geq {y}_{0} \), we have\n\n\[ \mathop{\sum }\limits_{{\gamma \notin {\Gamma }_{0}}}\frac{1}{{\left\lbrack 1 + u\left( z,\gamma {z}^{\prime }\right) \right\rbrack }^{\sigma }} \l... | Proof. Write \( \gamma \notin {\Gamma }_{0} \) as\n\n\[ \gamma = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right) ,\;c \neq 0.\]\n\nThen\n\n\[ {4y}{y}^{\prime }u\left( {z,{z}^{\prime }}\right) = {cz}{z}^{\prime } + {dz} - a{z}^{\prime } - {b}^{2} \]\n\n\[ = {\left( cx{x}^{\prime } + dx - a{x}^{\prime } - b\ri... | Yes |
For \( \sigma > 3, N\left( s\right) \) maps \( {\mathcal{B}}_{1} \) into \( {\mathcal{B}}_{-1 - \delta } \) for some \( \delta > 0 \), and \[ N\left( s\right) : {\mathcal{B}}_{1} \rightarrow {\mathcal{B}}_{-1} \] is compact. Also \( N\left( s\right) \) maps \( H \) into \( {\mathcal{B}}_{-1} \) continuously. | Proof. This comes from freshman integration applied to operators of type \( {\mathcal{B}}_{2 + \epsilon - \sigma } \) . For instance if \( f \) is in \( {\mathcal{B}}_{1}\left( {F}_{1}\right) \), then we evaluate the integral \[ {\int }_{a}^{\infty }{\left( y{y}^{\prime }\right) }^{2 + \epsilon - \sigma }{y}^{\prime }\... | Yes |
Lemma 1.\ni) For \( \sigma > \frac{1}{2}, Q\left( s\right) \) is a bounded operator on \( H \) .\n\nii) For \( 0 < \sigma < 2, Q\left( s\right) \) maps \( {\mathcal{B}}_{-1}\left( {F}_{1}\right) \) into \( {\mathcal{B}}_{1 - \sigma }\left( {F}_{1}\right) \) continuously, and\n\n\[ Q\left( s\right) : {\Re }_{-1}\left( {... | Proof. Except for the compactness assertion of (ii), the continuity assertions of (i) and (ii) are verified by the same elementary integrals as for the corresponding assertions concerning \( T\left( s\right) \) in Lemmas 4 and 5,§7. It is also clear from these integrals that \( Q\left( s\right) \) maps \( {\mathcal{B}}... | No |
Lemma 2. The operator relations \( \mathbf{Q}\mathbf{1},\mathbf{Q}\mathbf{2},\mathbf{Q}\mathbf{3} \) hold\ni) For \( \sigma > \frac{1}{2} \) on \( H \) ;\nii) For \( 0 < \sigma < 2 \) on \( {\mathcal{B}}_{-1} \) . | Proof. The integrals involving the operator \( Q\left( s\right) \) and \( T \) converge absolutely in the appropriate domains by Lemmas 0,4,5 of \( §7 \) . | No |
Lemma 3. Let \( M = M\left( \kappa \right) \) for \( \kappa > 3 \) . The composite operators \[ {MQ}\left( s\right) \text{ and }Q\left( s\right) M \] are defined on \( {\mathcal{B}}_{-1} \) and \( {\mathcal{B}}_{1} \) respectively, and \[ {MQ}\left( s\right) = O,\;Q\left( s\right) M = O. \] | Proof. The operator \( M\left( \kappa \right) \) has a kernel expressed as a Fourier series involving \( \cos {2\pi k}\left( {x - {x}^{\prime }}\right) \), which is therefore orthogonal to the kernel of the operator \( Q\left( s\right) \), which is independent of \( x \) . This concludes our list of relations for \( Q\... | No |
For \( \frac{1}{2} \leq \sigma < 2 \) and \( s \neq \frac{1}{2} \), the maps\n\n\[ I + \omega \left( s\right) Q\left( s\right) \;\text{ and }\;I - \omega \left( s\right) T\left( \kappa \right) \]\n\ngive inverse isomorphisms\n\n\[ {\mathcal{B}}_{-1}\left( {\omega \left( s\right), K\left( s\right) }\right) \leftrightarr... | The proof of Theorem 7 involves formal steps, with relations among our various operators, and also involves estimates which we prove as separate lemmas. We begin with the formal steps. We abbreviate \( Q\left( s\right) \) by \( Q \) .\n\nAssume first that \( f \in {\mathcal{B}}_{-1} \) and \( {\omega K}\left( s\right) ... | Yes |
Lemma 1. Let \( s = \frac{1}{2} + {it} \) but \( s \neq \frac{1}{2} \) . Let \( f \in {\mathcal{B}}_{-1} \) be an eigenvector for \( K\left( s\right) \) , \[ \omega \left( s\right) K\left( s\right) f = f. \] Then \( f \) is orthogonal on \( {F}_{1} \) to \( \theta \left( {y, s}\right) \), i.e. \[ {\int }_{{F}_{1}}\thet... | Proof. Let \( \psi = \left( {I + {\omega Q}\left( s\right) }\right) f \) . Then by assumption and the definition of \( K\left( s\right), f = {\omega V\psi } \) . The function \( \bar{\psi }\left( z\right) f\left( z\right) \) is in \( {\mathcal{L}}^{1}\left( F\right) \), because \( f \in {\mathcal{B}}_{-1} \) and \( \ps... | Yes |
Lemma 2. Let \( 0 < \sigma < 2 \) and \( s \neq \frac{1}{2} \) . If \( f \in {\mathcal{B}}_{-1} \) then\n\n\[ \nQ\left( s\right) f\left( z\right) = \frac{1}{{2s} - 1}{y}^{1 - s}{\int }_{{F}_{1}}\theta \left( {{y}^{\prime }, s}\right) f\left( {z}^{\prime }\right) d{z}^{\prime } + O\left( 1\right) .\n\]\n\nIf \( s \neq \... | Proof. We use the definition of the kernel \( q\left( {y,{y}^{\prime };s}\right) \) and\n\n\[ \n\left( {{2s} - 1}\right) Q\left( s\right) f\left( z\right) = {\int }_{-\frac{1}{2}}^{\frac{1}{2}}{\int }_{a}^{\infty }q\left( {y,{y}^{\prime };s}\right) f\left( {x + i{y}^{\prime }}\right) \frac{d{y}^{\prime }}{{y}^{\prime 2... | Yes |
Theorem 9. The map \( s \mapsto K\left( s\right) \) is an analytic family of operators from the strip \( 0 < \sigma < 2 \) into the space of compact operators on \( {\mathcal{B}}_{-1} \), except possibly at \( s = \frac{1}{2} \) where a pole may occur due to the factor \( {2s} - 1 \) in the denominator of the definitio... | Proof. The analyticity of \( s \mapsto K\left( s\right) \) is clear. A thorough discussion of analyticity of kernels and operators is given in Appendix 5 and in the end of the next section. The fact that \[ s \mapsto {\left\lbrack I - \omega K\left( s\right) \right\rbrack }^{-1} \] is meromorphic follows from a general... | No |
Lemma 1. Let \( \frac{1}{2} \leq \sigma < 2 \) but \( s \neq \frac{1}{2} \) . Assume also that \( s \) is non-singular. Then there exists at most one bounded operator \( X \) on \( H = {L}^{2}\left( {\Gamma \smallsetminus \mathfrak{H}}\right) \) such that\n\n\[ X - R = \omega \left( s\right) {RX}. \] | Proof. Let \( X,{X}^{\prime } \) be two solutions for the above equation. Then\n\n\[ X - {X}^{\prime } = \omega \left( s\right) R\left( {X - {X}^{\prime }}\right) \]\n\nIf \( X - {X}^{\prime } \neq O \), then any vector \( \psi = \left( {X - {X}^{\prime }}\right) h \neq 0 \), with \( h \in H \), is a solution of the eq... | Yes |
For \( s \) non-singular in the strip, the operator\n\n\[ I - {\omega K}\left( s\right) : {\mathcal{B}}_{-1} \rightarrow {\mathcal{B}}_{-1} \]\n\nis invertible, and there exists a unique bounded operator\n\n\[ B\left( s\right) : {\Re }_{-1} \rightarrow {\Re }_{-1} \]\n\nsuch that\n\n\[ \left\lbrack {I - {\omega K}\left... | Proof. We know that \( K\left( s\right) \) is compact, and hence that \( {\omega K}\left( s\right) \) is compact. Hence \( I - {\omega K}\left( s\right) \) is Fredholm. The assumption that \( s \) is non-singular implies that the kernel of \( I - {\omega K}\left( s\right) \) is 0 . It follows that \( I - {\omega K}\lef... | Yes |
Lemma 3. Let \( 0 < \sigma < 2 \) and assume \( s \) non-singular.\n\ni) Each component of the kernels \( v * v \) and \( v * {q}_{s} * v \) is of type \( {\% }_{-1} \), and hence each component of the kernel \( {b}_{0, s} \) is of this type.\n\nii) The family of functions \( {b}_{0, z, s} \) such that\n\n\[ \n{b}_{0, ... | Proof. Let us look first at \( v * v \), and more specifically at its component acting on \( {F}_{0} \times {F}_{0} \) . This component has a kernel of the form\n\n\[ \n{r}_{00}\left( {z,{z}^{\prime };\kappa }\right) = \frac{1}{2}\mathop{\sum }\limits_{{\gamma \in \Gamma }}\varphi \left( {u\left( {z,\gamma {z}^{\prime ... | Yes |
Lemma 4. If \( m\left( {z,{z}^{\prime };\kappa }\right) \) is a kernel as above, then \( m * m \) is of type \( {\Phi }_{-1} \) . | Proof. We have the orthogonality\n\n\[ \n{\int }_{-\frac{1}{2}}^{\frac{1}{2}}\cos {2\pi k}\left( {x - {x}^{\prime \prime }}\right) \cos {2\pi }{f}^{\prime }\left( {x - {x}^{\prime \prime }}\right) d{x}^{\prime \prime } = 0 \n\] \n\nunless \( k = \) f. Hence \n\n\[ \n\left| {m * m\left( {y,{y}^{\prime }}\right) }\right|... | Yes |
Lemma 5. Let \( 0 < \sigma < 2 \) and assume \( s \) non-singular. The operator \( {B}_{1}\left( s\right) \) defined by\n\n\[ \n{B}_{1}\left( s\right) = {\left\lbrack I - \omega K\left( s\right) \right\rbrack }^{-1}{B}_{0}\left( s\right) \n\]\n\ncan be defined by a kernel \( {b}_{1}\left( {z,{z}^{\prime };s}\right) \) ... | Proof. We know from Lemma 4 that the kernel \( {b}_{0}\left( {z,{z}^{\prime };s}\right) \) for \( {B}_{0}\left( s\right) \) is of type \( {}^{01} - 1 \cdot \) Write\n\n\[ \n{b}_{0}\left( {z,{z}^{\prime };s}\right) = {b}_{0,{z}^{\prime }, s}\left( z\right) , \n\]\n\nviewing \( {b}_{0,{z}^{\prime }, s} \) as a function o... | Yes |
Theorem 10. For each non-singular \( s \) in the strip let \( B\left( s\right) \) be defined by\n\n\[ B\left( s\right) = {\left\lbrack I - \omega \left( s\right) K\left( s\right) \right\rbrack }^{-1}V. \]\n\n i) The map \( s \mapsto B\left( s\right) \) is a meromorphic map from the strip into the Banach space of bounde... | Proof. The first assertion is immediate from Theorem 9 in the preceding section. The second follows from the lemmas, taking into account the equation\n\n\[ B\left( s\right) = {B}_{1}\left( s\right) + V = {B}_{1}\left( s\right) + M + N. \]\n\nThe third is clear from the sequence of equivalences (3) through (9), and the ... | Yes |
i) For \( 0 < \sigma < 1 \) and \( s,1 - s \) non-singular, we have, as operators on \( {}^{96} - 1 \)\n\n\[ B\left( s\right) - B\left( {1 - s}\right) = \omega {\left( s\right) }^{2}B\left( s\right) \left\lbrack {Q\left( s\right) - Q\left( {1 - s}\right) }\right\rbrack B\left( {1 - s}\right) . \]\n\nii) For \( \sigma ,... | Proof. We start with \( B = {\left( I - \omega K\right) }^{-1}V \). Let \( 0 < \sigma ,{\sigma }^{\prime } < 2 \). On \( {\mathcal{B}}_{-1} \) we get\n\n\[ B - {B}^{\prime } = \left\lbrack {{\left( I - \omega K\right) }^{-1} - {\left( I - {\omega }^{\prime }{K}^{\prime }\right) }^{-1}}\right\rbrack V \]\n\n\[ = {\left(... | Yes |
Theorem 12. Let \( U \) be an open set in \( \mathbf{C} \) . Let \( \mu ,\nu \in \mathbf{R} \) . For each \( s \) in \( U \) , assume given a continuous operator\n\n\[ R\left( s\right) : {\mathcal{B}}_{\mu } \rightarrow {\mathcal{B}}_{\nu } \]\n\nand assume that this operator is represented by a kernel \( r\left( {z,{z... | Proof. We use Theorem 1 of Appendix 5 in connection with a statement similar to Theorem 2 of that appendix. Namely, we first prove\n\nLemma 1. Let the hypoth | No |
Lemma 1. Let the hypotheses be as in Theorem 12. Then \( r\left( {z,{z}^{\prime };s}\right) \) is analytic in \( s \) for almost all \( \left( {z,{z}^{\prime }}\right) \) if and only if for all \( f \in {\mathcal{B}}_{\mu } \) and all \( z \in F \) the function\n\n\[ s \mapsto R\left( s\right) f\left( z\right) \]\n\nis... | \( \Rightarrow : \) Let \( C \) be a circle enclosing a disc in \( U \) . Let \( f \in {\mathcal{B}}_{\mu } \) and \( z \in F \) . The function\n\n\[ {z}^{\prime } \mapsto {r}_{C}\left( {z,{z}^{\prime }}\right) \left| {f\left( {z}^{\prime }\right) }\right| \]\n\nis integrable and majorizes \( {z}^{\prime } \mapsto r\le... | Yes |
Lemma 2. Let \( \Omega \) be a compact neighborhood in the strip of a point \( {s}_{0} \) . Let \( m \) be the order of the pole of the operator \[ {\left\lbrack I - \omega \left( s\right) K\left( s\right) \right\rbrack }^{-1} \] on \( {\mathfrak{B}}_{-1} \) . Then there exists \( C > 0 \) such that for all \( s \in \O... | Proof. The proof given for Lemma 5 in the preceding section works uniformly for \( s \) in the compact set \( \Omega \) . | No |
Lemma 3. Let \( \Omega \) be a compact neighborhood in the strip of a point \( {s}_{0} \), and assume \( \Omega \) is contained in \( 0 < {\sigma }_{1} < \sigma < {\sigma }_{2} \) . Let \( m \) be as in Lemma 2. There exist constants \( {c}_{i} \) such that for any \( s \in \Omega \) we have\n\n\[ \n{\left| s - {s}_{0}... | Proof. This follows by estimating the integrals entering into the convolutions defining \( \rho \left( {z,{z}^{\prime }{1s}}\right) \) . | No |
Theorem 13. For \( z \neq {z}^{\prime } \) and \( z,{z}^{\prime } \) not on the boundaries between the regions \( {F}_{i} \) of the fundamental domain \( F \), the functions \[ s \mapsto \rho \left( {z,{z}^{\prime };s}\right) \;\text{ and }\; s \mapsto b\left( {z,{z}^{\prime };s}\right) \] are meromorphic in the strip ... | The orders of the poles and the principal parts will be determined by relating the situation to the self-adjointness of the operator which \( \rho \left( {z,{z}^{\prime };s}\right) \) represents for \( \sigma > \frac{1}{2} \), and we shall obtain: | No |
Theorem 14. Let \( {s}_{0} \) be a singular point with \( {\sigma }_{0} \geq \frac{1}{2} \) and \( {s}_{0} \neq \frac{1}{2} \) . Let \( {\psi }_{1},\ldots ,{\psi }_{n} \) be a complete orthonormal system of eigenfunctions of the self-adjoint operator \( A \) in \( H \), chosen to be real. Let \( \lambda = s\left( {1 - ... | Proof. The difficulty, such as it is, which prevents us from dealing exclusively with the operators is that on the line \( \sigma = \frac{1}{2} \) we also have a continuous spectrum for \( A \), so that the resolvant of \( A \), as an operator, does not have a power series expansion in the neighborhood of a singular po... | No |
Lemma 4. Let \( {s}_{0} \) be a singular point \( \neq \frac{1}{2} \) and \( {\sigma }_{0} \geq \frac{1}{2} \) . Let\n\n\[ R\left( s\right) : {\mathcal{B}}_{0} \rightarrow {\mathcal{B}}_{3/4} \]\n\nbe the operator defined by the kernel \( r\left( {z,{z}^{\prime };s}\right) \), for \( s \) near \( {s}_{0} \). Then \( s ... | Proof. Let \( \left\{ {s}_{n}\right\} \) be a sequence of non-singular points with \( \operatorname{Re}{s}_{n} > \frac{1}{2} \) , converging to \( {s}_{0} \), and such that \( \operatorname{Re}\left( {{\lambda }_{{s}_{0}} - {\lambda }_{{s}_{n}}}\right) = 0 \). Thus the imaginary parts of the \( {\lambda }_{{s}_{n}} \) ... | Yes |
Lemma 5. Let \( {\mathcal{B}}_{-1}\left( {s}_{0}\right) \) be the \( \omega {\left( {s}_{0}\right) }^{-1} \) -eigenspace of \( K\left( {s}_{0}\right) \) in \( {\mathcal{B}}_{-1} \) . Let\n\n\[ \n{B}_{-1} = \mathop{\lim }\limits_{{s \rightarrow {s}_{0}}}\left( {\omega \left( {s}_{0}\right) - \omega \left( s\right) B\lef... | Proof. We have\n\n\[ \n\left\lbrack {I - \omega \left( s\right) K\left( s\right) }\right\rbrack B\left( s\right) = V.\n\]\n\nMultiply both sides by \( \omega \left( {s}_{0}\right) - \omega \left( s\right) \) and let \( s \rightarrow {s}_{0} \) . The limit makes sense and proves our lemma. | No |
Lemma 6. Define\n\n\[ \n{R}_{-1} = \mathop{\lim }\limits_{{s \rightarrow {s}_{0}}}\left( {\omega \left( {s}_{0}\right) - \omega \left( s\right) }\right) R\left( s\right) : {\mathcal{B}}_{0} \rightarrow {\mathcal{B}}_{3/4} \]\n\nas an operator relation among the spaces indicated. Then \( {R}_{-1} \) maps \( {\mathfrak{B... | Proof. We multiply the relation\n\n\[ \nR\left( s\right) = Q\left( s\right) + \left( {I + \omega \left( s\right) Q\left( s\right) }\right) B\left( s\right) \left( {I + \omega \left( s\right) Q\left( s\right) }\right) \]\n\nby \( \omega \left( {s}_{0}\right) - \omega \left( s\right) \) on both sides and let \( s \righta... | Yes |
Lemma 7. The operator \( {R}_{-1} \) maps \( {\mathcal{B}}_{0} \cap H{\left( {s}_{0}\right) }^{ \bot } \) into 0 . | Proof. Let \( g \in {\mathcal{B}}_{0} \cap H{\left( {s}_{0}\right) }^{ \bot } \) . Let \( \psi \in H\left( {s}_{0}\right) \) . For \( \sigma > \frac{1}{2} \) we have\n\n\[ \left\langle {\left( {{\lambda }_{0} - {\lambda }_{s}}\right) R\left( s\right) g,\psi }\right\rangle = \left\langle {g,\left( {{\lambda }_{0} - {\la... | Yes |
Theorem 14. For fixed \( z \) the functions \( \eta \left( {z, s}\right) \) are analytic in \( s \) in the strip \( 0 < \sigma < 2 \), except for singular points for which either \( \sigma < \frac{1}{2} \) or \( \frac{1}{2} \leq s \leq 1 \) . | Proof. If \( \sigma \neq \frac{1}{2} \), our assertion is clear from the analyticity property of the kernels and functions involved. Let us look at the line \( \sigma = \frac{1}{2} \). Let \( {s}_{0} \) be such that \( {\sigma }_{0} = \frac{1}{2} \) but \( {s}_{0} \neq \frac{1}{2} \). From the analytic expression for t... | Yes |
Theorem 15. For fixed s non-singular, with \( 0 < \sigma < 2 \), and \( y \rightarrow \infty \), we have\n\n\[ \eta \left( {z, s}\right) = {y}^{s} + \mathbf{c}\left( s\right) {y}^{1 - s} + O\left( 1\right) . \] | Proof. By definition,\n\n\[ {\eta }_{s} = \left\lbrack {I + \omega \left( {I + {\omega Q}\left( s\right) }\right) B\left( s\right) }\right\rbrack {\theta }_{s} \]\n\n\[ = {\theta }_{s} + \omega {b}_{s} * {\theta }_{s} + {\omega }^{2}{q}_{s} * {b}_{s} * {\theta }_{s}. \]\n\nWe know that \( B\left( s\right) = M + {N}^{B}... | Yes |
Theorem 16. Let \( s \) be non-singular. If \( 0 < \sigma < 1 \), there is one and only one solution to the equation\n\n\[ \omega \left( s\right) R\left( \kappa \right) \eta = \eta \]\n\nhaving the asymptotic behavior for fixed \( s \) and \( y \rightarrow \infty \)\n\n\[ \eta \left( z\right) = {y}^{s} + \mathbf{c}\lef... | Proof. The existence will be proved later, and we deal here only with the uniqueness. Take first the statement relating to the interval \( 0 < \sigma < 1 \) . If \( {\eta }_{1},{\eta }_{2} \) are two solutions of the given eigenvector equation for \( R\left( \kappa \right) \), then their difference \( \psi \) is bounde... | No |
Theorem 1. Let \( A \) be a hermitian operator. Then \( \left| A\right| \) is the greatest lower bound of all values \( c \) such that\n\n\[ \left| {\langle {Ax}, x\rangle }\right| \leq c{\left| x\right| }^{2} \]\n\nfor all \( x \), or equivalently, the sup of all values \( \left| {\langle {Ax}, x\rangle }\right| \) ta... | Proof. When \( A \) is hermitian we obtain\n\n\[ \left| {\langle {Ax}, y\rangle }\right| \leq c\left| x\right| \left| y\right| \]\n\nfor all \( x, y \in H \), so that we get \( \left| A\right| \leq c \) in the lemma. On the other hand, \( c = \left| A\right| \) is certainly a possible value for \( c \) by the Schwarz i... | No |
Lemma 1. Let \( p \) be a real polynomial such that \( p\left( t\right) \geq 0 \) for all \( t \in \left\lbrack {\alpha ,\beta }\right\rbrack \) . Then we can express \( p \) in the form\n\n\[ p\left( t\right) = c\left\lbrack {\sum {Q}_{i}{\left( t\right) }^{2}+\sum \left( {t - \alpha }\right) {Q}_{j}{\left( t\right) }... | Proof. We first factor \( p \) into linear and irreducible quadratic factors over the real numbers. If \( p \) has a root \( \gamma \) such that \( \alpha < \gamma < \beta \), then the multiplicity\n\nof \( \gamma \) is even (otherwise \( p \) changes sign near \( \gamma \), which is impossible), and then \( \left( {t ... | Yes |
Lemma 2. If \( p \) is positive on \( \left\lbrack {\alpha ,\beta }\right\rbrack \), then \( p\left( A\right) \) is a positive operator. If \( p, q \) are polynomials such that \( p \leq q \) on \( \left\lbrack {\alpha ,\beta }\right\rbrack \), then \( p\left( A\right) \leq q\left( A\right) \) . Finally, \[ \left| {p\l... | Proof. The first assertion comes from the remarks preceding our lemma. The second follows at once by considering \( q - p \) . Finally, if we let \[ q\left( t\right) = \parallel p\parallel \pm p\left( t\right) \] then \( q \geq 0 \) on \( \left\lbrack {\alpha ,\beta }\right\rbrack \) and hence \( q\left( A\right) \geq ... | Yes |
Theorem 2. If \( A \geq O \), then there exists \( B \in \overline{\mathbf{R}\left\lbrack A\right\rbrack } \) such that \( {B}^{2} = A \) . The product of two commuting positive hermitian operators is again positive. | Proof. The continuous function \( {t}^{1/2} \) maps on a square root of \( A \) in \( \overline{\mathbf{R}\left\lbrack A\right\rbrack } \), and it is clear that any element of \( \overline{\mathbf{R}\left\lbrack A\right\rbrack } \) commutes with \( A \) . If \( A, C \) commute and we write \( A = {B}^{2} \) with \( B \... | Yes |
Theorem 3. The map \( f \mapsto f\left( A\right) \) is a Banach-isomorphism from the algebra of continuous functions on \( \sigma \left( A\right) \) onto the Banach algebra \( \overline{\mathbf{R}\left\lbrack A\right\rbrack } \) . A continuous function \( f \) is \( \geq 0 \) on \( \sigma \left( A\right) \) if and only... | Proof. We had derived the norm inequality previously from the positivity statement. We do this again in the opposite direction. Thus we assume first that \( f\left( A\right) \geq O \) and prove that \( f \) is \( \geq 0 \) on the spectrum of \( A \) . Assume that this is not the case. Then \( f \) is negative at some p... | Yes |
Theorem 4. Let \( S \) be a set of operators on the Hilbert space \( H \), leaving no closed subspace invariant except \( \{ 0\} \) and \( H \) itself. Let \( A \) be a hermitian operator such that \( {AB} = {BA} \) for all \( B \in S \) . Then \( A = {cI} \) for some real number \( c \) . | Proof. It will suffice to prove that there is only one element in the spectrum of \( A \) . Suppose that there are two, \( {c}_{1} \neq {c}_{2} \) . There exist continuous functions \( f, g \) on the spectrum such that neither is 0 on the spectrum, but \( {fg} \) is 0 on the spectrum. For instance, we can take for \( f... | Yes |
Lemma 1. Let \( \alpha \) be real, and let \( \left\{ {A}_{n}\right\} \) be a sequence of Hermitian operators such that \( {A}_{n} \geq {\alpha I} \) for all \( n \), and such that \( {A}_{n} \geq {A}_{n + 1} \) . Given \( v \in H \), the sequence \( \left\{ {{A}_{n}v}\right\} \) converges to an element of \( H \) . If... | Proof. From the inequality\n\n\[ \left\langle {{A}_{n}v, v}\right\rangle \geq \alpha \langle v, v\rangle \]\n\nwe conclude that \( \left\langle {{A}_{n}v, v}\right\rangle \) converges, for each \( v \in H \) . Since\n\n\[ \left\langle {{A}_{n}v, w}\right\rangle = \frac{1}{2}\left\langle {{A}_{n}\left( {v + w}\right), v... | Yes |
Lemma 2. Let \( f \) be a function on the spectrum of \( A \), bounded from below, and which can be expressed as a pointwise convergent limit of a decreasing sequence of continuous functions, say \( \left\{ {h}_{n}\right\} \) . Then\n\n\[ \mathop{\lim }\limits_{{h \rightarrow \infty }}{h}_{n}\left( A\right) \]\n\nis in... | Proof. Say \( {g}_{n}\left( t\right) \) decreases also to \( f\left( t\right) \) . Given \( k \), for large \( n \) we have\n\n\[ \max \left( {{g}_{n},{h}_{k}}\right) \leq {h}_{k} + \epsilon \]\n\nso for all \( t \) we have \( {g}_{n}\left( t\right) < {h}_{k}\left( t\right) + \epsilon \), and hence\n\n\[ {g}_{n}\left( ... | Yes |
Lemma 3. Let \( {\psi }_{c}\left( A\right) = {P}_{c} \). If \( {\alpha I} \leq A \leq {\beta I} \), then:\ni) \( {P}_{c} = 0 \) if \( c \leq \alpha \), and \( {P}_{c} = I \) if \( c \geq \beta \).\nii) If \( c \leq {c}^{\prime } \), then \( {P}_{c} \leq {P}_{{c}^{\prime }}\) | Proof. Clear from Lemma 2. | No |
Theorem 5. Let \( {P}_{c} \) be the spectral family associated with \( A \) . If \( b \leq c \), then we have\n\n\[ \n{bI} \leq A \leq {cI}\text{, on}\operatorname{Im}\left( {{P}_{c} - {P}_{b}}\right) \text{.} \n\] | Proof. From (1) above, we have \( A - {bI} = {f}_{b}\left( A\right) \) on the orthogonal complement of \( {P}_{b} \), whence the inequality \( {bI} \leq A \) follows on this complement since \( {f}_{b} \geq 0 \) . From (3) above, we have\n\n\[ \nA - {cI} = - {g}_{c}\left( A\right) \n\]\n\non the image of \( {P}_{c} \),... | Yes |
Theorem 6. The family \( \left\{ {P}_{t}\right\} \) is strongly continuous from the right. | Proof. Let \( v \in H \) . Our assertion means that \( {P}_{c + \epsilon }v \rightarrow {P}_{c}v \) as \( \epsilon \rightarrow 0 \) . It suffices to prove that\n\n\[ \left\langle {{P}_{c + \epsilon }v, v}\right\rangle \rightarrow \left\langle {{P}_{c}v, v}\right\rangle \]\n\nbecause\n\n\[ \left\langle {\left( {{P}_{c +... | Yes |
Theorem 7 (Lorch). From the left,\n\n\[ \n\\mathop{\\lim }\\limits_{{\\epsilon \\rightarrow 0}}\\left( {{P}_{c} - {P}_{c - \\epsilon }}\\right) = {Q}_{c} \n\]\n\nis the projection on the \( c \) -eigenspace of \( A \) . | Proof. Using Theorem 5, we have\n\n\[ \n\\left( {c - \\epsilon }\\right) \\left( {{P}_{c} - {P}_{c - \\epsilon }}\\right) \\leq A\\left( {{P}_{c} - {P}_{c - \\epsilon }}\\right) \\leq c\\left( {{P}_{c} - {P}_{c - \\epsilon }}\\right) \n\]\n\nwhence\n\n\[ \n\\left| {\\left( {A - {cI}}\\right) \\left( {{P}_{c} - {P}_{c -... | Yes |
Lemma 1. If \( A \) is symmetric, closed, and \( \lambda \in \mathbf{C} \) is not real, then \( \left( {A + {\lambda I}}\right) {D}_{A} \) is closed. | Proof. Let \( \left\{ {u}_{n}\right\} \) be a sequence in \( {D}_{A} \) such that \( \left\{ {\left( {A + {\lambda I}}\right) {u}_{n}}\right\} \) is Cauchy. Since \( U \) is unitary, it follows that \[ \left\{ {\left( {A + \bar{\lambda }I}\right) {u}_{n}}\right\} \] is also Cauchy, hence \( \left\{ {\left( {\lambda - \... | Yes |
Theorem 1. Let \( A \) be symmetric, closed with dense domain. Let \( \lambda \in \mathbf{C} \) be not real, and such that \( \left( {A + {\lambda I}}\right) {D}_{A} \) and \( \left( {A + \bar{\lambda }I}\right) {D}_{A} \) are dense (whence equal to \( H \) by the lemma). Then \( A \) is self-adjoint. | Proof. Let \( v \in {D}_{{A}^{ * }} \) . It suffices to show that \( v \in {D}_{A} \) . We have by definition\n\n\[ \langle {Au}, v\rangle = \left\langle {u,{A}^{ * }v}\right\rangle \]\n\nall \( u \in {D}_{A} \) .\n\nSince \( \left( {A + {\lambda I}}\right) {D}_{A} = H \), there exists \( {u}_{1} \in {D}_{A} \) such th... | No |
Theorem 2. Let \( A \) be a self-adjoint operator. Let \( z \in \mathbf{C} \) and \( z \) not real. Then \( A - {zI} \) has kernel 0 . There is a unique bounded operator\n\n\[ R\left( z\right) = {\left( A - zI\right) }^{-1} : H \rightarrow {D}_{A} \]\n\nwhich establishes a bijection between \( H \) and \( {D}_{A} \), a... | Proof. Let \( z = x + {iy} \) . If \( u \) is in the domain of \( A \), then\n\n\[ {\left| \left( A - zI\right) u\right| }^{2} = {\left| \left( A - xI\right) u\right| }^{2} + {y}^{2}{\left| u\right| }^{2} \geq {y}^{2}{\left| u\right| }^{2} \]\n\nbecause \( A \) is symmetric, so the cross terms disappear. This proves th... | Yes |
Lemma 2. With the above notation, we have \( C = {AB} \) and \( {BA} \subset {AB} \). The kernel of \( B \) is 0, and \( O \leq B \leq I \) . | Proof. We have from \( R{\left( i\right) }^{ * } = R\left( {-i}\right) \) that\n\n\[ \n{\left( A - iI\right) }^{-1} - {\left( A + iI\right) }^{-1} = {2iB}.\n\]\n\nWe multiply this on the left with \( A \), noting that\n\n\[ \nA{\left( A - iI\right) }^{-1} = i{\left( A - iI\right) }^{-1} + I \n\]\n\nand\n\n\[ \nA{\left(... | No |
Theorem 3. Let \( \left\{ {H}_{n}\right\} \) be a sequence of Hilbert spaces. Let \( {A}_{n} \) be a bounded self-adjoint operator on \( {H}_{n} \) . Let \( H \) be the orthogonal direct sum of the \( {H}_{n} \) , so that \( H \) consists of all series \( \sum {u}_{n} \) with \( \sum {\left| {u}_{n}\right| }^{2} < \inf... | Proof. The uniqueness is clear from the property that if \( A, B \) are self-adjoint and \( A \subset B \), then \( A = B \) . It suffices now to prove that if we let \( {D}_{A} \) be the domain described above, and define \( {Au} \) by \( \sum {A}_{n}{u}_{n} \), then \( A \) is selfadjoint. It is clear that \( A \) is... | Yes |
Theorem 4. Let \( A \) be a self-adjoint operator and let \( B = \operatorname{Im}{\left( A - iI\right) }^{-1} \) . Let \( {Q}_{n} = {\theta }_{n}\left( B\right) \) be the projection operator defined by the function \( {\theta }_{n} \) above. Then \( A \) is defined on \( \operatorname{Im}{Q}_{n} \), and\n\n\[ \n{Q}_{n... | Proof. Since \( t{s}_{n}\left( t\right) = {\theta }_{n}\left( t\right) \), we get \( B{s}_{n}\left( B\right) = {\theta }_{n}\left( B\right) = {Q}_{n} \) . Then by Lemma 2,\n\n\[ \nA{Q}_{n} = {AB}{s}_{n}\left( B\right) = C{s}_{n}\left( B\right) = {s}_{n}\left( B\right) C.\n\]\n\nIn particular, \( A{Q}_{n} \) is everywhe... | Yes |
Theorem 5. Let \( A \) be a self-adjoint operator. There exists a unique association \( f \mapsto f\left( A\right) \) from \( {BM}\left( \mathbf{R}\right) \) into the bounded operators on \( H \) satisfying SPEC 1 through SPEC 5. | Proof. The existence is essentially obvious from the above. Note that for SPEC 4, in view of\n\n\[ \left| {{Af}\left( A\right) {v}_{n}}\right| \leq \parallel g{\parallel }_{\infty }\left| {v}_{n}\right| \]\n\nit follows that \( \sum f\left( A\right) {v}_{n} \) lies in \( {D}_{A} \) and hence SPEC 4 is valid. The unique... | No |
Theorem 6. Let \( A \) be a self-adjoint operator on \( H \) and let \( v \in H \) . Let \( R\left( z\right) = {\left( A - zI\right) }^{-1} \) for \( z \) not real. For any \( \psi \in {C}_{c}\left( \mathbb{R}\right) \) we have\n\n\[ \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}\frac{1}{2\pi i}{\int }_{\mathbf{R}}\l... | The proof is based on the following lemma.\n\nLemma. Let \( \mu \) be a positive regular measure on \( \mathbf{R} \) such that \( \mu \left( \mathbf{R}\right) \) is finite. Then for \( \psi \in {C}_{c}\left( \mathbb{R}\right) \) we have\n\n\[ \mathop{\lim }\limits_{{\epsilon \rightarrow 0}}\frac{1}{\pi }{\int }_{-\inft... | Yes |
Theorem 1. Let \( T \) be a compact operator on \( E \) . Let \( {\lambda }_{0} \neq 0 \) . Let \( W \) be the complement of the \( {\lambda }_{0} \) -Jordan space of \( T \) as above. Let\n\n\[ P = \frac{1}{2\pi i}{\int }_{C}{\left( \lambda I - T\right) }^{-1}{d\lambda } \]\n\nwhere \( C \) is a sufficiently small cir... | Proof. We first show that if \( {\left( T - {\lambda }_{0}I\right) }^{n}v = 0 \), then \( {Pv} = v \) . We have\n\n\[ v = \frac{1}{2\pi i}{\int }_{C}\frac{v}{\lambda - {\lambda }_{0}}{d\lambda } \]\n\nThen\n\n\[ {Pv} - v = \frac{1}{2\pi i}{\int }_{C}\left\lbrack {{\left( \lambda I - T\right) }^{-1} - {\left( \lambda - ... | Yes |
Theorem 2. Let \( E \) be a Banach space, \( U \) a connected open set in \( \mathbf{C} \), and \( T : U \rightarrow K\left( E\right) \) a holomorphic map. If there exists one point \( {z}_{0} \in U \) such that \( I - T\left( {z}_{0}\right) \) is invertible, then\n\n\[ z \mapsto {\left( I - T\left( z\right) \right) }^... | Proof. The set of \( z \) where \( {\left( I - T\left( z\right) \right) }^{-1} \) is meromorphic is open. We shall prove that the complement is also open, whence empty because of our assumption for \( z = {z}_{0} \).\n\nLet \( {z}_{1} \in U \) . There exists a small circle \( C \) around 1 such that 1 is the only possi... | Yes |
Theorem 3. Let \( S \) be a closed set, contained in an open set \( U \) of \( \mathbf{C} \) . Let \( C \) be a simple closed curve in \( U \) whose interior contains \( S \) . Let\n\n\[ \lambda \mapsto R\left( \lambda \right) \]\n\n\[ \lambda \in U - S \]\n\nbe a holomorphic family of operators in a Banach space \( E ... | Proof. Let \( {C}^{\prime } \) be a closed curve surrounding \( C \) as in Fig. 1. Then\n\n\[ R\left( f\right) R\left( g\right) = \frac{1}{{\left( 2\pi i\right) }^{2}}{\int }_{C}{\int }_{{C}^{\prime }}f\left( \lambda \right) g\left( {\lambda }^{\prime }\right) R\left( \lambda \right) R\left( {\lambda }^{\prime }\right)... | Yes |
Lemma 1. Let \( L \) have constant coefficients, and be homogeneous of order \( k \) .\n\nThen\n\n\[ \parallel \varphi {\parallel }_{s + k}^{2}\underset{L}{ \ll }\parallel {L\varphi }{\parallel }_{s}^{2} + \parallel \varphi {\parallel }_{{s}_{0}}^{2} \] | Proof. We have\n\n\[ \parallel {L\varphi }{\parallel }_{s}^{2} = \sum {\left| {\varphi }_{n}\right| }^{2}{\left| {\sigma }_{L}\left( n\right) \right| }^{2}{\left( 1 + {n}^{2}\right) }^{s} \]\n\n\[ \gg \sum {\left| {\varphi }_{n}\right| }^{2}{\left( {n}^{2}\right) }^{k}{\left( 1 + {n}^{2}\right) }^{s} \]\n\n\[ \parallel... | Yes |
Lemma 2. Given an elliptic operator \( L \) of order \( k \) on \( T \), there exists \( \delta = \delta \left( L\right) \) such that if \( \varphi \) has support in a ball of radius \( < \delta \), then\n\n\[ \parallel \varphi {\parallel }_{s + k}^{2} \ll \parallel {L\varphi }{\parallel }_{s}^{2} + \parallel \varphi {... | Proof. Let \( L = \sum {\alpha }_{p}{D}^{p} \) . By uniform continuity, we can pick \( \delta \) so small\n\nthat the oscillation of each \( {\alpha }_{p} \) in a ball of radius \( \delta \) is less than \( \epsilon \) . Let \( {x}_{0} \) be the center of the ball in which \( \varphi \) has its support. Let\n\n\[ {L}_{... | Yes |
Lemma 1. Let \( r < s \) . Then the unit ball in \( {H}_{s} \) is relatively compact in \( {H}_{r} \) , i.e. is totally bounded in \( {H}_{r} \) . | Proof. Let\n\n\[ f = \sum {f}_{n}{e}^{{in} \cdot x} \]\n\nbe in the unit ball of \( {H}_{s} \), so that\n\n\[ \sum {\left| {f}_{n}\right| }^{2}{\left( 1 + {n}^{2}\right) }^{s} \leq 1 \]\n\nPick \( N \) such that \( {\left( 1 + {N}^{2}\right) }^{r - s} < \epsilon \) . Write\n\n\[ f = \mathop{\sum }\limits_{{\left| n\rig... | Yes |
Theorem 2. Let \( L \) be an elliptic operator on \( \mathbf{T} \), of order \( k \) . Given \( s \) there exists \( N \) such that \( L \) induces a topological linear isomorphism from \( {H}_{s}\left( N\right) \) to its image in \( {H}_{s - k} \), i.e. we have\n\n\[ \parallel h{\parallel }_{s} \ll \parallel {Lh}{\par... | Proof. By Theorem 1 and the basic inequality, given \( \epsilon \) there is some \( N \) such that for \( h \in {H}_{s}\left( N\right) \) we have\n\n\[ \parallel h{\parallel }_{s} \leq {C}_{1}\parallel {Lh}{\parallel }_{s - k} + {C}_{2}\parallel h{\parallel }_{{s}_{0}} \]\n\n\[ \leq {C}_{1}\parallel {Lh}{\parallel }_{s... | Yes |
Theorem 3. Let \( h \in {H}_{t} \) for some \( t \), and let \( L \) be an elliptic operator on \( \mathbf{T} \), of order \( k \) . If \( L\langle h\rangle = \langle g\rangle \) as functionals on \( {C}^{\infty }\left( \mathbf{T}\right) \), and \( g \in {H}_{s} \), then \( h \in {H}_{s + k} \) . In particular, if \( g... | Proof. For sufficiently large \( N \), we have\n\n\[ \parallel f{\parallel }_{-s} \ll {\begin{Vmatrix}{L}^{ * }f\end{Vmatrix}}_{-s - k},\;\text{ all }f \in {H}_{-s}\left( N\right) . \]\n\nBy Theorem 2, for \( \varphi \in {H}_{-s}\left( N\right) \) we have\n\n\[ \left\langle {h,{L}^{ * }\varphi }\right\rangle = \langle ... | Yes |
Theorem 4. Let \( f \) be locally \( {L}^{2} \) on an open set \( U \) in \( {\mathbf{R}}^{d} \) . Let \( g \in {C}^{\infty }\left( U\right) \), and let \( L \) be an elliptic operator of order \( k \) on \( U \) with \( {C}^{\infty } \) coefficients. Assume that \( L\langle f\rangle = \langle g\rangle \) . Then \( f \... | Proof. Let \( {x}_{0} \in U \) . Let\n\n\[ L = \sum {\alpha }_{p}{D}^{p}\;\text{ and }\;{L}_{0} = \sum {\alpha }_{p}\left( {x}_{0}\right) {D}^{p}. \]\n\nIt suffices to prove that \( f \) is \( {C}^{\infty } \) in a neighborhood of \( {x}_{0} \) . Let \( \alpha \in {C}_{c}^{\infty }\left( U\right) \) be equal to 1 in a ... | No |
Theorem 5. Let \( T \) be a distribution on \( U \) . Let \( g \in {C}^{\infty }\left( U\right) \), and let \( L \) be an elliptic operator of order \( k \) on \( U \) with \( {C}^{\infty } \) coefficients. Assume that \( {LT} = g \) .\n\nThen \( T = {T}_{h} \) for some \( h \in {C}^{\infty }\left( U\right) \) . | Proof. The same as that of Theorem 4. Instead of writing the integral of a function times \( f \), i.e. instead say of writing\n\n\[ \int {\alpha }_{1}f{M}^{ * }\varphi \]\n\nwe write\n\n\[ T\left( {{\alpha }_{1}{M}^{ * }\varphi }\right) \text{.} \]\n\nThere is no other change. | No |
Theorem 1. Let \( E \) be a Banach space over the complex numbers, and let \( \Lambda \) be a subset of the dual space \( {E}^{\prime } \) which is norm determining, i.e. for \( v \in E \) we have\n\n\[ \left| v\right| = \sup \frac{\left| \lambda v\right| }{\left| \lambda \right| } \]\n\n\[ \lambda \in \Lambda ,\lambda... | Proof. (Cf. A. E. Taylor, Functional Analysis, p. 206.) We use the fact that we are in the complex case, and prove that the Newton quotients form a Cauchy family for \( h \rightarrow 0 \) . In other words, we prove that \( f \) is differentiable. It suffices to prove that\n\n\[ \left| {\frac{f\left( {s + h}\right) - f\... | Yes |
Theorem 2. Assume that:\n\ni) For \( \left( {\mu \otimes \mu }\right) \) -almost all \( z,{z}^{\prime } \) the function\n\n\[ \ns \mapsto r\left( {z,{z}^{\prime };s}\right) \n\]\n\nis analytic.\n\nii) For any compact subset \( K \subset U \) there exists a kernel\n\n\[ \n{M}_{K}\left( {z,{z}^{\prime }}\right) \geq \mat... | Proof. By the dominated convergence theorem applied to the function\n\n\[ \nr\left( {z,{z}^{\prime };s}\right) f\left( {z}^{\prime }\right) \overline{g\left( z\right) }, \n\]\n\ndominated by\n\n\[ \n{M}_{K}\left( {z,{z}^{\prime }}\right) \left| {f\left( {z}^{\prime }\right) }\right| \left| {g\left( z\right) }\right| \n... | Yes |
Theorem 3. Let \( U \) be an open set in \( {\mathbf{R}}^{d} \) and let\n\n\[ f : U \rightarrow E \]\n\nbe a mapping into a real or complex Banach space E. Let \( {H}_{1},{H}_{2} \) be real or complex Banach spaces, and let\n\n\[ E \times {H}_{1} \times {H}_{2} \rightarrow \mathbf{R}\text{ or }\mathbf{C},\;\left( {u, v... | Proof. The theorem is local, and so we may assume that \( U \) is a disc around the origin, with coordinates \( x = \left( {{x}_{1},\ldots ,{x}_{d}}\right) \) . We recall the trivial fact that a power series\n\n\[ \mathop{\sum }\limits_{p}{a}_{p}{x}_{1}^{{p}_{1}}\cdots {x}_{d}^{{p}_{d}} = \mathop{\sum }\limits_{p}{a}_{... | Yes |
Lemma 3. \( 1\;\varphi \rightarrow \neg \psi \vdash \left( {WNM}\right) \) | Proof \( {1}^{ \circ }\varphi \rightarrow \neg \psi \) assumption\n\n\( {2}^{ \circ }\varphi \rightarrow \left( {\psi \rightarrow 0}\right) \;{1}^{ \circ }, \) Def. 2. 3\n\n\( {3}^{ \circ }\varphi \& \psi \rightarrow 0\;{2}^{ \circ },{D}_{1} \n\n\( {4}^{ \circ }\left( {WNM}\right) \;{3}^{ \circ },{D}_{2} \) | Yes |
Lemma 3.2 \( \varphi \rightarrow \psi ,\varphi \rightarrow {\varphi }^{2} \vdash \left( {WNM}\right) \) . | Proof \( \;{1}^{ \circ }\varphi \rightarrow \psi \; \) assumption\n\n\( {2}^{ \circ }\varphi \rightarrow {\varphi }^{2}\; \) assumption\n\n\( {3}^{ \circ }{\varphi }^{2} \rightarrow \varphi \& \psi \;{1}^{ \circ },{D}_{3} \), Def. 2.3\n\n\( {4}^{ \circ }\varphi \rightarrow \varphi \& \psi \;{2}^{ \circ },{3}^{ \circ },... | Yes |
Lemma 3. \( 3 \rightarrow \psi \rightarrow \varphi ,{\varphi }^{2} \rightarrow 0,\psi \rightarrow \neg \psi \vdash \left( {WNM}\right) \) | Proof \( {1}^{ \circ } \) assumption\n\n\( {2}^{ \circ } \rightarrow \psi \rightarrow \varphi \; \) assumption\n\n\( {3}^{ \circ }{\varphi }^{2} \rightarrow 0\; \) assumption\n\n\( {4}^{ \circ }\psi \rightarrow \varphi \;{1}^{ \circ },{2}^{ \circ },{HS} \)\n\n\( {5}^{ \circ }\varphi \& \psi \rightarrow {\varphi }^{2}\;... | No |
Lemma 3.4 \( \;\left( {\psi \rightarrow \neg \psi }\right) \rightarrow \neg \psi ,\varphi \& \psi \rightarrow \neg \psi \vdash \left( {WNM}\right) \) . | Proof \( \;{1}^{ \circ }\left( {\psi \rightarrow \neg \psi }\right) \rightarrow \neg \psi \; \) assumption\n\n\( {2}^{ \circ }\varphi \& \psi \rightarrow \neg \psi \; \) assumption\n\n\( {3}^{ \circ }\varphi \rightarrow \left( {\psi \rightarrow \neg \psi }\right) \;{2}^{ \circ },{A}_{7b} \)\n\n\( {4}^{ \circ }\varphi \... | Yes |
Lemma 3.5 \( {\varphi }^{2} \rightarrow 0, \rightarrow \left( {\varphi \& \psi }\right) \rightarrow \varphi \& \psi \vdash \left( {WNM}\right) \) . | Proof \( \;{1}^{ \circ }{\varphi }^{2} \rightarrow 0\; \) assumption\n\n\( {2}^{ \circ } \rightarrow \left( {\varphi \& \psi }\right) \rightarrow \varphi \& \psi \; \) assumption\n\n\( {3}^{ \circ }\varphi \& \psi \rightarrow \varphi \;{A}_{2} \)\n\n\( {4}^{ \circ }\varphi \rightarrow \neg \varphi \;{1}^{ \circ },{A}_{... | Yes |
Lemma 3.6 \( \rightarrow \psi \rightarrow \varphi \& \psi , \rightarrow \left( {\varphi \& \psi }\right) \rightarrow {\left( -\left( \varphi \& \psi \right) \right) }^{2} \vdash \left( {WNM}\right) \) . | Proof \( \;{1}^{ \circ } \rightarrow \psi \rightarrow \varphi \& \psi \; \) assumption\n\n\( {2}^{ \circ } \rightarrow \left( {\varphi \& \psi }\right) \rightarrow {\left( \neg \left( \varphi \& \psi \right) \right) }^{2}\; \) assumption\n\n\( {3}^{ \circ } \rightarrow \left( {\varphi \& \psi }\right) \rightarrow \psi ... | Yes |
Theorem 3. \( 1 + {}_{{1}_{W}^{ * }}\left( {WNM}\right) \) . | Proof By Lemma 3. \( 5{\varphi }^{2} \rightarrow 0, \rightarrow \left( {\varphi \& \psi }\right) \rightarrow \varphi \& \psi \vdash \left( {WNM}\right) \) Lemma 3. \( 6 \rightarrow \psi \rightarrow \varphi \& \psi , \rightarrow \left( {\varphi \& \psi }\right) \rightarrow \)\n\n\( {\left( \rightarrow \left( \varphi \& ... | Yes |
Let \( V \) be the algebraic set in \( {\mathbb{A}}^{2} \) given by the single equation\n\n\[ \n{X}^{2} - {Y}^{2} = 1 \n\]\n\nClearly \( V \) is defined over \( K \) for any field \( K \) . Let us assume that \( \operatorname{char}\left( K\right) \neq 2 \) . Then the set \( V\left( K\right) \) is in one-to-one correspo... | one possible map being\n\n\[ \n{\mathbb{A}}^{1}\left( K\right) \smallsetminus \{ 0\} \rightarrow V\left( K\right) \n\]\n\n\[ \nt \mapsto \left( {\frac{{t}^{2} + 1}{2t},\frac{{t}^{2} - 1}{2t}}\right) \n\] | Yes |
The algebraic set\n\n\[ V : {X}^{n} + {Y}^{n} = 1 \]\n\nis defined over \( \mathbb{Q} \). Fermat’s last theorem, proven by Andrew Wiles in 1995 [291, 311], states that for all \( n \geq 3 \), | \[ V\left( \mathbb{Q}\right) = \left\{ \begin{array}{ll} \{ \left( {1,0}\right) ,\left( {0,1}\right) \} & \text{ if }n\text{ is odd,} \\ \{ \left( {\pm 1,0}\right) ,\left( {0, \pm 1}\right) \} & \text{ if }n\text{ is even. } \end{array}\right. \] | Yes |
The algebraic set\n\n\[ V : {Y}^{2} = {X}^{3} + {17} \]\n\nhas many \( \mathbb{Q} \) -rational points, for example\n\n\[ \left( {-2,3}\right) \;\left( {{5234},{378661}}\right) \;\left( {\frac{137}{64},\frac{2651}{512}}\right) . \] | In fact, the set \( V\left( \mathbb{Q}\right) \) is infinite. See (I.2.8) and (III.2.4) for further discussion of this example. | No |
The dimension of \( {\mathbb{A}}^{n} \) is \( n \), since \( \bar{K}\left( {\mathbb{A}}^{n}\right) = \bar{K}\left( {{X}_{1},\ldots ,{X}_{n}}\right) \) . Similarly, if \( V \subset {\mathbb{A}}^{n} \) is given by a single nonconstant polynomial equation\n\n\[ f\left( {{X}_{1},\ldots ,{X}_{n}}\right) = 0 \]\n\nthen \( \d... | (The converse is also true; see [111, I.1.2].) In particular, the examples described in (I.1.3.1), (I.1.3.2), and (I.1.3.3) all have dimension one. | No |
Let \( V \) be given by a single nonconstant polynomial equation\n\n\[ f\left( {{X}_{1},\ldots ,{X}_{n}}\right) = 0. \]\n\nThen (I.1.4) tells us that \( \dim \left( V\right) = n - 1 \), so \( P \in V \) is a singular point if and only if\n\n\[ \frac{\partial f}{\partial {X}_{1}}\left( P\right) = \cdots = \frac{\partial... | Since \( P \) also satisfies \( f\left( P\right) = 0 \), this gives \( n + 1 \) equations for the \( n \) coordinates of any singular point. Thus for a \ | No |
Consider the two varieties\n\n\\[ \n{V}_{1} : {Y}^{2} = {X}^{3} + X\\;\\text{ and }\\;{V}_{2} : {Y}^{2} = {X}^{3} + {X}^{2}.\n\\]\n\nUsing (I.1.5), we see that any singular points on \\( {V}_{1} \\) and \\( {V}_{2} \\) satisfy, respectively,\n\n\\[ \n{V}_{1}^{\\text{sing }} : 3{X}^{2} + 1 = {2Y} = 0\\;\\text{ and }\\;{... | The graphs of \\( {V}_{1}\\left( \\mathbb{R}\\right) \\) and \\( {V}_{2}\\left( \\mathbb{R}\\right) \\) illustrate the difference; see Figure 1.1. | No |
Proposition 1.7. Let \( V \) be a variety. A point \( P \in V \) is nonsingular if and only if\n\n\[{\dim }_{\bar{K}}{M}_{P}/{M}_{P}^{2} = \dim V\] | Proof. [111, I.5.1]. (See Exercise 1.3 for a special case.) | No |
Consider the point \( P = \left( {0,0}\right) \) on the varieties \( {V}_{1} \) and \( {V}_{2} \) of (I.1.6). In both cases, \( {M}_{P} \) is the ideal of \( \bar{K}\left\lbrack V\right\rbrack \) generated by \( X \) and \( Y \), and \( {M}_{P}^{2} \) is the ideal generated by \( {X}^{2},{XY} \), and \( {Y}^{2} \) . Fo... | \[ X = {Y}^{2} - {X}^{3} \equiv 0\;\left( {\;\operatorname{mod}\;{M}_{P}^{2}}\right) \] so \( {M}_{P}/{M}_{P}^{2} \) is generated by \( Y \) alone. On the other hand, for \( {V}_{2} \) there is no nontrivial relationship between \( X \) and \( Y \) modulo \( {M}_{P}^{2} \), so \( {M}_{P}/{M}_{P}^{2} \) requires both \(... | Yes |
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